(i) Mathematics & Statistics Conteഀ渀琀猀 ⸀瘀

Code Book Name Authors Page No.

B.Sc. Mathematics Text Books

616 Algebra A.R. Vasishtha & Others 01

617 Trigonometry A.R. Vasishtha & Others 01

618 Differential Calculus A.R. Vasishtha & Others 02

619 Integral Calculus A.R. Vasishtha & Others 02

620 Geometry (2D & 3D) A.R. Vasishtha & Others 02

621 Vector Calculus A.R. Vasishtha & Others 02

719 Linear Algebra A.R. Vasishtha & Others 02

720 Matrices A.R. Vasishtha & Others 02

721 Differential Equations A.R. Vasishtha & Others 03

722 Integral Transforms A.R. Vasishtha & Others 03

723 Statics A.R. Vasishtha & Others 03

799 Dynamics A.R. Vasishtha & Others 03

744 Real Analysis A.R. Vasishtha & Hemlata Vasishtha 03

745 Complex Analysis A.R. Vasishtha & Hemlata Vasishtha 04

746 Numerical Analysis & Programming in C A.R. Vasishtha & Hemlata Vasishtha 04

756 Linear Programming A.R. Vasishtha & R.K. Gupta 04

757 Differential Geometry & Tensor Analysis Batuk Prasad Singh & Chauhan 04

796 Analysis A.R. Vasishtha & Others 05

797 Linear Programming A.R. Vasishtha & Others 05

844 Mathematical Methods A.R. Vasishtha & Others 05

848 Abstract Algebra A.R. Vasishtha & Others 05

850 Differential Geometry & Tensor Analysis A.R. Vasishtha & Others 06

B.Sc. Statistics Text Books

689 Probability Dr. Arun Kumar & Dr. Alka Chaudhary 06

690 Probability Distribution & Numerical Analysis Dr. Arun Kumar & Dr. Alka Chaudhary 06

691 Probability Distribution & Theory of Attributes Dr. Arun Kumar & Dr. Alka Chaudhary 07

692 Statistical Methods Dr. Arun Kumar & Dr. Alka Chaudhary 07

693 Statistical Inference Dr. Arun Kumar & Dr. Alka Chaudhary 07

694 Survey Sampling Dr. Arun Kumar & Dr. Alka Chaudhary 08

695 Analysis of Variance & Design of Experiments Dr. Arun Kumar & Dr. Alka Chaudhary 08

698 Applied Statistics Dr. Arun Kumar & Others 08

697 Non-Parametric Methods & Numerical Analysis Dr. Arun Kumar & Dr. Alka Chaudhary 09

700 Linear Programming & Computational Techniques Dr. Arun Kumar & Dr. Alka Chaudhary 09

ii

ATHEMATICS & TATISTICSM SINDEX

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M.Sc. Mathematics

211 Analytical Solid Geometry (Analytical Geometry of Three Dimensions)

A.R. Vasishtha & D.C. Agarwal 10

212 Advanced Differential Calculus J. N. Sharma 10

213 Advanced Integral Calculus D.C. Agarwal 11

214 Calculus of Finite Differences & Numerical Analysis P.P. Gupta, G.S. Malik & J.P. Chauhan 11

215 Differential Equations (Gen) J.N. Sharma & R.K. Gupta 12

215 Advanced Differential Equations R.K. Gupta & J.N. Sharma 13

216 Differential Geometry S.C. Mittal & D.C. Agarwal 15

217 Dynamics of a Particle A.R. Vasishtha & D.C. Agarwal 15

218 Fluid Dynamics Shanti Swarup 16

219 Functional Analysis J.N. Sharma & A.R. Vasishtha 17

220 Functions of a Complex Variable J.N. Sharma 17

221 Complex Analysis A.R. Vasishtha, Vipin Vasishtha & A.K. Vasishtha 18

222 Hydrodynamics Shanti Swarup 19

223 Infinite Series & Products J.N. Sharma & J.P. Chauhan 20

224 Integral Transforms (Transform Calculus) A.R. Vasishtha & R.K. Gupta 21

225 Linear Algebra (Finite Dimensional Vector Spaces) J.N. Sharma, A.R. Vasishtha & A.K. Vasishtha 21

226 Linear Difference Equations Dr. R.K. Gupta & D.C. Agarwal 22

227 Integral Equations Shanti Swarup & Shiv Raj Singh 22

228 Linear Programming R.K. Gupta 23

229 Mathematical Analysis-I (Metric Spaces) J.N. Sharma 24

231 Mathematical Analysis-II J.N. Sharma & A.R. Vasishtha 24

232 Measure & Integration(Measure Theory & Functional Analysis)

K.P. Gupta & Ashutosh Shanker Gupta 26

233 Real Analysis (General) J.N. Sharma & A.R. Vasishtha 27

234 Vector Calculus J.N. Sharma & A.R. Vasishtha 28

235 Modern Algebra (Abstract Algebra) A.R. Vasishtha & A.K. Vasishtha 28

236 Matrices A.R. Vasishtha & A.K. Vasishtha 29

237 Mathematical Methods(Special Functions & Boundary Value Problems)

J.N. Sharma & R.K. Gupta 29

238 Special Functions (Spherical Harmonics) J.N. Sharma & R.K. Gupta 30

239 Vector Algebra A.R. Vasishtha 31

240 Mathematical Statistics J.N. Sharma & J.K. Goyal 32

241 Operations Research R.K. Gupta 33

242 Rigid Dynamics-I (Dynamics of Rigid Bodies) P.P. Gupta & G.S. Malik 35

243 Rigid Dynamics-II (Analytical Dynamics) P.P. Gupta & Sanjay Gupta 36

iii


244 Set Theory and Related Topics K.P. Gupta 37

245 Spherical Astronomy S.K. Sharma, R.K. Gupta & D. Kumar 37

246 Statics (With Attraction & Potential) J.K. Goyal & K.P. Gupta 38

247 Tensor Calculus and Riemannian Geometry D.C. Agarwal 39

248 Theory of Relativity J.K. Goyal & K.P. Gupta 39

249 Topology (General & Algebraic) J.N. Sharma & J.P. Chauhan 40

250 Discrete Mathematics M.K. Gupta 41

251 Advanced Mathematics for Pharmacists A.R. Vasishtha & Others 41

252 Basic Mathematics for Chemists A.R. Vasishtha & A.K. Vasishtha 42

254 Number Theory Hari Kishan 44

255 Bio-Mathematics Bhupendra Singh & Neenu Agarwal 45

336 Cryptography and Network Security Dr. Manoj Kumar 45

526 Partial Differential Equations Dr. R.K. Gupta 46

529 Advanced Abstract Algebra S.K. Pundir 47

538 Spherical Astronomy & Space Dynamics J.P. Chauhan 48

539 Space Dynamics J.P. Chauhan 48

592 Advanced Mathematical Methods Shiv Raj Singh 49

679 Fuzzy Set Theory Shiv Raj Singh & Chaman Singh 49

851 Advanced Numerical Analysis Prof. P.P. Gupta, G.S. Malik & J.P. Chauhan 50

852 Analysis-I (Real Analysis) J.P. Chauhan 50

260 Real Analysis A.R. Vasishtha & Vipin Vasishtha 51

864 Calculus of Variations Mukesh Kumar Singh 51

Fully Solved Series for IAS/PCS and Other Competitions

442 Series: Trigonometry A.R. Vasishtha, S.K. Sharma & A.K. Vasishtha 52

443 Series: Matrices A.R. Vasishtha & A.K. Vasishtha 52

444 Series: Algebra A.R. Vasishtha, R.K. Gupta & A.K. Vasishtha 53

446 Series: Differential Calculus A.R. Vasishtha, S.K. Sharma & A.K. Vasishtha 53

447 Series: Integral Calculus A.R. Vasishtha, S.K. Sharma & A.K. Vasishtha 54

448 Series: Differential Equations A.R. Vasishtha & S.K. Sharma 55

449 Series: Analytical Geometry of Two Dimensions A.R. Vasishtha, D.C. Agarwal & A.K. Vasishtha 56

450 Series: Analytical Geometry of Three Dimensions A.R. Vasishtha, D.C. Agarwal & A.K. Vasishtha 56

451 Series: Modern Algebra A.R. Vasishtha & Kiran Vasishtha 57

452 Series: Vector Calculus A.R. Vasishtha & A.K. Vasishtha 57

455 Series: Statics A.R. Vasishtha, R.K. Gupta & A.K. Vasishtha 58

456 Series: Dynamics A.R. Vasishtha, R.K. Gupta & A.K. Vasishtha 58

457 Series: Real Analysis A.R. Vasishtha, A.K. Vasishtha & Hemlata Vasishtha 59

458 Series: Numerical Analysis A.R. Vasishtha, S.K. Sharma & Hemlata Vasishtha 60

459 Series: Hydrostatics A.R. Vasishtha, A.K. Vasishtha 60

iv

Y Sequence and its convergence (basic idea) Y Convergence of infinite series Y Comparison test Y Ratio test Y Root test Y Raabe’s test Y Logarithmic ratio

test Y Cauchy’s condensation test Y De Morgan and Bertrand test and higher logarithmic ratio test Y Alternating series Y Leibnitz test Y Absolute and

conditional convergence Y Congruence modulo m relation Y Equivalence relations and partitions Y Definition of a group with examples and simple

properties Y Permutation groups Y Subgroups Y Centre and normalizer Y Cyclic groups Y Coset decomposition Y Lagrange’s theorem and its

consequences Y Homomorphism and Isomorphism Y Cayley’s theorem Y Normal subgroups Y Quotient group Y Fundamental theorem of

homomorphism Y Conjugacy irrelation Y Class equation Y Direct product Y Introduction to rings Y Subrings Y Integral domains and fields Y Characteristic

of a ring Y Homomorphism of rings Y Ideals Y Quotient rings.

Y Complex Numbers Y Inverse Circular Functions Y General and Principal Values of Inverse Circular Functions Y Relations between Inverse Functions

Y Some Important Results about Inverse Functions Y Complex Numbers Y Addition of Complex Numbers Y Multiplication of Complex Numbers

Y Difference of two Complex Numbers Y Division in C Y The Symbol i and its Powers Y Conjugate of a Complex Numbers Y Modulus of a Complex

Number Y Some Important Results about Complex Numbers Y Modulus-Argument Form or Polar Standard Form or Trigonometric Form of a Complex

Numbers Y De Moivre's Theorem Y Exponential, Trigonometric and Hyperbolic functions of a Complex Variable (Separation into real &

Imaginary parts) Y The Exponential Function of a Complex Variable Y Index Law for the Exponential Functions Y Trigonometrical Functions or

Circular Functions of a Complex Variable Y Euler's Theorem Y Periodicity of Functions Y De Moivre's Theorem for Complex Argument Y Standard

Trigonometrical Results for Complex Arguments Y Hyperbolic Functions Y Relations between Hyperbolic and Circular Functions Y Properties of

Hyperbolic Functions Y Expansions in Series for sinh x and cosh x Y Periods of Hyperbolic Functions Y Separation into Real and Imaginary Parts Y

Logarithms of Complex Numbers Y Logarithms in the Set of Real Numbers Y Logarithms of Complex Numbers Y Principal and General Values of

Logarithm of a Non-Zero Complex Number Y Properties of the Logarithmic Function Y Working Rule to Evaluate Log (x + iy) i.e., to Express Log (x + iy)

in the Form A+iB Y Logarithm of a Positive Real Number in the Set of Complex Numbers Y Logarithm of a Negative Real Number Y The General

Exponential Function az Y To Separate ( )α + β +i p iq into Real and Imaginary Parts Y Inverse Circular and Hyperbolic Functions of Complex

Numbers Y Inverse Circular Functions of Complex Numbers Y Inverse Hyperbolic Functions Y Relations between Inverse Hyperbolic Functions and

Inverse Circular Functions Y Gregory’s Series Y General Theorem on Gregory's Series Y Value of π Y Summation of Trigonometrical Series Y

C i S+ Method for Summing Up Trigonometric Series Y Series Based on Geometric Progression or Arithmetico-Geometric Series Y Series Based on

Binomial Expansions Y Series Based on Exponential Series Y Series Based on Logarithmic Series and its Sub-Case Gregory's Series Y The Difference

Method Y Angles in Arithmetical Progression.

1

–A.R. Vasishtha & Others616-23 (B) Algebra

[Fully Solved Series Available]

–A.R. Vasishtha & Others617-23 Trigonometry


ATHEMATICSM& TATISTICS S

Krishna's

B.Sc. Mathematics

Contents

Text Cum Reference Books

Y ε-δ definition of the limit of a function Y Continuous functions and classification of discontinuities Y Differentiability Y Chain rule of Differentiability Y

Rolle’s theorem Y First and second mean value theorems Y Taylor’s theorems with Lagrange’s and Cauchy’s forms of remainder Y Successive differentiation

and Leibnitz’s theorem Y Expansion of functions (in Taylor’s and Maclaurin’s series) Y Indeterminate forms Y Partial differentiation and Euler’s theorem Y

Jacobians Y Maxima and Minima (for functions of two variables) Y Tangents and normals (polar form only) Y Curvature Y Envelopes and evolutes Y

Asymptotes Y Tests for concavity and convexity Y Points of inflexion Y Multiple points Y Tracing of curves in Cartesian and Polar coordinates.

Y Reduction Formulae (For Trigonometric Functions) Y Reduction Formulae Continued (For Irrational Algebraic and Transcendental Functions) Y Beta

and Gamma Functions Y Multiple Integrals (Double and Triple Integrals, Change of Order of Integration) Y Dirichlet's and Liouville's Integrals Y Areas of

Curves Y Rec ti fi ca tion (Lengths of Arcs and In trin sic Equa tions of Plane Curves) Y Vol umes and Sur faces of Sol ids of Rev o lu tion.

Y General equation of second degree Y Tracing of conics Y System of conics Y Confocal conics Y Polar equation of a conic and its properties Y Three

dimensional system of co-ordinates Y Projection and direction cosines Y Plane Y Straight line Y Sphere Y Cone and cylinder Y Central conicoids Y Reduction

of general equation of second degree Y Tangent plane and normal to a conicoid Y Pole and polar Y Conjugate diameters Y Generating lines Y Plane sections.

Y Vector differentiation and integration Y Gradient Y Divergence and curl and their properties Y Line integrals Y Theorems of Gauss Y Green and Stokes

and problems based on these.

Y Vector spaces and their elementary properties Y Subspaces Y Linear dependence and independence Y Basis and dimension Y Direct sum Y Quotient

space Y Linear transformations and their algebra Y Range and null space Y Rank and nullity Y Matrix representation of linear transformations Y Change of

basis Y Linear functionals Y Dual space Y Bi-dual space Y Natural isomorphism Y Annihil-ators Y Bilinear and quadratic forms Y Inner product spaces

Y Cauchy-Schwarz's inequality Y Bessel's inequality and orthogonality.

Y Symmetric and skew-symmetric matrices Y Hermitian and skew-Hermitian matrices Y Orthogonal and unitary matrices Y Triangular and diagonal

matrices Y Rank of a matrix Y Elementary transformations Y Echelon and normal forms Y Inverse of a matrix by elementary transformations

Y Characteristic equation Y Eigen values and eigen vectors of a matrix Y Cayley-Hamilton's theorem and its use in finding inverse of a matrix

Y Application of matrices to solve a system of linear (both homogeneous and non-homogeneous) equations Y Consistency and general solution

Y Diagonalization of square matrices with distinct eigen values Y Quadratic forms.

2


–A.R. Vasishtha & Others619-22 Integral Calculus


–A.R. Vasishtha & Others620-19 Geometry (2D & 3D)


–A.R. Vasishtha & Others621-19 Vector Calculus


–A.R. Vasishtha & Others719-13 (B) Linear Algebra


–A.R. Vasishtha & Others618-22 (B) Differential Calculus


–A.R. Vasishtha & Others720-13 (B) Matrices

Y Formation of a differential equation (D.E.) Y Degree, order and solution of D.E. Y Equations of first order and first degree : Separation of variables

method Y Solution of homogeneous equations Y Linear equations and exact equations Y Linear differential equations with constant coefficients

Y Homogeneous linear differential equations Y Differential equations of the first order but not of the first degree Y Clairaut's equations and singular

solutions Y Orthogonal trajectories Y Simultaneous linear differential equations with constant coefficients Y Linear differential equations of the second

order (including the method of variation of parameters) Y Series solutions of second order differential equations Y Legendre and Bessel functions (Pn and

Jn only) and their properties Y Order, degree and formation of partial differential equations Y Partial differential equations of the first order Y Lagrange's

equations Y Charpit's general method Y Linear partial differential equations with constant coefficients Y Partial differential equations of the second order

Y Monge's method.

Y The concept of transform Y Integral transforms and kernel Y Linearity property of transforms Y Laplace transform Y Inverse Laplace transform

Y Convolution theorem Y Applications of Laplace transform to solve ordinary differential equations Y Fourier transforms (finite and infinite) Y Fourier

integral Y Applications of Fourier transform to boundary value problems Y Fourier series.

Y Common catenary Y Centre of gravity Y Stable and unstable equilibrium Y Virtual work Y Forces in three dimensions Y Poinsot's central axis

Y Wrenches Y Null line and null plane.

Y Velocity and acceleration along radial and transverse directions and along tangential and normal directions Y Simple harmonic motion Y Motion under

other laws of forces Y Earth attraction Y Elastic strings Y Motion in resisting medium Y Constrained motion (circular and cycloidal only) Y Motion on

smooth and rough plane curves Y Rocket motion Y Central orbits and Kepler's law Y Motion of a particle in three dimensions.

Y Axiomatic study of real numbers Y Completeness property in R Y Archimedean property Y Countable and uncountable sets Y Neighbourhood Y Interior

points Y Limit points Y Open and closed sets Y Derived sets Y Dense sets Y Perfect sets Y Bolzano-Weierstrass theorem Y Sequences of real numbers

Y Subsequences Y Bounded and monotonic sequences Y Convergent sequences Y Cauchy's theorems on limit Y Cauchy sequence Y Cauchy's general

principle of convergence Y Uniform convergence of sequences and series of functions Y Weierstrass M-test Y Abel's and Dirichlet's tests Y Sequential

continuity Y Boundness and intermediate value properties of continuous functions Y Uniform continuity Y Meaning of sign of derivative Y Darboux

theorem Y Limit and continuity of functions of two variables Y Taylor's theorem for functions of two variables Y Maxima and minima of functions of three

variables Y Lagrange's method of undetermined multipliers Y Riemann integral Y Integrability of continuous and monotonic functions Y Fundamental

theorem of integral calculus Y Mean value theorems of integral calculus Y Improper integrals and their convergence Y Comparison test Y µ-test Y Abel's

test Y Dirichlet's test Y Integral as a function of a parameter and its differentiability and integrability Y Definition and examples of metric spaces

Y Neighbourhoods Y Interior points Y Limit points Y Open and closed sets Y Subspaces Y Convergent and Cauchy sequences Y Completeness Y Cantor's

intersection theorem.

3


–A.R. Vasishtha & Others721-17 (B) Differential Equations


–A.R. Vasishtha & Others722-17 (B) Integral Transforms

–A.R. Vasishtha & Others723-13 (B) Statics


–A.R. Vasishtha & Others799-13 Dynamics


–A.R. Vasishtha & Hemlata Vasishtha744-02 (B) Real Analysis


Y Functions of a complex variable Y Concepts of limit Y Continuity and differentiability of complex functions Y Analytic functions Y Cauchy-Riemann

equations (Cartesian and polar form) Y Harmonic functions Y Orthogonal system Y Power series as an analytic function Y Elementary functions

Y Mapping by elementary functions Y Linear and bilinear transformations Y Fixed points Y Cross ratio Y Inverse points and critical points Y Conformal

transformations Y Complex Integration Y Line integral Y Cauchy's fundamental theorem Y Cauchy's integral formula Y Morera's theorem Y Liouville

theorem Y Maximum Modulus theorem Y Taylor and Laurent series Y Singularities and zeros of an analytic function Y Rouche's theorem Y Fundamental

theorem of algebra Y Analytic continuation Y Residue theorem and its applications to the evaluation of definite integrals Y Argument principle.

Y Shift operator Y Forward and backward difference operators and their relationships Y Fundamental theorem of difference calculus Y Interpolation Y

Newton-Gregory's forward and backward interpolation formulae Y Divided differences Y Newton's divided difference formula Y Lagrange's interpolation

formula Y Central differences Y Formulae based on central differences : Gauss, Striling's, Bessel's and Everett's interpolation formulae Y Numerical

differentiation Y Numerical integration Y General quadrature formula Y Trapezoidal and Simpson's rules Y Weddle's rule Y Cote's formula Y Numerical

solution of first order differential equations : Euler's method Y Picard's method Y Runge-Kutta method and Milne's method Y Numerical solution of linear,

homogeneous and simultaneous difference equations Y Generating function method Y Solution of transcendental and polynomial equations by iteration,

bisection Y Regula-Falsi and Newton-Raphson methods Y Algebraic eigen value problems : Power method Y Jacobi's method Y Given's method Y

Householder's method and Q-R method Y Approximation : Different types of approximations Y Least square polynomial approximation Y Polynomial

approximation using orthogonal polynomials Y Legendre approximation Y Approximation with trigonometric functions Y Exponential functions Y

Rational functions Y Chebyshev polynomials Y Programmer's model of computer Y Algorithms Y Data type Y Arithmetic and input/output instruction Y

Decisions Y Control structures Y Decision statements Y Logical and conditional operators Y Loop case control structures Y Functions Y Recursion Y

Preprocessors Y Arrays Y Puppetting of strings Structures Y Pointers Y File formatting.

Y Linear programming problems Y Statement and formation of general linear programming problems Y Graphical method Y Slack and surplus variables

Y Standard and matrix forms of linear programming problem Y Basic feasible solution Y Convex sets Y Fundamental theorem of linear programming

Y Simplex method Y Artificial variables Y Big-M method Y Two phase method Y Resolution of degeneracy Y Revised simplex method Y Sensitivity

Analysis Y Duality in linear programming problems Y Dual simplex method Y Primal-dual method, Integer programming Y Transportation problems

Y Assignment problems.

Y Local theory of Curves- Space curves Y Examples Y Plane curves Y Tangent and normal and binormal Y Osculating plane Y Normal plane and rectifying

plane Y Helices Y Serret-Frenet apparatus Y Contact between curve and surfaces Y Tangent surfaces Y Involutes and evolutes of curves Y Intrinsic equations

Y Fundamental existence theorem for space curves Y Local theory of Surfaces-Parametric patches on surface curve of a surface Y Surfaces of revolutions Y

Helicoids Y Metric-first fundamental form and arc length Y Local theory of surfaces (Contd.) Y Direction coefficients Y Families of curves Y Intrinsic properties

Y Geodesics, canonical geodesic equations, normal properties of geodesics, geodesics curvature, geodesics polars Y Gauss-Bonnet theorem Y Gaussian

curvature Y Normal curvature Y Meusneir's theorem Y Mean curvature Y Gaussian curvature Y Umbilic points Y Lines of curvature Y Rodrigue's formula Y

Euler's theorem Y The fundamental equation of surface theory - The equation of Gauss, the equation of Weingarten, the Mainardi-Codazzi equation Y

Tensor algebra : Vector spaces Y The dual spaces Y Tensor product of vector spaces Y Transformation formulae Y Contraction Y Special tensor Y Inner

product Y Associated tensor Y Differential Manifold-examples Y Tangent vectors Y Connexions Y Covariant differentiation Y Elements of general

Riemannian geometry-Riemannian metric Y The fundamental theorem of local Riemannian Geometry Y Differential parameters Y Curvature tensor Y

Geodesics Y Geodesics curvature Y Geometrical interpretation of the curvature tensor and special Riemannian spaces Y Contravariant and covariant vectors

and tensors Y Mixed tensors Y Symmetric and skew-symmetric tensors Y Algebra of tensors Y Contraction and inner product Y Quotient theorem Y

Reciprocal tensors Y Christoffel's symbols Y Covariant differentiation Y Gradient Y divergence and curl in tensor notation.

4

– Batuk Prasad Singh & Chauhan757-01 (C) Differential Geometry & Tensor Analysis

–A.R. Vasishtha & Hemla ta Vasishtha746-03 Numerical Analysis & Progr amming in C


–A.R. Vasishtha & R.K. Gupta756-02 Linear Programming


–A.R. Vasishtha & Hemla ta Vasishtha745-02 (B) Complex Analysis


Y Axiomatic study of real numbers Y Completeness property in R Y Archimedean property Y Countable and uncountable sets Y Neighbourhoods

Y Interior points Y Limit points Y Open and closed sets Y Derived sets Y Dense sets Y Perfect sets Bolzano-Weierstrass theorem Y Sequences of real

numbers Y Subsequences Y Bounded and monotonic sequences Y Convergent sequences Y Cauchy's theorems on limit Y Cauchy sequence Y Cauchy's

general properties of convergence Y Sequential continuity Y Boundness and intermediate value properties of continuous functions Y Uniform continuity

Y Meaning of sign of derivative Y Riemann integral Y Integrability of continuous and monotonic functions Y Fundamental theorem of integral calculus

Y Mean value theorems of integral calculus Y Improper integrals and their convergence Y Comparison test Y µ-test Y Abel's test Y Dirichlet's test Y Integral

as a function of a parameter and its differentiability and integrability Y Functions of a complex variable Y Concepts of limit Y Continuity and

differentiability of complex functions Y Analytic functions Y Cauchy Riemann equations (Cartesian and polar form) Y Harmonic functions Y Orthogonal

system Y Power series as an analytic function Y Elementary functions Y Mapping by elementary functions Y Linear and bilinear transformations Y Fixed

points Y Cross ratio Y Inverse points and critical points Y Conformal transformations.

Y Linear programming problems Y Statement and formation of general linear programming problems Y Graphical method Y Slack and surplus variables

Y Standard and matrix forms of linear programming problem Y Basic feasible solution Y Convex sets Y Fundamental theorem of linear programming

Y Simplex method Y Artificial variables Y Big-M method Y Two phase method Y Resolution of degeneracy Y Revised simplex method Y Sensitivity

Analysis Y Duality in linear programming problems Y Dual simplex method Y Primal-dual method, Integer programming Y Transportation problems

Y Assignment problems Y Goal Programming; Concept of goal programming, formulation and methodology for solution of goal programming.

Y Definition of a sequence Y Theorems on limits of sequences Y Bounded and Monotonic sequences Y Cauchy's convergence criterion Y Cauchy

sequence Y Limit superior and limit inferior of a sequence Y Subsequence Y Series of non-negative terms Y Comparison tests Y Cauchy's integral test

Y Ratio tests Y Root test Y Raabe's logarithmic Y De Morgan and Bertrand's tests Y Alternating series Y Leibnitz's theorem Y Absolute and conditional

convergence Y The concept of transform Y Integral transform Y Kernel Y Laplace Transformation- Linearity of the Laplace transformation Y Existence

theorem for Laplace transforms Y Laplace transforms of derivatives and integrals Y Shifting theorems Y Differentiation and Integration of Laplace

transforms Y Convolution theorem Y Inverse Laplace transforms Y Solution of system of differential equations using the Laplace transformation Y Fourier

transforms (finite and infinite) Y Fourier integral Y Applications of Fourier transform to boundary value problems Y Fourier series Y Calculus of

variations-Variational problems with fixed boundaries- Euler's equation for functionals containing first order derivative and one independent variable

Y Extremals Y Functionals dependent on higher order derivatives Y Functionals dependent on more than one independent variable Y Variational

problems in parametric form Y Invariance of Euler's equation under coordinates transformation Y Partial differential equations of the first order Y Lagrange's

solution Y Some special types of equations which can solve easily by methods other than the general methods Y Charpit's general method of solution Y

Partial differential equations of the second and higher orders Y Classification of linear partial differential equations of second order Y Homogeneous and

non-homogeneous equations with constant coefficients Y Partial differential equations reducible to equations with constant coefficients Y Monge's method.

Y Automorphism Y Inner automorphism Y Automorphism groups and their computations Y Conjugacy relations Y Normaliser Y Counting principle and

the class equation of a finite group Y Center of group of prime power order Y Sylow's theorems Y Sylow p-subgroup Y Prime and maximal ideals

Y Euclidean Rings Y Principal ideal rings Y Polynomial Rings Y Polynomial over the Rational Field Y The Eisenstein Criterion Y Polynomial Rings over

Commutative Rings Y Unique factorization domain Y R is unique factorization domain implies so is R [x1, x2, …, xn] Y Direct sum Y Quotient space

Y Linear transformations and their representation as matrices Y The Algebra of linear transformations Y Rank nullity theorem Y Change of basis Y Linear

5

–A.R. Vasishtha & Others844-01 (B) Mathematical Methods

– A.R. Vasishtha & Others796-02 Analysis


– A.R. Vasishtha & Others797-02 Linear Programming


–A.R. Vasishtha & Others848-01 Abstract Algebra

functional Y Dual space Y Bidual space and natural isomorphism Y Transpose of a linear transformation Y Characteristic values Y Annihilating

polynomials Y Diagonalisation Y Cayley Hamilton Theorem Y Invariant subspaces Y Primary decomposition theorem Y Inner product spaces Y

Cauchy-Schwarz inequality Y Orthogonal vectors Y Orthogonal complements Y Orthonormal sets and bases Y Bessel's inequality for finite dimensional

spaces Y Gram-Schmidt orthogonalization process Y Bilinear Y Quadratic and Hermitian forms.

Y Local theory of curves- Space curves Y Examples Y Plane curves Y Tangent and normal and binormal Y Osculating plane Y Normal plane and rectifying

plane Y Helices Y Serret-Frenet apparatus Y Contact between curve and surfaces, tangent surfaces Y Involutes and evolutes of curves Y Intrinsic equations

Y Fundamental existence theorem for space curves Y Local theory of surfaces- Parametric patches on surface curve of a surface Y Surfaces of revolutions

Y Helicoids Y Metric-first fundamental form and arc length Y Local theory of surfaces (Contd.) Y Direction coefficients Y Families of curves Y Intrinsic

properties Y Geodesics Y Canonical geodesic equations Y Normal properties of geodesics Y Geodesics curvature Y Geodesics polars Y Gauss-Bonnet

theorem Y Gaussian curvature Y Normal curvature Y Meusneir's theorem Y Mean curvature Y Gaussian curvature Y Umbilic points Y Lines of curvature Y

Rodrigue's formula Y Euler's theorem Y The fundamental equation of surface theory - The equation of Gauss Y The equation of Weingarten Y The

Mainardi-Codazzi equation Y Tensor algebra : Vector spaces Y The dual spaces Y Tensor product of vector spaces Y Transformation formulae Y

Contraction Y Special tensor Y Inner product Y Associated tensor Y Differential Manifold-examples Y Tangent vectors Y Connexions Y Covariant

differentiation Y Elements of general Riemannian geometry-Riemannian metric Y The fundamental theorem of local Riemannian Geometry Y Differential

parameters Y Curvature tensor Y Geodesics Y Geodesics curvature Y Geometrical interpretation of the curvature tensor and special Riemannian spaces Y

Tensor Analysis Y Contravariant and covariant vectors and tensors Y Mixed tensors Y Symmetric and skew-symmetric tensors Y Algebra of tensors Y

Contraction and inner product Y Quotient theorem Y Reciprocal tensors Y Christoffel's symbols Y Covariant differentiation Y Gradient Y Divergence and

curl in tensor notation.

Y Introduction to Probability Theory Y History and Relevance of Probability Theory Y Some Basic Definitions of Probability Y Sample Space and

Algebra of Events Y Rules of Counting Y Three Approaches to Probability Y Probability Rules Y Independent Events Y Baye's Theorem Y Random

Variables and Mathematical Expectation Y Random Variables Y Discrete Probability Distributions Y Continuous Probability Distributions Y Joint

Probability Distributions Y Marginal Distributions Y Conditional Distributions Y Independence of Random Variables and Mathematical

Expectation Y Independence of Random Variables Y Mathematical Expectation Y Laws of Expectation Y Moments Y Correlation Coefficient of Two

Random Variables Y Conditional Expectation Y Generating Functions and Law of Large Numbers Y Introduction of Generating Functions Y

Chebyshev's Inequality Y Law of Large Numbers Y Central Limit Theorem Y Tables.

Y Discrete Univariate Distributions Y Introduction Y Discrete Uniform Distribution Y Bernouli Distribution Y Binomial Distribution YPoisson

Distribution Y Negative Binomial Distribution Y Hypergeometric Distribution Y Continuous Univariate Distributions Y Introduction Y Uniform

Distribution Y Normal Distribution Y Exponential Distribution Y Gamma Distribution Y Beta Distribution Y Cauchy Distribution Y Laplace Distribution Y

Pareto Distribution Y Exact Sampling Distributions Y Introduction Y Distributions of Functions of Random Variable Y Chi-Square Distribution Y

t-Distribution Y F-Distribution Y Inter-Relationships between χ2, t and F-Distribution Y Bivariate Normal Distribution Y Finite Differences Y

Introduction Y Symbol used in Finite Differences Calculus Y The Operators E and ∆ Y Relationship between E and D Y The Difference Table Y Factorial

Functions Y Differences of Zero Y Effect of an Error in a Tabular Value Y Interpolation Y Introduction Y Interpolation with Equal Intervals Y

Interpolation with Unequal Intervals Y Central Differences Y Numerical Integration Y Introduction Y General Quadrature Formula for Equidistant

6

B.Sc. Statistics

–A.R. Vasishtha & Others850-01 (B) Differential Geometry & Tensor Analysis

–Dr. Arun Kumar & Dr. Alka Chaudhary689-08 Probability

–Dr. Arun Kumar & Dr. Alka Chaudhary690-06 (B) Probability Distribution & Numerical Analysis

Ordinates Y The Trapezoidal Rule Y Simpson' One-Third Rule Y Simpson's Three Eighth's Rule Y Weedle's Rule Y Error in Quadrature Formula Y Cote's

Method Y Nu mer i cal Dif fer en ti a tion Y In tro duc tion Y De riv a tives Us ing For ward Dif fer ence For mula Y De riv a tives Us ing Back ward Dif fer ence

For mula Y De riv a tives Us ing Cen tral Dif fer ence For mu lae Y De riv a tives of a Func tion when the given Ar gu ments are not Equally Spaced Y Ta bles.

Y Discrete Univariate Distributions Y Introduction Y Discrete Uniform Distribution Y Bernoulli Distribution Y Binomial Distribution Y Poisson

Distribution Y Negative Binomial Distribution Y Hypergeometric Distribution Y Continuous Univariate Distributions Y Introduction Y Uniform

Distribution Y Normal Distribution Y Exponential Distribution Y Gamma Distribution Y Beta Distribution Y Cauchy Distribution Y Exact Sampling

Distributions Y Introduction Y Distributions of Function of Random Variables Y Chi-Square Distribution Y t-Distribution Y F-Distribution Y

Inter-Relationships between χ2 , t and F-Distributions Y Theory of Attributes Y Introduction Y Concept and Definitions Y An Important Notation Y

Consistency of Data Y Independence of Attributes Y Association of Attributes Y Coefficient of Association Y Coefficient of Colligation Y Contingency Table

Y Association in a Contingency Table Y Tables.

Y Definition, Functions, Limitations and Importance of Statistics Y Introduction Y Definition of Statistics Y Other Popular Definitions of

Statistics Y Function of Statistics Y Limitation of Statistics Y Distrust of Statistics Y Importance of Statistics Y Statistical Tools Used in Economic Analysis Y

Types of Data and Scales Y Introduction Y Census and Sampling Y Types of Data Y Collection and Scrutiny of Data Y Introduction Y Primary and

Secondary Data Y Method of Collection Y Scrutiny of the Data Y Organisation of Data Y Introduction Y Classification Y Object of Classification Y Basis

of Classification Y Frequency Distribution Y Method of Construction of Discrete Frequency Distribution Y Method of Construction of Continuous

Frequency Distribution Y Basic Principles for Forming Grouped Frequency Distribution Y Sturges Rule for Number of Classes and Size of Class Interval Y

Cumulative Frequency Distribution Y Tabulation Y Types of Tables Y Difference between Classification and Tabulation Y Diagrammtic

Representation of Data Y Introduction Y Importance and Utility of Diagrams Y Limitations of Diagrams Y Rules for Constructing Diagrams Y Types of

Diagrams Y Limitations of Diagrammatic Representation Y Graphic Representation of Data Y Introduction Y Graphs of Frequency Distribution Y

Stem and Leaf Diagram Y Box Plot or Box and Whisker Diagram Y Mea sures of Cen tral Ten dency Y In tro duc tion Y Ob jec tives of Av er age Y

Char ac ter is tics of a Good Av er age Y Var i ous Mea sures of Cen tral Ten dency Y Par ti tion Val ues or Quantiles Y Mode Y Mea sures of Dis per sion,

Skew ness and Kurtosis Y In tro duc tion Y Ob jects and Im por tance of Dis per sion Y Char ac ter is tics for Sat is fac tory Mea sures of Dis per sion Y Ab so lute

and Rel a tive Mea sure of Vari a tion Y Mea sures of Dis per sion Y Root Mean Square De vi a tion Y Re la tion be tween σ and S Y Ef fect of Change of Or i gin and

Scale on Stan dard De vi a tion Y Com bined Stan dard De vi a tion Y Math e mat i cal Prop er ties of Stan dard De vi a tion Y Mo ments Y Re la tion be tween µ r and

µ r' Y Ef fect of Change of Or i gin and Scale on Mo ments Y Sheppard's Cor rec tion on Mo ments Y Charlier's Check Y Pearson's β Co ef fi cients and Fisher's γCo ef fi cients Y Skew ness Y Mea sures of Skew ness Y Kurtosis Y Method of Least Squares and Curve Fit ting Y In tro duc tion Y Method of Least Squares

Y Sys tem of Lin ear Equa tions Y Curve Fit ting Y Cor re la tion and Re gres sion Y Univariate and Bivariate Dis tri bu tions Y Cor re la tion Y Types of

Cor re la tion Y Scat ter Di a gram or Dot Di a gram Y Karl Pearson's Co ef fi cient of Cor re la tion Y As sump tions Y Prop er ties of Cor re la tion Co ef fi cient (r) Y

Cor re la tion in Grouped Data Y Co ef fi cient of De ter mi na tion Y Rank Cor re la tion Y Re gres sion Y An gle be tween Two Re gres sion Lines Y Prop er ties of

Re gres sion Co ef fi cients Y Method of Fit ting Re gres sion Lines from a Bivariate Data Y Biserial Cor re la tion Y Mul ti ple and Par tial Cor re la tion Y

In tro duc tion Y Mul ti ple Cor re la tion Y Par tial Cor re la tion Y Mul ti ple Re gres sion Equa tion Y Math e mat i cal No ta tions (Yule's No ta tion) Y Prop er ties of

Re sid u als Y Vari ance of Re sid u als Y Ex pres sion for Co ef fi cient of Mul ti ple Cor re la tion Y Ex pres sion for Co ef fi cient of Par tial Cor re la tion Y The ory of

At trib utes Y In tro duc tion Y Con cepts and Def i ni tions Y An Im por tant No ta tion Y Con sis tency of Data Y In de pend ence of At trib utes Y As so ci a tion of

At trib utes Y Co ef fi cient of As so ci a tion Y Co ef fi cient of Colligation Y Con tin gency Ta ble Y As so ci a tion in a Con tin gency Ta ble Y Log a rithm and

An ti log a rithm Table.

Y Point Estimation Y Introduction Y Population and Sample Y Parameter and Statistic Y Theoretical Population and its Random Sample Y Sampling

Distribution Y Standard Error Y Statistical Inference: An Overview Y Estimation Y Criteria of a Good Estimator Y Methods of Estimation Y Testing of

Hypothesis Y Introduction Y Hypothesis and its Types Y Critical and Acceptance Region Y Two Types of Error Y Level of Significance Y Power Function

7

–Dr. Arun Kumar & Dr. Alka Chaudhary691-07 (B) Probability Distribution & Theory of Attributes

–Dr. Arun Kumar & Dr. Alka Chaudhar y692-08 Statistical Methods

–Dr. Arun Kumar & Dr. Alka Chaudhary693-05 Statistical Inference

and Power of a Test Y p-value Y Procedure of Testing a Hypothesis Y Best Critical Region Y Unbiased Tests Y Neyman-Pearson Fundamental Lemma

Y Tests of Significance Y Introduction Y Tests of Significance Y Tests of Significance for Attributes (Large Samples) Y Test of Significance in Case of

Variables (Large Samples) Y Tests of Significance Based on χ2-Distribution Y Test of Significance Based on t-Distribution Y Tests of Significance Based on

F-Distribution Y Likelihood Ratio Tests and Interval Estimation Y Likelihood Ratio Test Y Reduction of L.R. Tests to Standard Tests Y Interval

Estimation Y Tables.

Y Introduction and Basic Concepts of Sampling Y Introduction Y Advantages of Sample Survey Y Disadvantages of Sample Survey Y The Principal

Steps in a Sample Survey Y Concepts, Definitions and Terminology Y Desirable Properties of an Estimator Y Sampling Errors Y Non-Sampling Errors

Y Sampling Distribution Y Various Sampling Procedures Y Simple Random Sampling Y Introduction Y Various Probabilities of Selection Y How to

Select a Simple Random Sample? Y Different Sets of Random Numbers Y Modified Procedure Based on Random Numbers Y Notation and Terminology

Y Some Theorems Relating to Simple Random Sampling Without Replacement (SRSWOR) Y Some Theorems Relating to Simple Random Sampling

with Replacement (SRSWR) Y Confidence Interval Y Stratified Random Sampling Y Introduction Y Advantages of Stratified Random Sampling Over

Simple Random Sampling Y Notations and Terminology Y Two Estimates of Population Mean Y Confidence Limits Y Allocation of Sample Size

Y Systematic Sampling Y Difference between Stratified Random Sampling and Systematic Sampling Y Advantages of Systematic Sampling Y

Disadvantages of Systematic Sampling Y Uses of Systematic Sampling Y Notation and Terminology Y Estimator of the Population Mean Y Systematic

Sampling versus Simple Random Sampling Y Comparison of Systematic with Simple and Stratified Random Sampling Y Population with Linear Trend Y

Population with Periodic Variation Y Auto-Correlated Population Y Population in Random Order Y Ratio and Regression Method of Estimation Y

Introduction Y Concept of r and R Y Notations and Terminology Y Ratio Estimator Y Bias of the Ratio Estimator Y First and Second Order Approximation

to Bias Y Mean Square Error of Ratio Estimate Y Conditions for which Ratio Estimate is Better than SRS Y Properties of Ratio Estimate Y Why Ratio

Estimation is Used ? Y Difference Estimate Y Value of k for which Variance of yD is Minimum Y Regression Estimate Y Bias of Regression Estimate Y Mean

Square Error of Regression Estimate Y Some other Sampling Schemes Y Cluster Sampling Y Notations and Terminology Y Difference between

Stratified Random Sampling and Cluster Sampling Y Efficiency with Respect of SRS Y Clusters of Unequal Size Y Double Sampling in Ratio Method of

Estimation Y Regression Estimate in Double Sampling Y Double Sampling for Stratification Y Non-Sampling and Sampling Errors Y Introduction Y

Classification of Errors Y Type of Non-Sampling Errors Y Control Measure Y Statistical Organisation in India Y Central Statistical Organisation (C.S.O) Y

National Sample Survey Organisation (NSSO) Y Governing Council Y Working Groups (WG) Y Socio-Economic Surveys Y Sarvekshana, NSSO Bulletin

Y United Nations World Food Programme (UNWFP) Y Agricultural Statistics Y Price Data Collection Y Urban Frame SurveyY Industrial Statistics Y

Annual Survey of Industries Y Labour Bureau Y Army Statistical Organisation (ASO) Y Some Non-Government Statistical Organisations Y Statistical

Organisation in States Y Statistical Organisation in U.P. and Uttaranchal Y Tables.

Y Analysis of Variance Y Introduction Y Meaning Y Assumptions Y Analysis of Variance (One-way Classified Data) Y Analysis of Two Way Classified

Data with one Observation Per Cell Y Design of Experiments Y Introduction Y Meaning and Need Y Nomenclature Used in Design of Experiments Y

Some Basic Points Regarding the Planning of an Experiment Y Three Principles of Design of Experiment Y Size and Shape of Plots Y Size and Shape of

Blocks Y Different Experimental Designs Y Missing Plot Technique in R.B.D. Y Two Missing Observations Y Missing Plot Technique in L.S.D. Y Factorial

Experiments Y Two Factors Each at 2 Levels (22 Factorial) Y Main Effects and Interactions Y Sum of Squares due to Factorial Effects Y Tests for Factorial

Effects Y Yate’s Method of Computing Factorial Effects Total Y The Case of 3 Factors Y The General Case Y Confounding Y Situations where Partial

Confounding is Preferable Y How to find Confounded Effects? Y Partial Confounding in a 23 Experiment Y Split Plot Design Y Comparison of a Split Plot

Design with R.B.D. Y Strip Plot Design Y Analysis of Covariance Y Tables.

Y Time Series Y Introduction Y Definitions Y Applications of Time Series Analysis Y Components of a Time Series Y Analysis of Time Series

Y Measurement of Trend Y Measurement of Seasonal Variations Y Residual Method for Isolation of Cyclic Variations Y Index Numbers Y Introduction

Y Characteristics of Index Numbers Y Uses of Index Numbers Y Points to be Considered in the Construction of Index Numbers Y Types of Index Numbers

8

–Dr. Arun Kumar & Dr. Alka Chaudhary694-05 (B) Survey Sampling

–Dr. Arun Kumar & Dr. Alka Chaudhary695-05 Analysis of Variance & Design of Experiments

–Dr. Arun Kumar & Others698-02 Applied Statistics

Y Methods of Constructing Index Numbers Y Test for Index Numbers Y Quantity Index Number Y Value Index Number Y Chain Based Index Y Relative

Merits and Demerits of Chain Base an Fixed Base Method Y Base Conversion Y Cost of Living Index Numbers Y Construction of Cost of Living Index

Numbers Y Limitation of Index Numbers Y Statistical Methods for Psychology & Education Statistics Y Introduction Y Scaling Y Some Scaling

Procedures Y Linear Model of Test Theory Y Reliability Y Methods of Estimating Test Reliability Y Validity Y Parallel Tests Y Point-Biserial and Biserial

Correlation Coefficients Y Demographic Methods Y In tro duc tion Y Uses of Vi tal Sta tis tics Y Col lec tion of Vi tal Sta tis tics Y Rates Y Fer til ity

Mea sure ments Y Fac tors Af fect ing Fer til ity Vari a tion Y Crude Birth Rate Y Gen eral Fer til ity Rate Y Age-Spe cific Fer til ity Rate Y To tal Fer til ity Rate Y

Mor tal ity Mea sure ment Y Mea sure ment of Pop u la tion Growth Y Sta tion ary and Sta ble Pop u la tions Y Life Ta ble or Mor tal ity Ta ble Y As sump tions of a

Life Ta ble Y Con struc tion of a Life Ta ble Y Uses of a Life Ta ble Y Es ti mates of the Var i ous Func tions of the Life Ta ble and their Inter-Re la tion ship Y

Sta tis ti cal Or gani sa tion in In dia Y Cen tral Sta tis ti cal Of fice (C.S.O.) Y Na tional Sam ple Sur vey Of fice (NSSO) Y Gov ern ing Coun cil Y Work ing Groups

(WG) Y Socio-Eco nomic Sur veys Y Sarvekshana, NSSO Bul le tin Y United Na tions World Food Programme (UNWFP) Y Ag ri cul tural Sta tis tics Y Price

Data Col lec tion Y Ur ban Frame Sur vey YIn dus trial Sta tis tics Y An nual Sur vey of In dus tries Y La bour Bu reau Y Army Sta tis ti cal Or gani sa tion (ASO) Y

Cen sus of In dia Or gani sa tion Y Some Non-Gov ern ment Sta tis ti cal Or gani sa tions Y Sta tis ti cal Or gani sa tion in States Y Sta tis ti cal Or gani sa tion in U.P. and

Uttaranchal Y Sta tis ti cal Qual ity Con trol Y In tro duc tion Y What is Qual ity? Y How Qual ity is Mea sured? Y What is Qual ity Con trol? Y Sta tis ti cal

Qual ity Con trol (S.Q.C.) Y Causes of Vari a tion Y Tech niques of Sta tis ti cal Qual ity Con trol Y Con trol Charts and their Ba sis Y Types of Con trol Charts Y

Con trol Charts for Vari ables Ver sus Con trol Charts for At trib utes Y Ad van tages of Sta tis ti cal Qual ity Con trol Y Lim i ta tion of Sta tis ti cal Qual ity Con trol Y

Sam pling In spec tion (Ac cep tance Sam pling) Y Pro ducer's Risk Y Con sumer's Risk Y Ac cep tance Qual ity Level (A.Q.L.) Y Rejectable Qual ity Level

(R.Q.L.) or Lot Tol er ance Pro por tion De fec tive (L.T.P.D.) Y Av er age Out go ing Qual ity Limit (A.O.Q.L) Y Average Sample Number (ASN) Y Operating

Characteristic (OC) Y Single Sampling Plan Y Double Sampling Plan Y Single Sampling vs. Double Sampling Plans.

Y Non-Parametric Inference Y Introduction Y Advantages and Disadvantages of Non-Parametric Tests/Methods Y The Sign Test Y Wilcoxon Signed

Rank Test Y Mann-Whitney U Test Y The Runs Test Y The Median Test Y Kolmogorov-Smirnov Test (K-S Test) Y Spearman's Rank Correlation

Coefficient TestY Finite Differences Y Introduction Y Symbols used in Finite Differences Calculus Y The Operators E and ∆ Y Relationship between E

and D Y The Difference Table Y Factorial Functions Y Differences of Zero Y Effect of an Error in a Tabular Value Y Interpolation Y Introduction Y

Interpolation with Equal Intervals Y Interpolation with Unequal Intervals Y Central Differences Y Numerical Integration Y Introduction Y General

Quadrature Formula for Equidistant Ordinates Y The Trapezoidal Rule Y Simpson's One Third Rule Y Simpson's Three-Eigth's Rule Y Weddle's Rule Y

Error in Quadrature Formula Y Cote's Method Y Numerical Differentiation Y Introduction Y Derivatives using Forward Difference Formula Y

Derivatives using Backward Difference Formula Y Derivatives using Central Difference Formula Y Derivatives of a Function when the given Arguments

are not Equally Spaced Y Tables.

Y Linear Programming Y Introduction Y Different Kinds of Linear Programming Problems Y Basic Requirements of a C.P. Problem Y Assumptions of

Linear Programming Y Applications of Linear Programming in Different Areas Y Steps Involved in the Formulation of C.P. Problem Y Solution of a Linear

Programming Problem Y Some Special Cases Y Minimisation of Objective Function Y Problem of Converting Minimise into Maximise Y Transportation

Problem Y Introduction Y Test for Optimility (Modi Method) Y Unbalanced Transportation Problem Y Degeneracy in Transportation Problem

Y Transportation Problem of Maximum Profit Y Assignment Problem Y Mathematical Formulation of Assignment Problem Y Basic Theorem in

Assignment Problem Y Unbalanced Assignment Problem Y Max-type Assignment Problems Y Restrictions on Assignments (Prohibitive Assignment) Y

Fundamentals of Computers Y Introduction Y Characteristics of a Computer Y History of Computers Y The Computer Generations Y Limitations and

Applications of Computer Y Number Systems Y Computer Organisation Y Input Devices Y Output Devices Y Memory Devices YClassification of

Secondary Storage Devices Y Hardware and Software Y Operating Systems Y Classification of Computers Y Communication and Computer

Languages Y Introduction Y Communication Y Digital and Analog Signals Y Modem Y Networking Y Internet Y Computer Languages Y Introduction to

Database Management System (DBMS) Y Introduction to C Language Y Algorithm and Flow Chart Y Introduction to C Language Y Learning

Fundamentals Y Operators and Expressions Y Library Functions Y C Instructions Y Writing a C Program Y Input and Output Statements Y Control

Statements and Arrays Y Introduction Y Selection Statements Y Repetitive Iterative Statements Y Jumping Statements Y Arrays Y Functions

Y Appendices.

9

–Dr. Arun Kumar & Dr. Alka Chaudhary697-02 Non-Parametric Methods & Numerical Analysis

–Dr. Arun Kumar & Dr. Alka Chaudhary700-02 Linear Programming & Computational Techniques

Y Central Conicoids Y The El lip soid Y The Hyperboloid of one sheet Y The Hyperboloid of two sheets Y The tan gent plane Y The con di tion of

tan gency Y The Di rec tor Sphere Y The Po lar Plane Y Prop er ties of the po lar planes and the po lar lines Y Lo cus of chords bi sected at a given point Y

Nor mal to a Conicoid Y Num ber of nor mals Y Cu bic curve through the feet of the nor mals Y To find the equa tion of the cone through six con cur rent

nor mals (the six nor mals drawn from a point to an el lip soid) Y Di am et ral plane Y Con ju gate diameers and con ju gate diameteral planes Y The re la tion ship

be tween the co-or di nates of the points P Q R, , where OP OQ, and OR are the con ju gate semi-di am e ter of an el lip soid Y Prop er ties of con ju gate

semi-di am e ters of an el lip soid Y The Cone Y The Pa rabo loids Y The el lip tic paraboloid Y The hy per bolic paraboloid Y The gen eral equa tion Y The

nor mal Y Cu bic curve through the feet of the nor mals Y Gen er at ing Lines Y The gen er a tion lines of a hyperboloid of one sheet Y Prop er ties of the

gen er at ing lines of hyperboloid of one sheet Y Prop er ties of the gen er at ing lines of hyperboloid of one sheet Y Per pen dic u lar gen er a tors Y The gen er at ing

lines of a hy per bolic paraboloid Y Prop er ties of gen er a tors of a hyperboloid Y Per pen dic u lar gen er a tors Y To show that the gen er a tors of the λ-and

µ-sys tems of the hy per bolic paraboloid x a y b z c2 2 2 2 2/ / /− = are par al lel to the planes x a y b/ /± = 0 Y The Plane Sec tions of Conicoids Y Na ture

of a plane sec tion Y Lengths and di rec tion ra tios of the axes of a cen tral sec tion Y Non-cen tral Plane Sec tion Y To Find the Lengths and Di rec tion Co sines

of the Axes of Non-Cen tral Plane Sec tion of a Cen tral Conicoid Y Plane Sec tions of a Paraboloid Y To de ter mine the na ture of a given sec tions of a

paraboloid Y To find the lengths and di rec tion ra tios of the axes of the sec tion of the paraboloid Y Cir cu lar Sec tions Y To de ter mine the cir cu lar sec tions

of an el lip soid Y To show that any two cir cu lar sec tions of an el lip soid which are not par al lel lie on a sphere Y The cir cu lar sec tions of any cen tral conicoid

Y The cir cu lar sec tions of the parabolid Y Umbilics Y Umbilics Y Def i ni tion Y To de ter mine the real umbilics of the el lip soid Y To de ter mine the real

umbilics of the paraboloid Y Con fo cal Conicoids Y Con fo cal conicoids Y Def i ni tion Y Three coafocals through a given point Y Three pa rabo loids through

a given point Y To prove that one conicoid con fo cal with a given conicoid touches a given plane Y To prove that the con fo cal conicoids cut one an other at

right an gles at all their com mon points, i.e., the tan gent planes at any com mon point are at right an gles Y El lip tic co-or di nates Y Fo cal Conics Y Prop er ties

re gard ing the nor mals to three con fo cal conicoids through a given point P Y The foci of conicoids Y Re duc tion of Gen eral Equa tion of Sec ond

De gree Y Points of In ter sec tion Y The tan gent plane Y The Nor mal Y The Po lar Plane Y The en vel op ing cone Y The en vel op ing cyl in der Y The

en vel op ing cyl in der Y To find the lo cus of the chords which are bi sected at a given point ( , , )α β γ Y The di am et ral plane Y Prin ci pal planes and the prin ci pal

di rec tions Y Orthogonallty of the prin ci pal di rec tions Y Trans for ma tion of f x y z( , , ) Y The cen tre of the sur face F x y z( , , ) = 0 Y Pro cess of re duc ing a

gen eral equa tion to the stan dard form and to dis cuss the na ture Y Sur face of Rev o lu tion Y To find the con di tion that a gen eral equa tion of sec ond

de gree namely F x y z( , , ) = 0, may rep re sent a sur face of rev o lu tion.

Y Change of Independent Variables Y To change the in de pend ent vari able into the de pend ent vari able Y To change the in de pend ent vari able x into

an other vari able z; where x z= φ ( ) Y Dif fer en tials Y To tal and par tial dif fer en tial co ef fi cients Y Trans for ma tion in the case of two in de pend ent vari ables Y

Trans for ma tion from Car te sian to po lar co-or di nates and vice versa Y Or thogo nal tansformation of ∇2 V Y Max ima and Min ima (Sev eral In de pend ent

Vari ables) Y Nec es sary con di tion for the ex is tence of max ima or min ima Y Algebraic lemma regarding the sign of quadratic expressions Y Lagrange’s

con di tion for two in de pend ent vari ables Y Three in de pend ent vari ables Y Sev eral in de pend ent vari ables Y Lagrange’s meth ods of un de ter mined mul ti pli ers

Y Jacobians Y Def i ni tion Y Case of func tion of func tions; Jacobian implicitl func tions Y Nec es sary and suf fi cient con di tion for a Jacobian to van ish;

Covariants and invariants Y Con ti nu ity and Differentiability Y Func tions, Lim its, Con ti nu ity Y The four func tional lim its at a point Y Kinds of

dis con tin u ous Saltus, The o rems on con ti nu ity Y The o rems on dis con tin u ous func tions, Pointwise dis con tin u ous func tion Y Uni form con ti nu ity, Ab so lute

con ti nu ity Y Con ti nu ity of a func tion of more than one vari able Y Differentiability Y Mean ing of the sign of the de riv a tive, Geo met ri cal mean ing of a

de riv a tive Y The chain rule, Darboux The o rem Y Rolle's, Tay lor's and Al lied The o rems Y Rolle’s the o rem Y Geo met ri cal in ter pre ta tion of Rolle’s

the o rem Y Lagrange’s mean value the o rem Y Cauchy’s mean value the o rem Y Tay lor’s de vel op ment of a func tion in a fi nite form with Lagrange’s form of

re main der, Tay lor’s the o rem with Cauchy’s form of re main der Y Tay lor’s the o rem with Schomilch and Roche’s from of re main der Y Fail ure of Tay lor’s

and Maclaurin’s ex pan sions in more than one vari ables.

10

–A.R. Vasishtha & D.C. Agarwal211-12Analytical Solid Geometry (Analytical Geometry of Three Dimensions)

–J. N. Sharma212-21 Advanced Differential Calculus

M.Sc. Mathematics(Books for Honours & Post-Graduate Students of All Indian Universities and Competitive Examination)

Y Definite Integrals Y Def i ni tion Y The def i nite integrals as the limit of a sum Y Geo met ri cal In ter pre ta tion Y Prop er ties of def i nite integrals Y Method of

dif fer en ti a tion un der the sign of in te gra tion Y Method of in te gra tion un der the sign of in te gra tion Y Prin ci pal and Gen eral Val ues of a def i nite in te gral

Y In fi nite lim its Y − ∞

∞

∫ f x

F xdx

( )

( ) Y

0

2

21

∞

∫ +x

xdx

m

n Y

0

2

21

∞

∫ −x

xdx

m

n Y Im por tant de duc tions Y log ( cos )1 2 2+ +∫ a x a dx Y

0

21 2π

∫ − +log ( cos )a x a dx Y

Eu ler’s Integrals Y Def i ni tion (Beta and Gamma func tions) Y Sym met ri cal prop erty of Beta func tion Y To eval u ate Beta func tion Y To eval u ate Gamma

func tion Y Trans for ma tion of Gamma func tion Y An other form of Beta func tion Y B l ml m

l m( , )

/

( )=

+Γ ΓΓ

Y Other trans for ma tion of Beta func tions Y

Γ Γ Γ( ) .( )

( )m mx

mm

+

= √−

1

2 22

2 1 Y To eval u ate Γ Γ Γ1 2 1

n n

n

n

−

... Y 0

1∞

− −∫ e x x dxax mcos .β and 0

1∞

− −∫ e x x dxax msin .β Y Mul ti ple

Integrals Y Dou ble In te gra tion (Car te sian co-or di nates) Y Dou ble In te gra tion (Po lar co-or di nates) Y Mul ti ple Integrals Y Change of or der of in te gra tion Y

Trans for ma tion of mul ti ple integrals Y Trans for ma tion of im plicit func tions Y Trans for ma tion of the el e ment of a sur face Y Beta and Gamma Func tions

(contd.) Y Dirichlets The o rem Y Liouvilles Ex ten sion of Dirichlet’s The o rem Y Vol umes and Sur faces Y Vol ume (Car te sian Co-or di nates) Y Vol ume

(Po lar-Co-or di nates) Y Area of the sur face Y Cen tre of Grav ity Y Mo ment of in er tia Y Con ver gence of Im proper Integrals Y Def i ni tion and kinds of

im proper in te gral Y Con ver gence of im proper in te gral of first kind Y Nec es sary and suf fi cient con di tion for the con ver gence of 0

∞

∫ f x dx( ) Y Tent for the

con ver gence of c

f x dx∞

∫ ( ) Y Com par i son Test Y The µ-test Y Abel’s Test Y Dirichlet’s test Y Ab so lute con ver gence Y Con ver gence of im proper integrals of

sec ond kind Y Com par i son Test Y c

d

n

dx

x c∫ −( ) Y The µ-test Y Abel’s test and Dirichlet’s test Y Ab so lute con ver gence Y lmproper integrals con tain ing a

pa ram e ter uni form con ver gence Y Weierstrass M-test Y Dirichlet’s Test Y Fou rier Se ries Y Pe ri odic func tions Y Some re sults of def i nite integrals Y

Fou rier se ries Y Eu ler’s For mu lae Y Dirichlet’s Con di tion Y Some integrals Y Even and Odd func tions of x Y Half Range Ex pan sions Y Fou rier’s se ries in

( , )a b Y The Riemann In te gral Y Def i ni tions Y Riemann In te gra tion (Def i ni tion end Ex is tence) The o rems (1 to 8) Y Os cil la tory sum The o rem 9 Y Mesh

of a par ti tion The o rem 10 Y Some classes of Riemann integrable func tions The o rems (11 to 14) Y Al ge bra of Riemann integrable func tions The o rems (15

to 21)Y The in ter val of in te gra tion (The o rems 22, 23) Y The integrability of the prod uct (The o rems 24, 25) Y The integrability of quo tient (The o rem 26) Y

In te gra tion and Dif fer en ti a tion (The o rem 27) Y Fun da men tal The o rem of In te gral Cal cu lus (28) Y Mean Value The o rems (The o rems 29 to 32) Y Change

of Vari ables (The o rem 33).

Y The Calculus of Finite Differences Y Fi nite Dif fer ences Y Dif fer ences Y Dif fer ence For mu lae Y Fun da men tal The o rem of Dif fer ence Cal cu lus Y The

Dif fer ence Ta ble Y The Op er a tor E Y Prop er ties of the Op er a tors E and ∆ Y Re la tion be tween Op er a tor E of Fi nite Dif fer ences and Dif fer en tial Co ef fi cient D

of Dif fer en tial Cal cu lus Y One or More Miss ing Terms Y Fac to rial No ta tion Y Meth ods of Rep re sent ing any Given Poly no mial in Fac to rial No ta tion Y

Dif fer ences of Zero Y Leibnitz’s Rule Y Ef fect of an Er ror in a Tab u lar Value Y Stirling Num bers Y In ter po la tion with Equal In ter vals Y The Fol low ing

In ter po la tion Meth ods are Used Y Sub-di vi sion of In ter vals Y In ter po la tion with Un equal In ter vals Y Di vided Dif fer ences Y Prop er ties of Di vided

Dif fer ences Y New ton’s For mula for Un equal In ter vals Y Re la tion be tween Di vided Dif fer ences and Or di nary Dif fer ences Y Sheppard’s Rule Y Lagrange’s

In ter po la tion For mula for Un equal In ter vals Y It er a tive Method Y Hermite In ter po la tion For mula Y Spline In ter po la tion Y Cen tral Dif fer ence

In ter po la tion For mu lae Y Gauss’s In ter po la tion For mu lae Y Stirling’s For mula Y Bessel’s For mula Y Laplace-Everett For mula Y Use of Var i ous

In ter po la tion For mu lae Y Nu mer i cal Dif fer en ti a tion Y Di rect Meth ods (us ing for mula) Y Max ima and Min ima of a Tab u lated Func tion Y Nu mer i cal

In te gra tion Y A Gen eral Quad ra ture For mula for Equi dis tant Or di nates Y The Trap e zoidal Rule Y Simpson’s One-Third Rule Y Simpson’s Three-Eighth’s

Rule Y Boole’s Rule Y Weddle’s Rule Y Er ror in Quad ra ture For mu lae Y Cote’s Method Y The Eu ler-Maclaurin’s Sum ma tion For mula Y Stirling’s

For mula for Ap prox i ma tion to Fac to ri als Y Method of Un de ter mined Co ef fi cients Y In te gra tion For mula Y Loz enge Di a grams For Quad ra ture For mu lae

Y Romberg In te gra tion Y Hardy’s For mula Y Nu mer i cal Dou ble In te gra tion Y Gaussi an In te gra tion Y In verse In ter po la tion Y Lagrange's Method Y

It er a tion or Suc ces sive Ap prox i ma tion Method Y Method of Re ver sion of Se ries Y Sum ma tion of Se ries Y To Find the Sum to n Terms of a Se ries Whose

gen eral term is the first dif fer ence of an other func tion Y A Se ries with Gen eral Term of the Form u x arxx= φ +( ) , where φ( )x is Some Ra tio nal In te gral

Func tion of x of De gree n Y Sum ma tion by Parts Y To Prove that u ux x

n

x

n

= ∆

−+

=∑ 1

1

1

1

where ∆ ≡∆

− 1 1 Y Re la tion ship in be tween ∆ and Σ Y If

11

–P.P. Gupta, G.S. Malik & J.P. Chauhan214-42Calculus of Finite Differences & Numerical Analysis

–D.C. Agarwal213-22 Advanced Integral Calculus

ψ = =( )( )

( )x

x

x

d

dx

Γ′Γ

[log Γ( )]x Y If f x( ) is Some Func tion of x and φ( )E is a Poly no mial in E then Prove that φ = φ( ) [ ( )] ( ) [ ( )]E a f x a aE f xx x Y Dif fer ence

Equa tions Y Def i ni tion of a Dif fer ence Equa tion Y Var i ous Types of Lin ear Dif fer ence Equa tions Y Ex is tence and Unique ness The o rem Y Method of

Vari a tion of Pa ram e ters Y Meth ods of Gen er at ing Func tions Y Non-ho mo ge neous Lin ear Dif fer ence Equa tions with Vari able Co ef fi cients Y So lu tion of

Some Spe cial Types of Dif fer ence Equa tions Y Ap pli ca tion of Dif fer ence Equa tions to So cial Sci ences Y Ma trix Method for Solv ing the Sys tem of

two Si mul ta neous Lin ear Dif fer ence Equa tions Y Cob web Phe nom e non Y Ap pli ca tion to De flec tion of a Loaded String Y Ap prox i ma tions and Er rors

in Com pu ta tion Y Num bers and their Ac cu racy Y Er rors and their Anal y sis Y Gen eral Method of Find ing Re main der Term Y Ab so lute, Rel a tive and

Per cent age Er rors Y Er ror in the Ap prox i ma tion of a Func tion Y Er ror Com mit ted in a Se ries Ap prox i ma tion Y Or der of Ap prox i ma tion Y Re main der

Term of Var i ous In ter po la tion For mu lae Y Er rors in Dif fer ent Quad ra ture For mu lae Y Nu mer i cal So lu tions of Or di nary Dif fer en tial Equa tions of

First and Sec ond Or der Y Picard’s Method of Suc ces sive Ap prox i ma tions Y Eu ler’s Method Y Im proved Eu ler’s Method Y Mod i fied Eu ler’s Method Y

Tay lor’s Se ries Method Y Runge’s Method Y Runge Kutta Method Y Pre dic tor and Corrector Method Y Milne’s Method Y Ad ams-Bash Forth Method Y

Gen eral Ap proach to Pre dic tors and Correctors Y Si mul ta neous Dif fer en tial Equa tion (first or der) Y Dif fer en tial Equa tion of Sec ond Or der Y Numerov’s

Method Y Bound ary Value Prob lems Y Er ror Anal y sis Y Con ver gence of a Method Y Sta bil ity Anal y sis Y So lu tion of Al ge braic and Tran scen den tal

Equa tions Y Quo tient & Re main der by Syn thetic Di vi sion (Prob lem) Y Nearly Equal Roots Y Rate of Con ver gence of New ton’s Method When there Ex ist

Dou ble Roots Y So lu tion of Nu mer i cal Equa tions (Contd.) Y Contracton of Hor ner’s Method Y So lu tion of Si mul ta neous Lin ear Al ge braic

Equa tions Y Dif fer ent Meth ods of Ob tain ing the So lu tions Y New ton-Raphson Method for Solv ing Non-lin ear Si mul ta neous Equa tions Y Ma trix

In ver sion Y Gauss Elim i na tion Method Y Gauss-Jor dan Method Y Triangularization Method Y Crout’s Triangularization Method Y Doolittle Method Y

Choleski’s Method Y It er a tive Method Y Es ca la tor Method for Ma trix In ver sion Y Com plex Ma tri ces and In ver sion Y Eigen Val ues and Eigen Vec tors Y

It er a tive Method for Dom i nant La tent Root (or Power Se ries Method) Y Eval u a tion of All the Eigen Val ues Y Com plex Eigen Val ues Y Bounds for Eigen

Val ues Y Eigen Val ues of Real Sym met ric Ma trix Y Jacobi’s Method Y Given’s Method Y House-Holder’s Method Y Bernoulli and Eu ler Poly no mi als Y

The φ Poly no mial Y The β Poly no mi als Y Bernoulli’s Poly no mi als and Bernoulli’s Num bers Y Bernoulli’s Poly no mi als and Num bers of the First Or der

Y Eu ler Poly no mi als and Eu ler Num bers Y Prop er ties of Eu ler’s Poly no mi als Y Com ple men tary Ar gu ment The o rem for Eu ler’s Poly no mi als Y Eu ler’s

Poly no mi als of Suc ces sive Or ders Y Eu ler Poly no mi als and Eu ler’s Num ber of First Or der Y Curve Fit ting and Prin ci ple of Least Squares Y Scat ter

Di a gram Y Curve Fit ting Y Method of Curve Fit ting Y Par tic u lar Cases Y Change of Or i gin and Scale for Sim pli fy ing the Cal cu la tions Y Most Plau si ble

So lu tion of a Sys tem of Lin ear Equa tions Y Fit ting of the Curve of the Type y abx= and y = ax b Y Fit ting of the Curve pv kr = Y Fit ting of the Curve of

Type xy b ax= + Y Method of Group Av er ages Y Laws Con tain ing Three Con stants Y Method-Mo ments Y Nu mer i cal So lu tion of Par tial Dif fer en tial

Equa tions Y Bound ary-Value Prob lems Y To Ob tain Fi nite-Dif fer ence Ap prox i ma tions of Par tial De riv a tives Y To Solve Laplace’s Equa tion ( / )∂ ∂2 2u x

+ ∂ ∂ =( / )2 2 0u y in the Bounded Re gion R with Bound ary C Y Par a bolic Equa tions Y It er a tive Meth ods Y So lu tion of Laplace’s Equa tion by It er a tion

(Leibmann’s Pro cess) Y Pois son’s Equa tion Y Par a bolic Equa tions [So lu tion by Bender Schmidt Re cur rence Re la tion] Y De rive the Crank Nichol son

Dif fer ence Scheme for the Par a bolic Equa tion u auxx t= with Bound ary Con di tions as u t T( , ) ,0 0= u l t T( , ) = 1 and the Ini tial Con di tion as u x f x( , ) ( )0 = Y

Hy per bolic Equa tions Y So lu tion of El lip tic Equa tions by Re lax ation Method Y Com puter Fun da men tals with Pro gram ming in C Y Float ing Points

Num bers Y Denormalized Num ber Y Rep re sen ta tion Er ror Y In tro duc tion to C Lan guage.

Y Elementary Concepts Y The com plete so lu tion of a dif fer en tial equa tion of the nth or der con tains n-ar bi trary in de pend ent con stants Y If y y1 2, , ...., yn are

so lu tions of an equa tion then y c y c y c yn n= + + +1 1 2 2 ... is also a so lu tion Y In de pend ence of con stants of in te gra tion Y Necc. and Suff. cond. for pa ram e ters c1

and c2 to be in de pend ent Y Lin ear de pend ence and in de pend ence of so lu tions of equa tions Y Necc. and Suff. cond. for n so lu tions to form a sys tem of lin early

in de pend ent integrals Y Lin ear Equa tions of Sec ond Or der Y Com plete so lu tion in terms of a known in te gral Y To find par tic u lar in te gral of

d y

dxP

dy

dxQy

2

20+ + = Y Re moval of the first de riv a tive Y Trans for ma tion of the equa tion by chang ing the in de pend ent vari able Y Method of Vari a tion of

pa ram e ters Y Meth ods of Op er a tional Fac tors Y Or di nary Si mul ta neous Dif fer en tial Equa tions Y Si mul ta neous lin ear diff. equa tions with con stant

co ef fi cient Y Si mul ta neous equa tions in a dif fer ent form Y So lu tion of si mul ta neous equa tions of the form dx P dy Q dz R/ / /= = Y Geo met ri cal in ter pre ta tion of

equa tion dx P dy Q dz R/ / /= = Y To tal Dif fer en tial Equa tions (Pfaffian Dif fer en tial Forms and Equa tions) Y Pfaffian Dif fer en tial form Y Pfaffian

Dif fer en tial Equa tions Y To tal Dif fer en tial Equa tion for Pfaffian Dif fer en tial Equa tion in three vari ables Y Necc. and Suff. con di tion for integrability of sin gle diff.

equa tion P dx Q dy R dz+ + = 0 Y The con di tion for ex act ness Y Meth ods for solv ing P dx Qdy R dz+ + = 0 Y So lu tion of P dx Q dy R dz+ + = 0, when it

is ex act and ho mo ge neous of de gree n ≠ − 1 Y Geo met ri cal in ter pre ta tion of the equa tion P dx Q dy R dz+ + = 0. The lo cus of P dx Qdy R dz+ + = 0 is

or thogo nal to the lo cus of dx P dy Q dz R/ / /= = Y The non-integrable sin gle equa tion Y Equa tions con tain ing more than three vari ables Y Gen eral method of

so lu tion of the equa tions con tain ing more than three vari ables Y In te gra tion in Se ries Y Gen eral method of solv ing a diff. eqn. Y Case I. Roots of in di cial equa tion

equal Y Case II. Roots of in di cial equa tion, un equal and dif fer ing by quan tity not an in te ger Y Case III. Roots of in di cial equa tion dif fer ing by an in te ger, mak ing a

co ef fi cient of y de ter mi nate Y Some cases where the meth ods fails Y Se ries so lu tion about a par tic u lar point Y The par tic u lar In te gral Y Method of dif fer en ti a tion Y

12

–J.N. Sharma & R.K. Gupta215-50 Differential Equations (Gen)

Picard's It er a tion Meth ods, Unique ness and Ex is tence The o rem Y Picard’s It er a tion method Y Ex is tence and unique ness of so lu tions Y The Lipschitz

con di tion Y Ex is tence the o rem Y Unique ness the o rem Y Ex is tence and unique ness the o rem Y The o rem Y Par tial Dif fer en tial Equa tions of the First Or der Y

De riv a tive of par tial dif fer en tial equa tion Y Def i ni tions Y Lin ear par tial dif fer en tial equa tion of or der one Y Lagrange’s Lin ear equa tion Y Largrange’s so lu tion of the

lin ear equa tion Y Geo met ri cal In ter pre ta tion of Lagrange’s lin ear equa tion Y The lin ear equa tion with n in de pend ent vari ables Y Spe cial types of equa tions Y

Stan dard I. Equa tion of the form f p q( , ) = 0 Y Stan dard II. Equa tion of the form f z p q( , , ) = 0 Y Stan dard III. Equa tions of the form f x p f y p( , ) ( , )= Y Stan dard

IV. Equa tions of the form z px qy f p q= + + ( , ) Y Gen eral Method of so lu tion Y Two independent Vari ables Charpit’s Method Y Three or more in de pend ent

Vari ables Jacobi’s Meth ods Y Par tial Dif fer en tial Equa tions with Con stant Co ef fi cients Y Ho mo ge neous lin ear equa tions with con stant co ef fi cients Y

So lu tion of the lin ear par tial dif fer en tial equa tions Y To find the com ple men tary func tion Y When the aux il iary equa tion has equal (re peated) roots Y The par tic u lar

in te gral Y Short Meth ods Y Ex cep tional case when f a b( , ) = 0 Y Gen eral Meth ods Y Non-ho mo ge neous Lin ear equa tions with con stant co ef fi cients Y Par tic u lar

In te gral Y Equa tion re duc ible to ho mo ge neous lin ear form Y Par tial Dif fer en tial Equa tions of the Second Or der Y Monge's Meth ods Y Monge’s

method of in te grat ing Rr Ss Tt V+ + = Y Monge’s method of in te grat ing Rr Ss Tt V rt s V+ + + − =( )2 Y Clas si fi ca tion of Lin ear Par tial

Dif fer en tial Equa tions Y Clas si fi ca tion of lin ear par tial dif fer en tial Equa tions of sec ond or der Y Ho mo ge neous Lin ear Equa tions with

Vari able Co ef fi cients Y Ho mo ge neous lin ear Equa tions Y Meth ods of So lu tion Y Equa tions re duc ible to ho mo ge neous form Y Sin gu lar

So lu tion Y Discriminant Y Ex tra ne ous Loci Y Ex act Dif fer en tial Equa tions and Equa tions of other Par tic u lar Forms Y Ex act Diff. Eqn.

(Def i ni tion) Y Con di tion of ex act ness of a lin ear equa tion of or der n Y In te grat ing fac tor Y Non-lin ear equa tion Y An equa tion which does not con tain y di rectly Y

An equa tion which does not con tain x di rectly Y An equa tion of the form d y

dxf x

n

n= ( ) Y An equa tion of the form

d y

dxf y

2

2= ( ) Y An equa tion of the form

d y

dx

d y

dxx

n

n

n

n, ,

−

−

=

2

20 Y Equa tion in which or der of the dif fer en tial co ef fi cients dif fer by unity Y Nu mer i cal In te gra tion Y Simpson’s Rule Y Nu mer i cal

ap prox i ma tion Y Legendre Poly no mi als Y Legendre’s equa tion Y So lu tion of Legendre’s Equa tion Y Def i ni tion of P xn( ) and Q xn( ) Y Gen eral so lu tion of

Legendre’s equa tion Y To show that P xn( ) is the co ef fi cient of hn in the ex pan sion in as cend ing pow ers of h of ( ) /1 2 2 1 2− + −xh h Y Laplace’s Def i nite Integrals

for P xn ( ) Y Or thogo nal prop er ties of Legendre’s Poly no mi als Y Re cur rence for mu lae Y Beltrami’s Re sult Y Christoffel’s Ex pan sion Y Christoffel’s Sum ma tion

For mu lae Y Rodrigues For mu lae Y Some Bounds on P xn( ) Y Even and odd func tions Y Ex pan sions of x n in Legendre’s Poly no mi als Y Gen eral Re sults Y An

im por tant Case Y Trig o no met ri cal se ries for P xn( ) Y Legendre's Func tion of the Sec ond Kind Q xn( ) Y Legendre’s func tions of the Sec ond Kind Y

Neumann’s In te gral Y Re cur rence for mu lae for Q xn( ) Y Re la tion be tween P xn( ) and Q xn( ) Y Christoffel’s Sec ond Sum ma tion for mula Y Com plete so lu tion of

Legendre’s equa tion (other form) Y Bessel Func tions Y Bessel’s equa tion (Def.) Y So lu tion of Bessel’s Gen eral Dif fer en tial Equa tions Y Gen eral so lu tion of

Bessel’s Equa tion Y In te gra tion of Bessel’s equa tion in se ries for n = 0 Y Def i ni tion of J xn( ) Y Re cur rence for mu lae for J xn( ) Y Gen er at ing func tion for J xn( ) Y

Some Trig o no met ric ex pan sion in volv ing Bessel’s func tions Y A sec ond so lu tion of Bessel’s Equa tion Y Hermite Poly no mi als Y Hermite Dif fer en tial

Equa tion Y So lu tion of Hermite Equa tion Y Hermite’s Poly no mi als Y Gen er at ing func tion Y Other forms for Hermite Poly no mi als Y To find first few Hermite

Poly no mi als Y Or thogo nal prop er ties of Hermite poly no mi als Y Recc. For mula for Hermite Poly no mi als Y Laguerre Poly no mi als Y Laguerre’s Dif fer en tial

Equa tion Y So lu tion of Laguerre Equa tion Y Laguerre Poly no mi als Y Gen er at ing func tion Y Rodrigues for mula Y To find first few Laguerre Poly no mi als Y

Orhtogonal Prop. of Laguerre Poly no mi als Y Recc. for mula for Laguerre Poly no mi als Y Chebyshev Poly no mi als Y Chebyshev’s Equa tion Y Chebyshev’s

Poly no mi als Y To prove that T x U xn n( ), ( ) are in de pend ent so lu tions of Chebyshev Equa tion Y Im por tant Re la tions for T xn( ) and U xn( ) Y To find first few

Chebyshev Poly no mi als Y Gen er at ing func tion Y Orhtogonal prop er ties of Chebyshev’s poly no mi als Y Recc. for mula for T xn( ) and U xn( ).

Y Ordinary Differential Equations and Wronskian Y Dif fer en tial Equa tions Y Or der and De gree of a Dif fer en tial Equa tion Y Lin ear and Non-lin ear

Dif fer en tial Equa tions Y So lu tion of a Dif fer en tial Equa tion Y Prop er ties of So lu tion of a Dif fer en tial Equa tion Y In de pend ence of Con stants of In te gra tion Y

Nec es sary and Suf fi cient Con di tion for the In de pend ence of Con stants of In te gra tion Y Wronskian Y Lin ear De pend ence and In de pend ence of So lu tions of an

Equa tion Y Nec es sary and Suf fi cient Con di tion for the In de pend ence of n So lu tions of a Lin ear Dif fer en tial Equa tion of Or der n Y Fun da men tal set of So lu tion Y

Qual i ta tive Prop er ties of So lu tions (Os cil la tion, Adjoint and Sturm The ory) Y Sec tion 1: Os cil la tory Equa tion Y Non-os cil la tory Equa tion Y

Os cil la tory and Non-os cil la tory So lu tions Y Some The o rems on Os cil la tion Y Sec tion 2: The Adjoint The ory Y Adjoint Op er a tor and Adjoint Equa tion Y

Lagrange Iden tity Y Green's For mula (Corrollary to Lagrange Iden tity)Y Some The o rems Y Self-adjoint Equa tion Y The o rems on Self-adjoint Equa tions

of the Sec ond Or der Y Sec tion 3 : The Sturm The ory Y The o rem I Y The o rem II (Abel's For mula) Y The o rem III Y The o rem IV (Sturm Sep a ra tion

The o rem) Y The o rem V (Sturm's Fun da men tal Com par i son The o rem) Y Sturm Com par i son The o rem Y In te gra tion in Se ries (Or di nary Points,

Reg u lar, Sin gu lar Points and Frobenius Se ries So lu tion) Y Some Im por tant Def i ni tions Y Or di nary and Sin gu lar Points Y The o rem Y Method of Se ries

So lu tion about an Or di nary Point x x= 0 Y Se ries So lu tion in Pow ers of ( ) ,x − a where x = ≠a 0 is an Or di nary Point Y Se ries So lu tion about a Reg u lar Sin gu lar

Point x = 0 (Frobenius Method) Y Case I. Roots of In di cial Equa tion Un equal and Dif fer ing by a Quan tity not an In te ger i.e., m m1 2− ≠ 0, 1, 2, ...

(Frobenius Method) Y Case II. Roots of In di cial Equa tion Un equal Dif fer ing by an In te ger Mak ing a Co ef fi cient of y In de ter mi nate (Frobenius Method) Y

13

–R.K. Gupta & J.N. Sharma215-49 Advanced Differential Equations

Case III. Roots of In di cial Equa tion Un equal Dif fer ing by an In te ger, One Root Mak ing a Co ef fi cient of y In fin ity (Frobenius Method) Y Roots of In di cial

Equa tion Equal [Frobenius Method] Y So lu tion of Legendre's Equa tion (In Decending Pow ers of x) Y The Par tic u lar In te gral Y Some Cases where the

Frobenius Method Fails Y Beta and Gamma Func tions Y Eu ler's Integrals Y El e men tary Prop er ties of Gamma Func tion Y To Show that Γ( )1 2 = π Y

Trans for ma tion of Gamma Func tion Y Sym met ric Prop erty of Beta Func tions i.e., Β Β( , ) ( , )m n n m= Y Trans for ma tion of Beta Func tion Y Re la tion be tween

Beta and Gamma Func tions Β Γ ΓΓ

( , )( ) ( )

( )m n

m n

m n=

+, m n> >0 0, Y

0

21

2

1

2

22

2

πθ θ θ

/

sin cos∫ =

+

+

+ +

p q d

p q

p q

Γ Γ

Γ

Y Legendre Du pli ca tion For mula Y To

Prove that Γ Γ Γ Γ1 2 3 1 2

n n n

n

n

−

=. . ....( )(π n

n

− 1 2) /

where n is a pos i tive in te ger Y Gauss Hypergeometric Equa tion Y The Pochhammer

Sym bol (Def.) Y Iden ti ties Sat is fied by Pochhammer Sym bol ( )α n Y Hypergeometric Se ries Y Hypergeometric Func tion Y Dif fer ent Forms of Hypergeometric

Func tion Y Con flu ent Hypergeometric Func tion (Kummer Func tion) Y Gen eral Hypergeometric Func tion Y Sym met ric Prop erty of Hypergeometric Func tion Y

Par tic u lar Cases of Hypergeometric Se ries Y Gauss's Hypergeometric Equa tion or Gauss's Equa tion or Hypergeometric Equa tion Y So lu tion of the

Hypergeometric Equa tion Y De riv a tives of Hypergeometric Func tions Y n-th De riv a tive of Hypergeometric Func tion Y De riv a tives of Hypergeometric Func tion at

x = 0 Y In te gral For mula for Hypergeometric Func tion Y Kummer's The o rem (For Hypergeometric Func tion) Y Gauss's The o rem Y Vandermonde's The o rem Y

Con flu ent Hypergeometric Equa tion (Or Kummer's Equa tion) and its So lu tion Y Se ries So lu tion of Con flu ent Hypergeometric Dif fer en tial Equa tion near x = 0

when γ is not an In te ger Y De riv a tives of Con flu ent Hypergeometric Func tion Y In te gral For mula for Con flu ent Hypergeometric Func tion 1 1F x( ; ; )α γ Y Kummer's

The o rem (for Con flu ent Hypergeometric Func tion) Y Whittaker's Con flu ent Hypergeometric Func tion Y Con tig u ous Hypergeometric Func tions Y The o rem :

Con ti gu ity Re la tion ship Y Hermite Poly no mi als Y Hermite Dif fer en tial Equa tion Y So lu tion of Hermite Dif fer en tial Equa tion Y Hermite Poly no mial Y

Gen er at ing Func tion for Hermite Poly no mial H xn( ) Y Other Forms for the Hermite Poly no mi als Y To find First Few Hermite Poly no mi als Y Or thogo nal

Prop er ties of Hermite Poly no mi als Y Re cur rence For mu lae for Hermite Poly no mi als Y Par tial Dif fer en tial Equa tions of the First Or der (Or i gin of

First Or der Par tial Dif fer en tial Equa tions and Clas si fi ca tion) Y Or der and De gree of a Par tial Dif fer en tial Equa tion Y Clas si fi ca tion of First Or der Par tial

Dif fer en tial Equa tions into Lin ear, Semi-lin ear, Quasi-lin ear and Non-lin ear Y Or i gin (Der i va tion) of First Or der Par tial Dif fer en tial Equa tion Y Some Def i ni tions Y

Lagrange's Lin ear Par tial Dif fer en tial Equa tion Y Lagrange's So lu tion of the Lagrange's Lin ear Equa tion (Lagrange's Method of Solv ing the Lin ear Par tial

Dif fer en tial Equa tion of Or der One Namely Pp Qq R+ = ) Y Work ing Method Y The Lin ear Par tial Dif fer en tial Equa tion with n In de pend ent Vari ables Y In te gral

Sur faces Pass ing Through a Given Curve Y Sur face Or thogo nal to a Given Sys tem of Sur face Y Com pat i ble Sys tem of First Or der Equa tions Y Non-lin ear Par tial

Dif fer en tial Equa tions of First Or der (Charpit's and Jacobi's Meth ods) Y So lu tion of Par tial Dif fer en tial Equa tions of First Or der and any De gree in Some

Stan dard Forms Y Stan dard Form I : Equa tion In volv ing Only p and q and no x y, and z Y Stan dard Form II : Equa tions In volv ing Only p q, & z Y Stan dard Form

III : Equa tions of the Form f x p f y q1 2( , ) ( , )= Y Stan dard Form IV : Equa tions of the Form z px qy f p q= + + ( , ) Y Charpit's Method : Gen eral Method of

So lu tion of Non-lin ear Par tial Dif fer en tial Equa tion of Or der One with Two In de pend ent Vari ables Y Jacobi's Meth ods Y Jacobi's Method of Solv ing a Non-lin ear

First Or der Par tial Dif fer en tial Equa tion in Two In de pend ent Vari ables Y Par tial Dif fer en tial Equa tions of the Sec ond Or der with Vari able Co ef fi cients

(Or i gin and Clas si fi ca tion) Y Or i gin (Der i va tion) of Sec ond Or der Par tial Dif fer en tial Equa tion Y Spe cial Types of Sec ond Or der Par tial Dif fer en tial

Equa tions Y So lu tions of Equa tions un der Given Con di tions Y Clas si fi ca tion of Lin ear Par tial Dif fer en tial Equa tions of Sec ond Or der in n-In de pend ent Vari ables

Y Clas si fi ca tion of Lin ear Par tial Dif fer en tial Equa tion of Sec ond Or der in Two In de pend ent Vari ables Y Lin ear Par tial Dif fer en tial Equa tions with

Con stant Co ef fi cients Y Ho mo ge neous and Non-ho mo ge neous Lin ear Par tial Dif fer en tial Equa tions with Con stant Co ef fi cients Y So lu tion

of a Ho mo ge neous Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y Meth ods of Find ing the Com ple men tary Func tion (C.F.)

of the Ho mo ge neous Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y Work ing Method of Find ing C.F. of a Ho mo ge neous

Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y De ter mi na tion of the Par tic u lar In te gral (P.I.) of a Ho mo ge neous Lin ear

Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y Short Method to Find P.I. when φ( , )x y is a Func tion of the form x ym n or a Ra tio nal

In te gral Al ge braic Func tion of x and y Y Short Meth ods of Find ing P.I. When φ( , )x y is a Func tion of ax by+ Y Gen eral Method of Find ing the P.I. of

Ho mo ge neous Lin ear Dif fer en tial Equa tion with Con stant Co ef fi cients Y Non-ho mo ge neous Lin ear Dif fer en tial Equa tions with Con stant Co ef fi cients Y

Meth ods of Find ing the Com ple men tary Func tion (C.F.) of Re duc ible Non-ho mo ge neous Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients

Y Work ing Method of Find ing C.F. of Re duc ible Non-ho mo ge neous Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y Method of Find ing

C.F. of Ir re duc ible Non-ho mo ge neous Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y Work ing Method of Find ing C.F. of Ir re duc ible

Non-ho mo ge neous Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients Y De ter mi na tion of the Par tic u lar In te gral (P.I.) of Non-ho mo ge neous

Lin ear Par tial Dif fer en tial Equa tion (Re duc ible or Ir re duc ible) with Con stant Co ef fi cients Y Equa tion Re duc ible to Ho mo ge neous Lin ear Form Y So lu tion

of Lin ear Par tial Dif fer en tial Equa tion with Con stant Co ef fi cients un der given Geo met ri cal Con di tions Y Re duc tion of Sec ond Or der Par tial

Dif fer en tial Equa tion into Ca non i cal Forms (Non-lin ear Equa tions of Sec ond Or der)Y Laplace Trans for ma tion (Ca non i cal Forms)

Y Work ing Method of Re duc ing a Hy per bolic Equa tion to Ca non i cal Form Y Work ing Method of Re duc ing a Par a bolic Equa tion to Ca non i cal

Form Y Work ing Method of Re duc ing El lip tic Equa tion to Ca non i cal Form Y Wave Equa tions (By Method of Sep a ra tion of Vari ables) Y

Wave Equa tion Y So lu tion of One Di men sional Wave Equa tion by Us ing the Method of Sep a ra tion of Vari ables Y So lu tion of One Di men sional

Wave Equa tion Un der the Given Con di tions Y Some Im por tant and Use ful Dif fer en tial Equa tions and Their So lu tions Y So lu tion of Two Di men sional

14

Wave Equa tion by the Method of Sep a ra tion of Vari ables Y Vi bra tion of a Cir cu lar Mem brane (So lu tion of Two Di men sional Wave Equa tion in Po lar

Co or di nates) Y So lu tion of Three Di men sional Wave Equa tion by the Method of Sep a ra tion of Vari ables Y Wave Equa tion in Cy lin dri cal Co or di nates Y

So lu tion of Wave Equa tion is Cy lin dri cal Co or di nates by the Method of Sep a ra tion of Vari ables Y Wave Equa tion in Spher i cal Co or di nates Y So lu tion of

Wave Equa tion in Spher i cal Po lar Co or di nates by the Method of Sep a ra tion of Vari ables Y Heat and Dif fu sion Equa tions (By Method of

Sep a ra tion of Vari ables) Y One Di men sional Heat Equa tion Y Heat Equa tion Y Dif fu sion Equa tion Y So lu tion of One Di men sional Heat

Equa tion by Sep a ra tion of Vari ables Y So lu tion of One Di men sional Heat Equa tion un der given Bound ary Con di tions Y So lu tion of Two

Di men sional Heat Equa tion in Car te sian Co or di nates Y Heat Equa tion in Plane Po lar Co or di nates Y So lu tion of Heat Equa tion in Plane Po lar

Co or di nates by Sep a ra tion of Vari ables Y So lu tion of Three Di men sional Heat Equa tion by the Method of Sep a ra tion of Vari ables Y Heat

(Dif fu sion) Equa tion in Cy lin dri cal Co or di nates Y So lu tion of Heat (Dif fu sion) Equa tion in Cy lin dri cal Co or di nates by the Method of

Sep a ra tion of Vari ables Y Heat (Dif fu sion) Equa tion in Spher i cal Po lar Co or di nates Y So lu tion of Heat (Dif fu sion) Equa tion in Spher i cal Po lar

Co or di nates by the Method of Sep a ra tion of Vari ables Y Laplace Equa tions (By Method of Sep a ra tion of Vari ables) Y Laplace

Equa tion Y So lu tion of Two Di men sional Laplace's (Har monic) Equa tion by Us ing the Method of Sep a ra tion of Vari ables Y So lu tion of Two

Di men sional Laplace's Equa tion un der the Given Con di tions Y Laplace Equa tion in Plane Po lar Co or di nates Y So lu tion of Laplace Equa tion

in Plane Po lar Co or di nates by Sep a ra tion of Vari ables Y So lu tion of Laplace's Equa tion in Rect an gu lar Car te sian Co or di nates (x, y, z) by the

Method of Sep a ra tion of Vari ables Y Laplace Equa tion in Cy lin dri cal Co or di nates Y So lu tion of Laplace's Equa tion in Cy lin dri cal Co or di nates by the

Method of Sep a ra tion of Vari ables Y Laplace Equa tion in Spher i cal Co or di nates Y So lu tion of Laplace's Equa tion in Spher i cal Co or di nates by the

Method of Sep a ra tion of Vari ables.

Y Curves in Space ( )R3 Y Space curves Y Path Y Arc length Y Tan gent Line Y Con tract of n th or der of a curve and sur face Y The os cu lat ing plane (or

plane of cur va ture) Y Tan gent plane at any pont of the sur face f x y z( , , )= 0 Y To find the os cu lat ing plane at a point of a space curve given by the

in ter sec tion of the sur face f( )r = 0, ψ ( )r = 0 Y The Prin ci pal nor mal and binormal Y Def i ni tions of cur va ture, Tor sion and screw cur va ture Y To find

cur va ture and Tor sion of curve Y He li ces Y In trin sic Equa tions (or Nat u ral Equa tions) Fun da men tal The o rems for space curves Y The cir cle of cur va ture

Y The os cu lat ing sphere (or sphere of cur va ture) Y Be hav iour of curve in the Neigh bour hood of a point Y In vo lute and Evolute Y The spher i cal indicatrics

or spher i cal im ages Y Bertnard curves Y Con cept of a Sur face and Fun da men tal Forms Y Con cept and Def i ni tion of a sur face Y Curvilinear

equa tions of the curve on the sur face Y Para met ric curves Y Tan gent plane and nor mal Y Fun da men tal Forms Y Two fun da men tal forms Y First

Fun da men tal form or Met ric Y Sec ond Fun da men tal Form Y Some Im por tant Prod ucts Y De riv a tives of N, Weingarton Equa tions Y An gle be tween

para met ric curves Y Di rec tion Co ef fi cients Y An gle be tween any two in ter sect ing curves on the sur face Y Fam i lies of curves Y Or thogo nal Tra jec to ries Y

Dou ble Fam ily of curves Y Lo cal Non-in trin sic Prop er ties of a Sur face, Curve on a Sur face Y Cur va ture of nor mal sec tion Y Prin ci ple Di rec tions

and Prin ci pal cur va tures Y Line of cur va ture Y Gen eral sur face of rev o lu tion Y Joachimsthal’s The o rem Y Dupin’s Indicatrix Y Third Fun da men tal Form Y

En ve lope, Edge of Re gres sion and Developable Y En ve lope of sys tem of sur faces whose equa tions in volves two pa ram e ters Y Ruled Sur faces

(Developable and Skew) Y Developable sur face Y Developables as so ci ated with space curves K = 0, for a developable sur face Y Monge’s The o rem Y

Con ju gate di rec tions Y As ymp totic Lines Y Fun da men tal co ef fi cients and gaussi an cur va ture for a ruled sur face Y The fun da men tal Equa tions of Sur face

The ory Gauss’s For mu lae Y The Fun da men tal Equa tions of Sur face The ory (Tenosr no ta tion) Y Par al lel Sur faces Y Whole cur va ture Y Geo de sic and

Map ping of Sur faces Y Geo de sics Y Dif fer en tial equa tion of geo de sics Y Nor mal Prop erty of geo de sics Y Geo de sic cur va ture Y Gauss Bon net The o rem

Y Tor sion of a geo de sic Y Bon net’s the o rem in re la tion to geo de sics Y Geo de sics on F x y z( , , ) = 0 Y Geo de sics par al lel Y Map ping of Sur faces Y Some

Def i ni tions Y Iso met ric lines and Iso met ric cor re spon dence Y Mind ing The o rem Y Conformal mapping Y Geodesic mapping Y Tissot’s theorem Y Dini’s

theorem Y Symbols and Abbreviations.

Y Central Orbits Y Cen tral Forces, El lip tic, Hy per bolic and Par a bolic Or bits, Apses and apsidal dis tances Y Plan e tary Mo tion Y Mo tion un der in verse

square law, Plan e tary Mo tion, Kep ler’s Laws, Peri he lion and Aph elion Points, Dis trib uted El lip tic Mo tion Y Anom a lies, Plan e tary Mo tion

(Con tin ued) Y Lam bert’s The o rem Y Tan gen tial and Nor mal Ac cel er a tion, Con ser va tion of En ergy, Sim ple Pen du lum and Con strained

Mo tion Y Mo tion in a smooth ver ti cal cir cle, Mo tion on a smooth plane curve, Mo tion on a gen eral curve, Mo tion on a cir cle, Elas tic string Y Mo tion on a

smooth cycloid, Mo tion on a rough cycloid Y Mo tion in a Re sist ing Me dium and Mo tion when Mass Var ies Y Mo tion in a Straight Line in a

Re sist ing Me dium, Mo tion of Pro jec tiles in a Re sist ing Me dium Y Re volv ing Curves Y Mo ment of In er tia Y D'Alembert's Prin ci ple and Mo tion

about a Fixed Axis.

15

–S.C. Mittal & D.C. Agarwal216-40 Differential Geometry

–A.R. Vasishtha & D.C. Agarwal217-16 Dynamics of a Particle

Y Basic Concepts Y Types of fluid Y Fluid prop er ties Y Den sity Y Spe cific weight Y Spe cific vol ume Y Spe cific grav ity Y Pres sure Y Vis cos ity Y Tem per a ture Y

Ther mal con duc tiv ity Y Spe cific heat Y Sur face ten sion Y Vapour pres sure Y Bulk modulus of Elas tic ity Y Ki ne mat ics of the Flow Field Y Lagrangian method Y

Eulerian method Y Re la tion ship be tween the Lagrangian and Eulerian method Y Ve loc ity of a fluid par ti cle at a point Y Lo cal, con vec tive and ma te rial de riv a tives Y

Equa tion of con ti nu ity Y Equa tion of con ti nu ity (stream tube con cept Y Equa tion of con ti nu ity (car te sian co or di nates) Y Equa tion of con ti nu ity (spher i cal po lar

co or di nates) Y Equa tion of con ti nu ity (cy lin dri cal po lar co or di nates) Y Equa tion of con ti nu ity (Lagrangian method) Y Equiv a lence of the two forms of the equa tion

of con ti nu ity Y Ve loc ity po ten tial, Irotational flow Y Ro ta tional flow Y Vorticity Y Vorticity vec tor, Vor tex lines, Vor tex tube Y Vor tex fil a ment Y Bound ary Sur face Y

Con ser va tion of Mo men tum Y Eu ler’s equa tion of mo tion along a stream line Y Equa tion of mo tion of an inviscid fluid Y Equa tion of mo tion of an inviscid fluid

(car te sian co or di nates) Y Cauchy’s in te gral Y Bernoulli’s equa tion (Stream tube method) Y Con ser va tive field of force Y In te gra tion of Eu ler’s equa tion Y Helmholtz

equa tions Y Sym met ri cal forms of the equa tion of con ti nu ity Y Spher i cal sym me try Y Cy lin dri cal sym me try Y Im pul sive mo tion of a fluid Y lmpulsive mo tion

of a fluid (Car te sian co or di nates) Y En ergy equa tion Y Ap pli ca tions of Bernoulli’s The o rem Y Flow over a pro tu ber ance in a closed chan nel Y Pitot tube Y

Venturi tube Y Or i fice plate Y Weirs Y Irrotational Mo tion Y Gen eral mo tion of a fluid el e ment Y Mo tion of a fluid el e ment (car te sian co or di nates) Y Vorticity Y

Body forces and sur face forces Y Flow and cir cu la tion Y Stoke’s the o rem Y Kel vin’s cir cu la tion the o rem Y Con nec tiv ity Y Cy clic con stants Y Irrotational mo tion in

mul ti ply-con nected space Y Acy clic and Cy clic mo tion Y Green’s the o rem Y De duc tions from Green’s the o rem Y Mean value of the ve loc ity po ten tial over a

spher i cal sur face Y Mo tion re garded as due to Sources and Sinks Y Liq uid ex tend ing to in fin ity Kel vin’s min i mum en ergy the o rem Y Mo tion in Two Di men sions

Y Stream func tion (Plane po lar co or di nates) Y Phys i cal in ter pre ta tion of Stream func tion Y Com plex po ten tial and com plex ve loc ity Y Uni form flows Y Two

di men sional Source and Sink Y Strength Y Com plex po ten tial of a source Y Two-di men sional dou blet Y Com plex po ten tial of a dou blet Y Im ages in

two-di men sion Y Im age of a source with re gard to a plane Y Im age of a dou blet with re gard to a plane Y The cir cle the o rem Y Im age of a Source with re gard to a

cir cle Y Im age of a dou blet with re gard to a cir cle Y Conformal rep re sen ta tion Y Ap pli ca tion to Fluid Dy nam ics Y Gen eral mo tion of a cyl in der in two-di men sions Y

Mo tion of a cir cu lar cyl in der in a uni form stream Y Liq uid Stream ing past a fixed cir cu lar cyl in der Y Two co-ax ial cyl in ders (Prob lem of ini tial mo tion) Y Cir cu la tion

about a cir cu lar cyl in der Y Blasius’s the o rem Y Stream ing and Cir cu la tion for a fixed cir cu lar cyl in der Y Equa tion of mo tion of a cir cu lar cyl in der with cir cu la tion Y

El lip tic co or di nates Y Mo tion of an el lip tic cyl in der Y Stream ing past a fixed el lip tic cyl in der Y El lip tic cyl in der ro tat ing in an in fi nite mass of liq uid at rest at in fin ity Y

Ki netic en ergy of ro tat ing el lip tic cyl in der Y Ki netic en ergy when the liq uid con tained in a ro tat ing el lip tic cyl in der Y Mo tion of a par a bolic cyl in der Y Ve loc ity

po ten tial and stream func tion for a liq uid stream ing past a fixed par a bolic cyl in der Y The aero foil Y Joukowski trans for ma tion Y Kutta-Joukowski’s the o rem Y

D’Alembert’s par a dox Y Schwarz-Christoffel the o rem Y Trans for ma tion of a semi-in fi nite strips Y Semi-in fi nite strip Y In fi nite strip Y Flow into a Chan nel through a

nar row slit in a wall Y Flow past a step in a deep stream Y Flow past a step in a chan nel Y Vor tex Mo tion Y Prop er ties of the vor tex Y Strength of the vor tex Y

Rec ti lin ear Vor ti ces Y Ve loc ity com po nents Y Cen tre of vor ti ces Y A case of two vor tex fil a ments Y Stream func tion when the strength of the vor tex fil a ments are

equal Y Vor tex pair Y Vor tex dou blet Y Vor tex in side an in fi nite cir cu lar cyl in der Y Vor tex out side a cir cu lar cyl in der Y An in fi nite sin gle row of par al lel rec ti lin ear

vor ti ces of the same strength Y Two in fi nite rows of par al lel rec ti lin ear vor ti ces Y Karman's vor tex street Y Kirchoff vor tex the o rem Y Rec ti lin ear vor tex with cir cu lar

sec tion Y Ran kine’s com bined vor tex Y Rec ti lin ear vor ti ces with el lip tic sec tion Y Vor tex sheets Y Routh the o rem Y Waves Wave mo tion Y Math e mat i cal

rep re sen ta tion of wave mo tion Y Stand ing or Sta tion ary waves Y Clas si fi ca tion of waves Y Sur face waves Y Pro gres sive waves on the sur face of a ca nal Y Waves on

a deep ca nal Y En ergy of pro gres sive wave Y Pro gres sive waves re duced to a steady mo tion Y Stand ing or Sta tion ary waves Y En ergy of Sta tion ary waves

Y Waves at the com mon sur face of two liq uids Y Waves at an in ter face with up per sur face free Y Group ve loc ity Y Rate of trans mis sion of en ergy Y Long

waves Y En ergy of a long wave Y Long waves at the com mon sur face of two liq uids bounded above and be low by two fixed hor i zon tal planes Y

Irrotational Mo tion in Three Di men sions Y But ler’s sphere the o rem Y So lu tion of Laplace equa tion Y Mo tion of a sphere in an in fi nite mass of liq uid at rest at

in fin ity Y Ideal flow round a sphere Y Liq uid stream ing past a fixed sphere Y Con cen tric spheres Y Equa tion of mo tion of a sphere Y Three di men sional source and

sink Y Three di men sional dou blet Y Im age of a source with re gard to a sphere Y Mo tion of a liq uid in side a ro tat ing el lip soi dal shell Y Mo tion of an el lip soid in an

in fi nite mass of liq uid Y Val ues of Stoke’s stream func tion Y Val ues of Stoke’s stream func tion Y A sim ple source on the X-axis Y A uni form line source along the axis

Y A dou blet along the axis Y Solid of rev o lu tion along their axes in an in fi nite mass of liq uid Y Vis cous Fluid Flow Y Stress anal y sis at a point Y State of a stress at a

point Y Sym me try of stress ten sor Y Al ter na tive proof Y Stress in a fluid at rest Y Stress in a fluid in mo tion Y Trans for ma tion of stress-com po nents Y Ten sor

char ac ter of stress ma trix Y Stress quad ric Y Or thogo nal prin ci pal di rec tions Y Prin ci pal stresses and Prin ci pal di rec tions Y Strain anal y sis Y Rate of Strain quad ric Y

Al ter na tive proof Y Trans for ma tion of the rates of strain Y Re la tion be tween stress and rate of strain Y Navier-Stokes equa tion of mo tion of a vis cous fluid (Car te sian

co or di nates) Y Lim i ta tions of the Navier-Stokes equa tion Y Equa tion of en ergy Y Dissipation of energy Y Vorticity and Cir cu la tion in vis cous flu ids Y Dif fu sion of

vorticity Y The equa tion of state Y Di men sional Anal y sis Y Reynolds num ber Y Buckinghuam’s the o rem Y Si mil i tude Y Froude num ber Y Pres sure co ef fi cient

(Eu ler’s num ber) Y Match num ber Y Reynolds num ber Y Grashof num ber Y Prandtl Num ber Y Peclet num ber Y Ex act So lu tions of the Navier-Stoke's

Equa tion Y Lam i nar flow be tween par al lel plates Y Plane couette flow Y Gen er al ised plane couette flow Y Plane Poiseuille flow Y Flow be tween par al lel plates

(Tem per a ture dis tri bu tion) Y Plane couette flow Y Gen er al ised Plane couette flow Y Plane Poiseuille flow Y Hagen-Poiseuille flow, through a cir cu lar pipe Y Steady

flow be tween co-ax ial cir cu lar pipes Y Steady flow in pipes of el lip tic cross-sec tion Y Steady flow in pipes of equi lat eral tri an gu lar sec tion Y Steady flow in pipes of

rect an gu lar sec tions Y Hagen-Poiseuille flow in a cir cu lar pipe (tem per a ture dis tri bu tion) Y Lam i nar flow be tween con cen tric ro tat ing cyl in ders Y Tem per a ture

dis tri bu tion Y Steady mo tion of a vis cous fluid due to a slowly ro tat ing sphere Y Flow in con ver gent and di ver gent chan nels Y Un steady mo tion of a flat plate Y Flow

due to an os cil lat ing flat plate Y Pulsatile flow be tween par al lel sur faces Y Un steady flow of vis cous in com press ible fluid be tween two par al lel plates Y Dif fu sion of a

16

–Shanti Swarup218-18 Fluid Dynamics

vor tex fil a ment Y Low Reynolds num ber so lu tion Y So lu tion of the Navier-Stokes equa tion at low Reynolds num ber Y Slow flow past a sphere Y Flow past

a sphere (Aliter) Y Flow past a cir cu lar cyl in der Y Lam i nar Bound ary-Layer Flow Y Two di men sional bound ary layer equa tions for flow over a plane wall Y

Bound ary layer flow along a flat plate Y Bound ary layer thick ness Y Prop er ties of the bound ary layer equa tions Y Bound ary layer flow past a wedge Y Po ten tial flow

past a wedge Y Po ten tial flow around a cor ner Y Flow in a con ver gent chan nel Y Mo men tum in te gral equa tion for the bound ary layer Y Mo men tum and en ergy

in te gral equa tion for the bound ary layer (Aliter) Y Ap pli ca tion of the mo men tum In te gral equa tion to bound ary lay ers Von Karman's Pohlhausen method

Y Dis con tin u ous Mo tion Y Prop er ties of the free stream lines Y Flow in jets and cur rents Y Mo tion of two im ping ing jets Y Di rect im pact of four equal jets Y

Borda’s Mouth piece Y Jet of a liq uid through a slit Y Im pact of a stream on a lamina Y Lu bri ca tion The ory Y The gen er al ised Reynolds equa tion Y Flow be tween

par al lel walls Y The Real bear ing Y One-di men sional jour nal bear ings (In fi nitely long bear ing) Y In fi nitely short bear ing Y One-di men sional Thrust bear ing Y Step

bear ing Y Ap pen dix: Or thogo nal curvilinear co or di nate.

Y Elements of Set Theory Y Sets and sub sets Y Ba sic op er a tions on sets Y Re la tions Y Func tions Y Or der Y Some Prop er ties of real num bers Y The ax iom of

choice and its equiv a lents Y Car di nal ity and denumerability Y Dec i mal, Ter nary and bi nary rep re sen ta tion Y Car di nal Arith me tic Y Can tor’s ter nary set Y

Or der types and or di nal num bers Y Met ric Spaces Y Met ric Y Eu clid ean spaces Y Some im por tant in equal i ties Y Bounded and un bounded Met ric

spaces Y Spheres (or balls) Y Open Sets Y Closed Sets Y Neigh bour hoods Y Ac cu mu la tion points: Ad her ent points Y Clo sure, in te rior, ex te rior and

bound ary of a set Y Subspaces Y Prod uct spaces Y Dense and non-dense sets; sep a ra ble spaces Y Se quences and sub se quenc es in a met ric space Y

Cauchy se quences Y Com plete met ric spaces Y Com plete ness and con tract ing mappings Y Some com plete met ric spaces Y Com ple tion of a met ric space

Y Lin ear Spaces Y Gen eral prop er ties of lin ear spaces Y Lin ear subspaces Y Al ge bra of subspaces Y Lin ear com bi na tion of vec tors, Lin ear span of a set

Y Lin ear sum of two subspaces Y Di rect sum of spaces Y Quo tient space Y Lin ear de pend ence and lin ear in de pend ence of vec tors Y Hamel Ba sis of lin ear

space Y Di men sion of a lin ear space Y Isomorphism of lin ear spaces Y Lin ear trans for ma tions or Homo morph ism of lin ear spaces Y Prop er ties of lin ear

trans for ma tions Y Some par tic u lar lin ear trans for ma tions Y Range and null space of a lin ear trans for ma tion Y Rank and nul lity of a lin ear trans for ma tion Y

Lin ear trans for ma tions as vec tors Y Prod uct of lin ear trans for ma tions Y Al ge bra or lin ear al ge bra Y Poly no mi als Y Invertible or non-sin gu lar lin ear

trans for ma tions Y Lin ear functionals Y Lin ear functionals in fi nite-di men sional spaces Y Ex ten sion the o rems for lin ear trans for ma tions Y Re flex ivi ty Y

Pro jec tions Y Banach Spaces Y Normed lin ear spaces Y The clas si cal Banach space Lp Y Subspaces and Quo tient spaces of Banach spaces Y

Con tin u ous lin ear trans for ma tions Y Equiv a lent norms Y Riesz-lemma Y Con vex ity Y Lin ear functionals and the Hahn Banach the o rem Y The nat u ral

im bed ding of N into N∗ ∗; Re flex ivi ty Y The open map ping the o rem Y Pro jec tions on Banach spaces Y The Closed graph the o rem Y Uni form bounded

prin ci ple Y The con ju gate of an op er a tor Y Hilbert Spaces Y In ner prod uct spaces Y Hilbert spaces Y Some prop er ties of Hilbert spaces Y Or thogo nal

com ple ments Y Orthonormal sets Y The Con ju gate space H ∗Y The Adjoint of an op er a tor Y Self adjoint op er a tors Y Nor mal and Uni tary op er a tors Y

Per pen dic u lar Pro jec tions Y Fi nite Di men sional Spec tral The ory Y Eigen val ues and Eigen vec tors Y Ex is tence of eigen val ues Y Ma trix of a Lin ear

trans for ma tion Y The spec tral the o rem.

Y Elements of Set Theory Y Sets and Their Ba sic Op er a tions Y Re la tions Y Func tions Y R as an Or dered Field Y Con ver gence of Se quences in R Y Com plex

Num bers and their Geo met ri cal Rep re sen ta tion Y Fun da men tal Laws of Ad di tion and Mul ti pli ca tion Y Dif fer ence of Two Com plex Num bers Y Modulus

and Ar gu ment of Com plex Num bers Y The Geo met ri cal Rep re sen ta tion of Com plex Num bers Y Vec tor Rep re sen ta tion of Com plex Num bers Y The Point on

the Ar gand Plane Rep re sent ing the Sum, Dif fer ence, Prod uct and Divsion of Com plex Num bers Y Con ju gate Com plex Num bers Y Prop er ties of Moduli Y

Prop er ties of the Ar gu ments Y Riemann Sphere and the Point at In fin ity Y An a lyt i cal Func tions Y Com plex Dif fer en ti a tion Y Limit and Con ti nu ity Y

Differentiability Y The Nec es sary and Suf fi cient Con di tion for f z( ) to be An a lytic Y Method of Con struct ing a Reg u lar Func tion Y A Sim ple Method of Con struct ing an

An a lytic Func tion Y Po lar form of Cauhy-Riemann Equa tions Y Com plex Equa tion of a Straight Land Cir cle Y Poly no mi als Y Ra tio nal Func tions Y Mul ti ple Val ued

Func tions Y Power Se ries and El e men tary Func tions Y Se quences Y Se ries Y Se quences and Se ries of Func tions Y Power Se ries Y El e men tary

Tran scen den tal Func tions Y Conformal Mappings Y Map ping or Trans for ma tions Y Jacobian of a Trans for ma tion Y Conformal Map ping Y Suf fi cient Con di tions

for w f z= ( ) to Rep re sent Con for ma tion Map ping Y Nec es sary Con di tions for w f z= ( ) to Rep re sent Con for ma tion Map ping Y Su per fi cial Mag ni fi ca tion Y The Cir cle

Y In verse Points with Re spect to a Cir cle Y Some El e men tary Trans for ma tions Y Lin ear Trans for ma tions Y Bilinear Trans for ma tion Y Re sul tant of two Bilinear

Trans for ma tions Y Bilinear Trans for ma tion as the re sul tant of El e men tary Bilinear Trans for ma tion with Sim ple Geo met ric Prop er ties Y Bilinear Trans for ma tions as the

Re sul tant of an Even Num ber of In ver sions Y The Lin ear Group Y Equa tion of a Cir cle Through Three Given Points Y Cross-ra tio Y Preservance of Cross-ra tio un der

Bilinear Trans for ma tions Y To Find Bilinear Trans for ma tions which Trans form Three Dis tinct Points into Three Spec i fied Points Y Preservance of the Fam ily of Cir cles

and Straight Lines un der Bilinear Trans for ma tions Y Two Im por tant Fam i lies of Cir cles Y Fixed Points of a Bilinear Trans for ma tion Y Nor mal Form of a Bilinear

17

–J.N. Sharma & A.R. Vasishtha219-37 Functional Analysis

–J.N. Sharma220-52 Functions of a Complex Variable

Trans for ma tions Y El lip tic, Hy per bolic and Par a bolic Trans for ma tions Y Some Spe cial Bilinear Trans for ma tions Y More about Conformal Mappings Y

The Trans for ma tion w zn= Y The Trans for ma tion w z= 2 Y The In verse Trans for ma tion z w= Y The Ex po nen tial Trans for ma tion w ez= Y The

Log a rith mic Trans for ma tion w z= log Y The Trig o no met ri cal Trans for ma tions Y The Trans for ma tion w z=

tan2

4

π Y The Trans for ma tion

w z= +

1

2

1

2 Y Some Gen eral Tech niques of Conformal Map ping Y Com plex In te gra tion Y Com plex Line Integrals Y Re duc tion of Com plex Integrals

to Real Integrals Y Some Prop er ties of Com plex Integrals Y An Es ti ma tion of a Com plex In te gral Y Line Integrals as Func tions of Arcs Y Cauchy’s

Fun da men tal The o rem Y Sec ond Proof of Cauchy-Goursat The o rem Y A Third proof of Cauchy-Goursat The o rem Y Cauchy’s In te gral For mula Y Pois son’s

In te gral For mula of a Cir cle Y De riv a tive of an An a lytic Func tion Y Higher Or der De riv a tives Y Morera’s The o rem Y In def i nite Integrals or Prim i tives Y

Cauchy’s In equal ity Y Liouville’s The o rem Y Ex pan sion of An a lytic Func tions as Power Se ries : Tay lor and Laurent’s The o rems Y The Ze ros of an An a lytic

Func tion Y Dif fer ent Types of Sin gu lar i ties Y Some The o rems on Poles and Other Sin gu lar i ties Y The Point at In fin ity Y Char ac ter iza tion of Ro ta tional

Func tions Y Max i mum Modulus Prin ci ple Y The Ex cess of Num ber of Ze ros Over Num ber of Poles of a Meromorphic Func tion Y Rouche’s The o rem Y

Schwarz Lemma Y In verse Func tion The o rem Y Fun da men tal The o rem of Al ge bra Y An a lytic Con tin u a tion Y Power Se ries Method of An a lytic Con tin u a tion

Y Schwart’s Re flec tion Prin ci ple Y Cal cu lus of Res i dues Y Res i due at Sim ple Pole Y Res i due at a Pole of Or der Greater than Unity Y Res i due at In fin ity Y

Cauchy’s Res i due The o rem Y Eval u a tion of Def i nite Integrals Y In te gra tion Round the Unit Cir cle Y Eval u a tion of the Integrals − ∞∫x

f x dx( ) Y Jor dan’s

In equal ity Y Jor dan’s Lemma Y Eval u a tion of the Integrals of the form −∞∫x P x

Q x

( )

( ) sin mx dx etc. Y Case of the Poles on the Real Axis Y Integrals of may Val ues

Func tions Y Rect an gu lar and Other Con tours Y Ex pan sion of Meormorhpic Func tions Y Uni form Con ver gence and In fi nite Prod ucts Y Uni form

Con ver gence of a Se quence Y Gen eral Prin ci ple of Uni form Con ver gence Y Uni form Con ver gence of a Se ries Y Weierstrass’s M-test Y Hardy’s Test Y

Con ti nu ity of the Sum Func tion Y Term by Term In te gra tion Y Analyticity of the Sum Func tion of a Se ries, Term by Term Dif fer en ti a tion Y Hurwitz The o rem

Y Uni form Con ver gence of Power Se ries Y A note on Ab so lute and Uni form Con ver gence Y In fi nite Prod ucts Y Three Im por tant The o rems on In fi nite

Prod ucts Y The Ab so lute Con ver gence of In fi nite Prod ucts Y Uni form Con ver gence of In fi nite Prod ucts Y En tire Func tions Y Mittag Leffler’s The o rem Y

The Weierstrass Factorization Theorem Y Canonical Products Y The Jenson and Poisson -Jenson Formulas Y Growth, Order and Convergence Exponents

of Entire Functions Y Hadmard’s Factorization Theorem Y The Gamma Function.

Y Complex Numbers and their Geometrical Representation Y Complex Numbers Y Properties of the Addition Of Complex Numbers Y

Properties Of The Multiplication Of Complex Numbers Y Difference Of Two Complex numbers Y Division In C Y Modulus Of A Complex Number Y

Conjugate Of A Complex Number Y Modulus-argument Form Or Polar Standard Form Or Trigonometric Form Of A Complex Number Y The

Geometrical Representation Of Complex Numbers Y The Points On The Argand Plane Representing The Sum, Difference, Product And Division Of Two

Complex Numbers Y More Properties Of Moduli And Arguments Y Theorem: The Order Relations Greater Than Or Less Than Do Not Apply To Complex

Numbers Y Some Important Results About Complex Numbers Y Integral And Rational Powers Of A Complex Number Y Geometrical Applications Of

Complex Number Y Complex Equation Of A Straight Line In The Complex Plane Y Equation Of A Circle In The Complex Plane Y The Spherical

Representation Of Complex Numbers And Stereographic Projection Y Analytic Functions Y Curves In The Argand Plane Y Functions Of a Complex

Variable Y Neighbourhood Of A Point Y Limits And Continuity Y Differentiability Y Analytic, Holomorphic And Regular Functions Y The Necessary

And Sufficient Conditions For f z( ) To Be Analytic Y Polar Form Of Cauchy-Riemann Equations Y Derivative of w f z= ( ) In Polar Form Y Orthogonal

System Y Harmonic Function Y Methods Of Constructing A Regular function (Milne-Thomson’s Method) Y Multiple Valued Functions Y Conformal

Mappings Y Mappings Or Transformations Y Jacobian Of A Transformation Y Conformal Mapping Y Necessary Conditions for w f z= ( ) To Represent

A Conformal Mapping Y Sufficient Conditions For w f z= ( ) To Represent A Conformal Mapping Y Superficial Magnification Y The Circle Y Inverse

Points With Respect To A Circle Y Some Elementary Transformations Y Linear Transformation Y Billinear Or Linear Fractional Transformation Y

Critical Points Y Resultant Or Product Of Two Bilinear Transformations Y Bilinear Transformation As The Resultant Of Elementary Bilinear Y

Transformations With Simple Geometric Properties Y Bilinear Transformation As The Resultant Of An Even Number Of Inversions Y The Linear

Group Y Equation Of A Circle Through Three Given Points Y Cross Ration Y Preservance Of Cross-Ratio Under Bilinear Transformation Y To Find

The Bilinear Transformation Which Transforms Three Distinct Points Z Z Z1 2 3, , Of z-Plane Respectively Into Three Specified Points W W W1 2 3, , of

w-Plane Y Two Important Families Of Circles Y Preservance Of The Family Of Circles And Straight Lines Under Bilinear Transformations Y Fixed

Points Or Invariant Points Of A Bilinear Transformation Y Normal Form Of A Bilinear Transformation Y Elliptic, Hyperbolic And Parabolic

Transformations Y Some Spe cial Bilinear Trans for ma tions Y Tay lor’s Se ries Y More about Conformal Mappings (Some Spe cial

Trans for ma tions) Y The Trans for ma tion w Z n= (Where n Is A Pos i tive In te ger) Y The Trans for ma tion w z= 2 Y The In verse Trans for ma tion z W= Y

18

–A.R. Vasishtha, Vipin Vasishtha & A.K. Vasishtha221-19 Complex Analysis

The Ex po nen tial Trans for ma tion w ez= Y The Log a rith mic Trans for ma tion w z= log Y The Trans for ma tion w zz

= +

1

2

1 Y The Trans for ma tion

w a b za b

z= − + +

1

2( ) Y The Trans for ma tion w z= 1/ α Y The Trans for ma tion w

aZ=

tan2

4

π Y The Trans for ma tion w c z= sin Y The Trans for ma tion

w z= tan Y The Trans for ma tion w z= cos Y The Trans for ma tion w z= cosh Y Power Se ries and El e men tary Func tions Y Se quences Y In fi nite

Se ries Y Se quences And Se ries Of Func tions Y Prin ci pal Of Uni form Con ver gence Of A Se quence Y Cauchy’s Cri te rion For Se ries Y Power Se ries Y

El e men tary Tran scen den tal Func tions Y Ex po nen tial Func tion Y Ad di tion The o rem For Ex po nen tial Func tion ez Y Trig o no met ri cal Func tions sin z

and cos z Y Ad di tion The o rem for sin z and cos z Y Hy per bolic Func tions sinh z and cosh z Y Re la tion Be tween Trig o no met ric And Hy per bolic

Func tions Y Pe ri od ic ity Y Pe ri od ic ity Of sin z and cos z Y Pe ri od ic ity Of ez Y Log a rith mic Func tion (In verse Of Ex po nen tial Func tion) Y Branches Of

log w Y Ad di tion The o rem For log w Y Analyticity of log w Y The Gen eral Power za Y In verse Trig o no met ric Func tions Y Com plex In te gra tion Y

Def i ni tions Y Rectifiable Arcs Y Func tions of Bounded Vari a tion Y Com plex Intetgrals Y Eval u a tion Of some Integrals, ab-in itio (By def i ni tion) Y

Re duc tion Of Com plex Integrals To Real Integrals Y Some El e men tary Prop er ties Of Com plex Integrals Y An Up per Bound For A Com plex In te gral Y

Line Integrals As Func tions Of Arcs Y Cauchy’s Fun da men tal The o rem Y Cauchy Goursat The o rem (Sec ond proof) Y Cauchy Goursat The o rem (Third

proof) Y Cauchy’s In te gral For mula Y De riv a tive Of An An a lytic Func tion Y Higher Or der De riv a tives Of An An a lytic Func tion Y Pois son’s In te gral

For mula For A Cir cle Y Morera’s The o rem Y Cauchy’s In equal ity Y In def i nite Integrals Y In te gral Func tion Y Ex pan sion Of An a lytic Func tions As Power

Se ries Y The Ze ros Of An An a lytic Func tion Y Sin gu lar i ties Of An An a lytic Func tion Y Poly no mi als Y Char ac ter iza tion Of Poly no mi als Y Ra tio nal

Func tion Y Char ac ter iza tion Of Ra tio nal Func tions Y The o rems On Poles And Other Sin gu lar i ties Y Max i mum Modulus Prin ci ple Y Min i mum Modulus

Prin ci ple Y The Ex cess Of The Num ber Of Ze ros Over The Num ber Of Poles Of A Meromorphic Func tion (The Ar gu ment Prin ci ple, Max i mum

Modulus Prin ci ple The o rem) Y Rouche’s The o rem Y Schwarz Lemma Y In verse Func tion The o rem Y Fun da men tal The o rem Of Al ge bra Y An a lytic

Con tin u a tion Y Power Se ries Method Of An a lytic Con tin u a tion Y Schwarz’s Re flec tion Prin ci ple Y The Cal cu lus of Res i dues Y Res i due At A Pole Y

Com pu ta tion Of Res i due At A Fi nite Pole Y Res i due At In fin ity Y Com pu ta tion Of Res i due At In fin ity Y Cauchy’s Res i due The o rem Y Eval u a tion Of

Real Def i nite Integrals By Con tour In te gra tion Y In te gra tion Round The Unit Cir cle Y Eval u a tion Of The In te gral f x dx( )– ∞

∞

∫ Y Jor dan’s In equal ity Y

Jor dan’s Lemma Y Eval u a tion Of The Integrals Of The Forms P x

Q xmx dx

( )

( )sin ,

– ∞

∞

∫ P x

Q xmx dx m

( )

( )cos , ,

–

>∞

∞

∫ 0 Where (i) P x Q x( ), ( ) Are Poly no mi als,

(ii) deg ( ) deg ( ),Q x P x> (iii) Q x( )= 0 Has No Real Roots Y Poles Lie On the Real Axis Y Integrals Of many Val ued Func tions Y Rect an gu lar And Other

Con tours Y To Find The Res i due By Know ing The In te gral First Y Ex pan sion Of Meromorphic Func tions Y Uni form Con ver gence and In fi nite

Prod ucts Y Uni form Con ver gence Of A Se quence Y Gen eral Prin ci pal Of Uni form Con ver gence Y Uni form Con ver gence Of Se ries Y Weierstrass’s

M-test Y Hardy’s Test For Uni form Con ver gence Y Con ti nu ity Of The Sum Func tion Of A Se ries Y Term By Term In te gra tion Y Analyticity Of The Sum

Func tion Of A Se ries Term By Term Dif fer en ti a tion (Weierstrass’s The o rem) Y Hurwitz The o rem Y Uni form Con ver gence Of Power Se ries Y In fi nite

Prod ucts Y Gen eral Prin ci ple Of Con ver gence Of An In fi nite Prod uct Y Im por tant The o rems Y The Ab so lute Con ver gence Of In fi nite Prod ucts Y Uni form

Con ver gence Of An In fi nite Prod uct Y En tire Func tions Y En tire Or In te gral Func tion Def i ni tion Y Mittag–Leffier’s The o rem Y Weierstrass Factorization

The o rem Y Ca non i cal Prod ucts Y The Jensen And Pois son–Jensen For mu lae Y Growth, Or der And Con ver gence Ex po nents Of En tire Func tions Y

Con vex Func tion Y Hadmard’s Factorization The o rem Y The Gamma Function.

Y Kinematics Y Introduction Y Ideal or Perfect fluid, Real or Actual fluid Y Pressure Y Density Y Viscosity Y Description of fluid motion Y Lagrangian

method Y Eulerian method Y Relationship between the Lagrangian and Eulerian method Y Definitions: Steady and Unsteady flows Y Uniform and

Non-uniform flows Y One-dimensional and Three dimensional flows Y Axi-Symmetric flow Y Line of flow Y Streamline Y Pathline Y Stream surface Y

Stream tube Y Streak lines Y Velocity of a fluid particle at a Point Y Local, Convective and Material derivatives Y Equation of Continuity: Vector form Y

Cartesian coordinates Y Stream tube concept Y Spherical polar coordinates Y Cylindrical polar coordinates Y Lagrangian method Y Equivalence of the

two form of the equation of continuity Y Velocity potential, Irrotational flow Y Boundary surface Y Conservation of Momentum Y Euler’s equation

of motion along a streamline Y Equation of motion of an inviscid fluid Y Helmholtz equations Y Cauchy’s integral Y Bernoulli’s equation (Stream tube

method) Y Conservative field of force Y Integration of Euler’s equation Y Symmetrical forms of the equation of Continuity, Spherical symmetry Y

Cylindrical symmetry Y Applications of Bernoulli’s theorem: Flow from a tank through a small orifice Y Trajectory of a free jet Y Pitot tube Y Venturi tube Y

Weirs Y Impulsive motion of a fluid Y Energy equation Y Motion in Two Dimensions Y Stream function Y Physical interpretation of stream function Y

Complex Potential and Complex velocity Y Uniform flows Y Two dimensional source and sink Y Complex potential of a source Y Two-dimensional

doublet Y Complex Potential of a doublet Y Images in two dimensions Y Image of a source with regard to a plane Y Images of a doublet with regard to a

plane Y Circle theorem Y Image of source with regard to a circle Y Image of a doublet with regard to a circle Y Conformal representation Y Application to

Hydrodynamics Y Irrotational Motion Y General motion of a fluid element Y Motion of a fluid element (Cartesian coordinates method) Y Vorticity Y

19

–Shanti Swarup222-18 Hydrodynamics

Body forces, Y Surface forces Y Stress analysis at a point Y Strain analysis Y Flow and circulation Y Stoke’s theorem Y Kelvin’s circulation theorem Y

Connectivity Y Cyclic constants Y Irrotational motion in multiple connected space Y Acyclic and cyclic motion Y Green’s theorem Y Deduction from

Green’s theorem Y Mean value of the velocity potential over a spherical surface Y Motion regarded as due to Sources and Sinks Y Liquid extending to

infinity Y Kelvin’s minimum energy theorem Y Irrotational Motion in Two Dimensions Y General motion of a cylinder in two dimensions Y

Motion of a circular cylinder in a uniform stream Y Liquid streaming past a fixed circular cylinder Y Two-coaxial cylinders Y Circulation about a circular

cylinder Y Blasius’s theorem Y Streaming and circulation for a fixed circular cylinder Y Equation of motion of a circular cylinder Y Elliptic coordinates Y

Motion of an elliptic cylinder Y Streaming past a fixed elliptic cylinder Y Elliptic cylinder rotating in an infinite mass of liquid at rest at infinity Y Kinetic

energy of rotating elliptic cylinder Y Kinetic energy when the liquid contained in a rotating elliptic cylinder Y The aerofoil Y Joukowski transformation Y

Kutta-Joukowski’s theorem Y D’Alembert’s Paradox Y Schwarz-Christoffel theorem Y Transformation of a semi-infinite and infinite strips Y Vortex

Motion Y Voricity vector Y Vortex line Y Vortex tube Y Properties of the vortex Y Strength of the vortex Y Rectilinear vortices Y Velocity components Y

Centre of vortices Y A case of two vortex filaments Y Stream function when the strength of the vortex filaments are equal Y Vortex pair Y Vortex doublet Y

Vortex inside an infinite circular cylinder Y Vortex outside a circular cylinder Y Image of a vortex filament in a plane Y Four vortices Y An infinite single row

of parallel rectilinear vortices of the same strength Y Two infinite rows of parallel rectilinear vortices Y Karman’n vortex street Y Kirchhoff vortex theorem

Y Rectilinear vortex with circular section Y Rankine’s combined vortex Y Rectilinear vortices with elliptic section Y Vortex sheets Y Irrotational Motion

in Three Dimensions Y Butler’s sphere theorem Y Solution of the Laplace equation in three dimensions Y Motion of a sphere in an infinite mass of

liquid at rest at infinity Y Ideal fluid flow round a sphere Y Liquid streaming past a fixed sphere Y Concentric spheres Y Equation of motion of a sphere Y

Three dimensional source and sink Y Three dimensional doublet Y Image of a source with regard to a sphere Y Motion of liquid inside a rotating ellipsoidal

shell Y Motion of an ellipsoid in an infinite mass of liquid Y Stoke’s strem functions Y Values of Stoke’s stream function Y Solid of revolution moving along

their axes in an infinite mass of liquid Y Waves Y Wave motion Y Mathematical representation of wave motion Y Standing of Stationary waves Y Surface

waves, Tidal waves Y Surface waves Y Progressive waves on the surface of a canal Y Energy of progressive waves Y Progressive waves reduced to a steady

motion Y Standing waves Y Energy of stationary waves Y Waves at the common surface of two liquids Y Waves at an interface with upper surface free Y

Group velocity Y Rate of transmission of energy Y Long waves Y Energy of a long wave Y Long waves of the common surface of two liquids bounded

above and below by two fixed horizontal planes Y Viscous Fluid Flow Y State of stress at a point Y Symmetry of stress tensor Y Stress in a fluid at rest Y

Stress in a fluid in motion Y Transformation of Stress Components Y Stress quadric Y Orthogonality of Principal Stresses Y The rate of strain quadric Y

Transformation of the rate of strain Y Relation between stress and rate of strain Y Navier Stoke’s equation of motion Y Limitations of the Navier-Stoke’s

equation Y Equation of energy Y Dissipation of energy Y Vorticity and circulation in viscous fluids Y Dimensional analysis Y Relation between a set of

variables Y The II theorem Y Similitude Y Exact Solutions Y Laminar flow between parallel plates Y Hagen-Poiseuille flow through a circular pipe Y

Laminar flow between concentric rotating cylinders Y Steady motion of a viscous fluid due to a slowly rotating sphere Y Flow in convergent and divergent

channels Y Unsteady motion of a flat plate Y Pulsatile flow between parallel surfaces Y Diffusion of vortex filament.

Y Sequences and Limits Y Sequences of real numbers Y Bounded sequences Y Monotonic sequences Y Series Y Limits of a sequence Y Convergence

of a sequence Y Cauchy’s general Principle of Convergence of a Series Y Upper and Lower bounds and limits Y Convergence of monotonic sequences

Y Theorems on limits Y Cauchy’s first theorem on limits Y Cauchy’s second theorem on limits Y Theorems on limits of quotients Y Harder examples of

sequences Y Series of Non-Negative Terms Y Pringsheim’s Theorem Y Changing the order of terms in a Series Y Removal and insertion of

brackets Y Some general remarks Y Tests for convergence Y Comparsion test Y The auxiliary series ∑ 1

np Y Tests : vn−Method Y Cauchy’s root Y

D’Alembert’s Ratio tests Y Cauchy’s Condensation test Y The auxiliary series ∑ 1

n n p(log ) Y Comparision of ratios Y Raabe’s test Y Logarithmic test

Y DeMorgan and Bertand’s Test Y An alternative to Bertrand’s test Y Absolute and Conditional Convergence Y General Principle of

Convergence Y Alternating Series Y Absolute and non-absolute convergence Y Rearrangement of terms of an absolutely convergent series Y Insersion of

Parentheses Y Removal of brackets Y Rearrangement of terms of a conditionally convergent series Y Pringsheim’s Method Y Multiplication of infinite

series Y Merten’s theorem Y Abel’s theorem Y Failure of multiplication rule Y Tests for absolute convergence Y Abel’s inequality Y Dirichlet’s Test Y

Abel’s test Y Eulers constant Y The integral test for series of positive terms Y Convergence of Infinite Products Y Convergence and divergence of

infinite products Y General principle of convergence of infinite products Y Weierstrass’s inequalities Y Absolutes Y Convergence of infinite products Y

Dearrangement of factors Y Semi-convergent infinite products Y Complex factors Y Uniform Convergence of Sequences and Series of

Functions Y Uniforms convergence Y Cauchy’s general principle of uniform convergence Y Dini’s criterion of uniform convergence of a sequence of

continuous functions Y Tests for uniform convergence Y Mn− test Y Weierstrass’s M-test Y Abel’s test Y Dirichlet’s test Y Uniform convergence and

continuity Y Uniform convergence and integration Y Uniform Convergence and differentiation Y Uniform convergence of infinite products Y The

20

–J.N. Sharma & J.P. Chauhan223-17 (C) Infinite Series & Products

Weierstrass’s Approximation Theorem Y Arzela’s Theorem an Equicontinuous Families Y Power Series Y Definition and some elementary

theorems Y Radius of convergence Y Uniform convergence of power sereis Y Properties of power series Y Abel’s summability Y Abel’s theorem Y

Tauber’s theorem Y Expansions of Trigonometrical Functions as Infinite Series and Products Y Infinite Product for sin x and cos x Y

Convergence of Infinite Products for sin x and cos x Y Weierstrass’s Formula for sine as an infinite product Y Series for secz.

Y The Laplace Transform Y Integral Transform (Definition) Y Laplace Transform (Definition) Y Linearity Property of Laplace Transform Y

Piece-wise (or sectionally) continuous functions Y Existence of Laplace Transform Y Functions of Exponential order Y A function of Class A Y Table

(Laplace Transforms of some elementary functions) Y First translation or shifting theorem Y Second translation or shifting theorem Y Change of scale

property Y Laplace transform of the derivative of F ( )t Y Laplace transform of nth order derivative of F ( )t Y Initial value theorem Y Final value theorem Y

Laplace transform of Integrals Y Multiplication by t Y Multiplication by tn Y Division by t Y Evaluation of Integrals Y Periodic Functions Y Some Special

functions Y Table Laplace Transform Theorems Y The Inverse Laplace Transform Y Null Function (Definition) Y Lerch’s Theorem Y Linearity

Property Y Table of Inverse Laplace transforms Y First translation or shifting theorem Y Second translation or shifting theorem Y Change of scale

property Y Use of-Partial Fractions Y Inverse Laplace transform of derivatives Y Inverse Laplace transform of Integrals Y Multiplication by powers of P Y

Division by powers of P Y Convolution (Definition) Y Convolution theorem Y Heaviside’s expansion theorem or formula Y The Beta function Y The

Complex Inversion formula Y Table of Inverse Laplace Transform theorems Y Application of Laplace Transform to Solutions of Differential

Equations Y Solution of ordinary Differential Equations with constant coefficients Y Solution of ordinary Differential Equations with variable

coefficients Y Solution of Simultaneous Ordinary Differential Equations Y Solution of partial Differential Equations Y Applications to Electrical circuits

Y Applications to Mechanics Y Application of Laplace Transform to Integral Equations Y Definitions Y Applications of L.T. to Integral

Equations Y Applications of Laplace Transforms in Initial and Boundary Value Problems Y A Boundary Value Problem Y Heat

Conduction Equation Y Wave Equation Y Laplace Equation Y Applications to Beams Y Miscellaneous Exercises Y Fourier Transforms Y Dirichlet’s

Conditions Y Fourier Series Y Fourier Integral formula Y Fourier Transform or Complex Fourier Transform Y Inversion Theorem for Complex Fourier

transform Y Fourier sine transform Y Inversion formula for Fourier sine transform Y Fourier cosine transform Y Inversion formula for Fourier cosine

transform Y Linearity property of Fourier transform Y Change of Scale property Y Shifting Property Y Modulation Theorem Y Multiple Fourier

Transforms Y Convolution Y The Convolution or Falting theorem for Fourier transforms Y Parseval’s Identity for Fourier Transforms Y Relationship

between Fourier and Laplace Transforms Y Fourier transforms of the derivatives of a function Y Problems related to Integral equations Y Finite

Fourier Transforms Y Finite Fourier sine transforms Y Inversion formula for sine transform Y Finite Fourier cosine transform Y Inversion formula for

cosine transform Y Multiple finite Fourier transforms Y Operational properties of finite Fourier sine transforms Y Theorem I Y Theorem II Y Operational

properties of finite Fourier cosine transforms Y Theorem I Y Theorem II Y Combined properties of finite Fourier sine and cosine transforms Y

Convolution Y Applications of Fourier Transforms in Initial and Boundary Value Problems Y Application of infinite Fourier transforms Y

Choice of infinite sine or cosine transforms Y Application of finite Fourier transforms Y Finite Fourier transforms of partialanivatives Y Choice of finite

sine or cosine transforms Y Hankel Transforms Y Hankel Transform (Def.) Y Inversion formula for the Hankel transform Y Some Important Results

for Bessel Functions Y Linearity Property Y Hankel Transform of the Derivatives of a function Y Hankel transform of d f

dx x

df

dx

n

xf

2

2

21

2+ − Y Parseval’s

Theorem Y The Finite Hankel Transforms Y Finite Hankel Transform (Def.) Y Another form of Hankel Transform Y Hankel Transform of df

dx Y

Hankel Transform of d f

dx x

df

dx

2

2

1+ where p is the root of the equation J apn ( ) = 0 Y Hankel Transform of d f

dx x

df

dx

n

xf x

2

2

21

2+ − ( ) where p is the root of

the equation J apn ( ) = 0 Y Applications of Hankel Transform in Initial and Boundary Value Problems Y Mellin Transforms Y Mellin

Transform (Def.) Y The Mellin Inversion theorem Y Linearity property Y Some elementary properties of Mellin transform Y Mellin transform of

derivatives Y Mellin transform of integrals Y Convolution (or Falting) theorem for Mellin transform.

Y Vector Spaces Y Binary operation on a set Y Group Definition Y Field Y Vector space Y General properties of vector spaces Y Vector subspaces Y

Algebra of subspaces Y Linear combination of vectors, Linear span of a set Y Linear sum of two subspaces Y Linear combination of vectors, Linear span

of a set Y Linear sum of two subspaces Y Linear dependence and linear independence of vectors Y Basis of a vector space Y Finite dimensional vector

spaces Y Dimension of a finitely generated vector spaces Y Dimension of subspace Y Homomorphism of vector spaces of Linear Transformation Y

21

–A.R. Vasishtha & R.K. Gupta224-34 Integral Transforms (Transform Calculus)

–J.N. Sharma, A.R. Vasishtha & A.K. Vasishtha225-48 Linear Algebra (Finite Dimensional Vector Spaces)

Isomorphism of vector spaces Y Quotient space Y Direct sum of spaces Y Disjoint subspaces Y Complementary subspaces Y Co-ordinates Y Linear

Transformations Y Linear operator Y Range and null space of a linear transformation Y Rank and nullity of a linear transformation Y Linear

transformations as vectors Y Product of linear transformations Y Algebra of Linear algebra Y Invertible linear transformations Y Singular and

Non-singular Transformations Y Matrix Y Representation of Transformations by Matrices Y Similarity Y Determinant of Linear transformations on a

finite dimensional Y Trace of matrix Y Trace of a Linear Transformations on a Finite Dimensional Y Linear functionals Y Dual spaces Y Dual bases Y

Reflexivity Y Annihilators Y Invariant direct sum decompositions Y Reducibility Y Projections Y The adjoint or transpose a linear transformation Y

Sylvester’s law of nullity Y Characteristics values and characteristics vectors or proper vectors Y The Cayley-Hamilton theorem Y Diagonalizable

operators Y Minimal polynomial and minimal equation Y Inner Product Spaces Y Euclidean and unitary spaces Y Norm or length of a vector Y

Schwarz’s inequality Y Orthogonality Y Orthonormal set Y Complete Orthonormal Set Y Gram-Schmidt Orthogonalization Process Y Projection

theorem Y Linear functionals and adjoints Y Self-adjoint transformation Y Positive operators Y Non-negative operators Y Positive matrix Y Unitary

operators Y Normal operators Y Characteristics of Spectra Y Perpendicular projections Y Spectral theorem Y Bilinear Forms Y Bilinear forms as

vectors Y Matrix of a bilinear forms Y Symmetric bilinear forms Y Skew-Symmetric bilinear forms Y Groups preserving bilinear forms Y Index.

Y Introduction Y Applications of Difference Equations Y Study of Period of Analysis Y Verbal Learning Experiment Y Panel Surveys

Y Social Sciences Y Psychology Y Physiology Y Economic Dynamics Y The Calculus of Finite Differences Y Definitions Y Operators Y Linear

Operators Y Algebra of Operators Y First Differences (or Forward Differences) of y Y Forward Difference Operator ∆ Y Backward Difference Operator

∇ Y Central Difference Operator δ Y Second and Higher Order Differences Y Identity Operator Y Second and Higher Order backward Differences Y The

Transition or Shifting Operator E Y Properties of ∆ and E Y Equivalence of Operators Y Some Important Theorems of ∆ and E Y To Express any

Functions In terms of Leading Term and the Leading Differences of Difference Table Y Leibnitz’s Rule for Differences Y Factorial Function Y The

Difference of Factorial Function Y Method of Representing any Polynomial in Factorial Notation Y Indefinite Summation, The operator ∆−1 Y

Theorem. To prove ∆ = +−1y Y t x( ) Y To find ∆− 1 x n( ) Y To prove ∆ =−

+−1

1c

c

ct xx

x

k( ) Y To prove ∆ ∆−1[ ( ). ( )]V x U x = − ∆−U x V x EU x( ). ( ) [ ( ).1

∆V x( )] Y Analogies between the Difference and Differential Calculus Y Difference Equations Y To Write a Difference Equation as a Relation among

the Value of y Y Linear Difference Equation Y Order of a Linear Difference Equation Y Solution of a Difference Equation Y An Existence and

Uniqueness Theorem Y Solution of the Equation y Ay Ch h+ = +1 Y Theorem Y Sequences Y Definitions Y Solution as Sequences Y Theorem Y A

Probability Model for Learning Y Approximating a Differential Equation by a Difference Y Equation Y Linear Difference Equations with

Constant Coefficients Y Basic Definitions Y Theorem 1 Y Finite Linear Combination of Solutions Y Theorem 2 Y Theorem 3 Y Fundamental Set of

Solutions (or Linearly Independent Sols.) Y Theorem Y General Solution of the Homo. Diff. Equation of Order 2 Y General Solution of the Homo. Diff.

Equation of Order n Y Particular Solution of the Complete Diff. Equation Y Method of Undetermined Coefficients to Find the Particular Solution Y

Special operator Method to Find the Particular Solution Y Method of Variation of Parameters Y Solution of Simultaneous Difference Equations Y

Matrix Method for Solving a system of linear diff. equations Y Working Method for Solving a Second Order Homo. Difference Equation with Constants

Coefficients by Matrix Method Y Examples from the Social Sciences Y The First Order Equations. Cobweb Cycles and Generating Functions

Y Solution of y b y rh h h h+ − =1 Y Cobweb Phenomenon (or Cobweb Cycles) Y Generating Functions Y Some Special Generating Functions Y The

Linearity Property of the Generating Function Transformation Y Generating Function Method for Solving a Linear Difference Equation.

Y Basic Concepts Y Integral Equation Y Differentiation of a Function Under an Integral Sign Y Relation Between Differential and Integral Equations

Y Solution of Integral Equations Y Solution of Non-homogeneous Volterra's Integral Equation of Second kind by the Method of Successive

Substitution Y Solution of Non-homogeneous Volterra's Integral Equation of Second Kind by the Method of Successive Approximation Y

Determination of Some Resolvent Kernels Y Volterra Integral Equation of the First Kind Y Solution of the Fredholm Integral Equation by the Method of

Successive Substitutions Y Iterated Kernels Y Solution of the Fredholm Integral Equation by the Method of Successive Approximation Y Reciprocal

Functions Y Volterra's Solution of Fredholm's Equation Y Fredholm Integral Equations Y Fredholm First Theorem Y Prove that the solution Y

Every Zero of Fredholm Function D( )λ is a Pole of the Resolvent Kernel Y If a Real Kernel K x( , )ξ has a Complex Eigen Value λ µ +0 = iv, then it Also

Contains the Conjugate Eigen Value to λ µ0 = – iv Y Hadamard's Lemma Y Convergence Proof Y Fredholm Second Theorem Y Fredholm's Associated

Equation Y Characteristic Solutions Y Fredholm's Third Theorem Y Solution of the Homogeneous Integral Equation Y If D( )λ0 0= and D x( , ; ) /ξ ≡λ0 0,

22

–Dr. R.K. Gupta & D.C. Agarwal226-07 (B) Linear Difference Equations

–Shanti Swarup & Shiv Raj Singh227-25 Integral Equations

then for a Proper Choice of ξ φ = ξ0 0 0, ( ) ( , ; )x D x λ is a Continuous Solution of the Homogeneous Integral Equation Y Fundamental Functions Y Integral

Equations with Degenerate Kernels Y Hilbert Schmidt Theory Y All Iterated Kernels of a Symmetric Kernel are also Symmetric Y Orthogonality Y

Orthogonality of Fundamental Functions Y Eigen Values of Symmetric Kernel are Real Y Real Characteristic Constants Y Expansion of a Symmetric

Kernel in Eigen Functions Y Symmetric Kernels with a Finite Number of Eigen Values Y Symmetric Kernels with a Finite Eigen Values λ λ+ +m 1 m 2... Y

Sequence of the pth Power of the Eigen Values of the Iterated Kernel Y Fourier Series of Power of the Eigen Values of the Iterated Kernel Y

Hilbert-Schmidt Theorem Y The inequalities of Schwarz and Minkowski Y Hilbert's Theorem Y Complete Normalized Orthogonal System of

Characteristic Functions Y Coefficients of the Continuous Function f x( ) Y Complete Normalized Orthogonal System of Fundamental Functions Y

Bessel Inequality Y Riesz-Fischer Theorem Y Representation by a linear Combination of the Characteristic Functions Y Schmidt's Solution of the

Non-Homogeneous Integral Equation Y Solution of the Fredholm Integral Equation of first Kind Y Application of Integral Equations Y Initial Value

Problem Y Boundary Value Problems Y Deformation of a Rod Y Determination of Periodic Solutions Y Green's Function Y Construction of Green's

Function Y Particular Case Y Influence Function Y Construction of Green's Function when the Boundary Value Problem Contains a Parameter Y

Longitudinal Vibrations of a Rod Y Singular Integral Equations Y Abel Integral Equation Y Particular Case Y Weakly Singular Kernel Y Iteration of

the Singular Equation Y Fredholm Operator Y Equivalence of the Fredholm Integral Equation and the Iterated Equation Y Prove that the Eigen values

λ0 and λp of the Kernels k and kp are of the Same Rank Y If a Number µ is an Eigen value of the Iterated Kernel k xp ( , ),ξ then atleast one of the Distinct

Numbers Y Integral Equation in an Infinite Interval Y Cauchy Principal Integral Y Cauchy Type Integral Y Cauchy Integral on the Path of Integration Y

Plemelj Formulae Y The Plemelj –Privalov Theorem Y Poincare'-Bertrand Transformation Formula for Iterated Singular Integrals Y Application of the

Calculus of Residues Y Hilbert Kernel Y Solution of the Cauchy-type Singular Integral Equation Y Integral Transform Methods Y Laplace Transform Y

Properties of the Laplace Transform Y Application to Volterra Integral Equation Y Fourier Transform Y Application of Fourier Transform Y Mellin

Transform Y Modified Green's Function Y Properties of Modified Green's Function Y Reciprocity Relation Y Higher Dimensional Green's

Function Y Construction of Green's Function for Dirichlet Problems Y Conformal Mapping Y Table of Conformal Mapping Y Dirac Delta Function

Y Sampling Property of Dirac Delta Y Derivatives of the Delta Function Y Integration of Delta Function Y Unit Step Function Y Derivative of Unit Step

Function Y Some Properties of the Dirac Delta Function Y Three dimensional δ-Function Y Perturbation Theory Y Perturbation Orders Y First-Order

Non-singular Perturbation Theory Y Example of Second-order Singular Perturbation Theory Y First Order Perturbation to an Harmonic One

Dimensional Oscillator Y Time Independent Perturbation Theory Y Time Dependent Perturbation Theory Y Eigen Value and Eigen Function of a

Perturbed Self Adjoint Operator Y Green's Function For Initial Value Problem Y Working Method to Construct Green's Function of Initial Value

Problem Y Eigen Function Expansions and Green's Function Y Application of Integral Equations.

Y Mathematical Preliminaries Y Ma tri ces and De ter mi nants Y Op er a tions of Ma trix and Ad di tion and Mul ti pli ca tion Y Sub- ma trix Y Mi nor of

or der k Y De ter mi nant Y Im por tant prop er ties of de ter mi nants Y Mi nors Y Co factors Y Rank of a Ma trix Y Adjoint of a Ma trix Y Sin gu lar and

Non-sin gu lar Ma tri ces Y In verse of a ma trix Y Vec tors and Vec tor Spaces Y Def i ni tions Y Eu clid ean space Y Lin ear De pend ence and

In de pend ence of vec tors Y Lin ear Com bi na tion (L.C.) of vec tors Y Span ning Set Y Ba sis Set Y Some use ful The o rems of Lin ear Al ge bra Y

Si mul ta neous Lin ear Equa tions Y Lin ear Pro gram ming Prob lems Y For mu la tion and Graph i cal So lu tion Y Gen eral Lin ear Pro gram ming

prob lems Y Math e mat i cal for mu la tion of a L.P.P. Y Ba sic So lu tion (B.S.) Y An Im por tant The o rem Y Some Im por tant The o rems Y So lu tion of a lin ear

pro gram ming prob lem Y Geo met ri cal (or graph i cal) method for the so lu tion of a L.P.P Y Con vex Sets and their Properties Y Definitons Y Con vex

Com bi na tion Y Some Im por tant The o rems Y Sim plex Method Y Slack and Sur plus Vari ables Y Some Definitons and No ta tions Y Fun da men tal

The o rem of Lin ear Pro gram ming Y To ob tain B.F.S. from F.S. Y To De ter mine Im proved B.F.S. Y Un bounded So lu tions Y Optimality Con di tions Y

Al ter na tive Op ti mal So lu tions Y In con sis tency and Re dun dancy in Con straint Equa tions Y To de ter mine start ing B.F.S. Y Com pu ta tional pro ce dure

of the sim plex method for so lu tion of a max i mi za tion L.P.P Y Ar ti fi cial Vari ables Tech nique Y L.P.P with un re stricted vari ables Y Sol. of sys tem of

si mul ta neous lin ear eqs. by sim plex method Y To com pute the in verse of a ma trix for which one col umn is dif fer ent from that of a ma trix whose

in verse is known Y In verse of a ma trix by Sim plex Method Y Res o lu tion of De gen er acy Y Con di tions for the occurence of de gen er acy in a L.P.P Y

Method of Re solv ing De gen er acy Y Charne's Pertubation Method Y Se lec tion of the out go ing (de part ing) vec tor Y Com pu ta tional Pro ce dure Y

Gen er al ized Sim plex Method Y Re vised Sim plex Method Y Re vised Sim plex Method in stan dard Form I (For mu la tion of a L.P.P. in the form of

re vised sim plex) Y No ta tions for Stan dard form I Y To find the in verse of the Ba sis and the Ba sic so lu tion in stan dard Form I Y Com pu ta tional

Pro ce dure of the re vised Sim plex Method in Stan dard Form I Y Re vised Sim plex Method in Stan dard Form II Y No ta tions, Ba sis and its In verse in

Stan dard form II Y Com pu ta tional Pro ce dure of the Re vised Sim plex Method in Stan dard Form II Y Ad van tages and Dis ad van tages of Re vised

Sim plex Method over the orig i nal Sim plex Method Y Du al ity Y Symmmetric Dual Prob lem Y Unsymmetric Dual Prob lem Y The dual of a mixed

sys tem Y Stan dard form of the pri mal Y The o rem Dual of the dual of a given pri mal is the pri mal it self Y Fun da men tal Prop er ties of Dual Prob lems Y

Com ple men tary Slack ness The o rem Y Cor re spon dence be tween pri mal and dual Y To read the so lu tion of the dual from the fi nal Sim plex ta ble of

the pri mal and vice versa Y Dual Sim plex Al go rithm Y Der i va tion of the Dual Sim plex Al go rithm Y Ini tial so lu tion for Dual Sim plex Al go rithm Y

Ad van tage of Dual Sim plex Al go rithm Y Com pu ta tional Pro ce dure of the Dual Sim plex Al go rithm Y Pri mal Dual Al go rithm Y To de ter mine the

23

–R.K. Gupta228-29 Linear Programming

ini tial Dual So lu tion Y To de ter mine the Re stricted Pri mal Prob lem Y To find the En ter ing and leav ing vec tors Y Method to ob tain New Dual So lu tion Y

Test of optimality Y Com pu ta tional Pro ce dure of Pri mal-Dual Al go rithm Y Sen si tiv ity Anal y sis Y Vari a tion of a price vector c Y Vari a tion in the

re quire ment vec tor b Y Vari a tion in the com po nent aij of the co ef fi cient ma trix A Y Ad di tion of a new vari able to the prob lem Y Ad di tion of a new

con straint to the prob lem Y Para met ric Lin ear Pro gram ming Y Lin ear Vari a tion in c Y Lin ear Vari a tion in b Y In te ger Pro gram ming Y

Im por tance (or need) of I..P.P. Y So lu tion of I.P.P. Y Gomory's all I.P.P. method Y Con struc tion of Gomory's con straint Y Com pu ta tion pro ce dure

for the so lu tion of all I.P.P. by Gomory method Y The Branch and Bound Tech nique Y Branch and Bound Al go rithm Y As sign ment Prob lem Y

Im por tant the o rems Y Method for solv ing an as sign ment Prob lem (As sign ment al go rithm) Y Un bal anced As sign ment Prob lem Y Trans por ta tion Prob lem Y

Dif fer ence be tween a trans por ta tion and an As sign ment prob lem Y Few Im por tant definitons Y So lu tion of trans por ta tion prob lem Y To find an Ini tial fea si ble

so lu tion Y Optimality Test Y The o rem Y Com pu ta tional Pro ce dure of optimality test Y Trans por ta tion Al go rithm or Modi Method Y De gen er acy in Trans por ta tion

Prob lems Y Un bal anced Trans por ta tion Prob lem Y Game The ory (Com pet i tive Strat e gies) Y Com pet i tive Games Y Fi nite or In fi nite Games Y Zero

sum game Y Two per son zero sum (or Rect an gu lar) Games Y Pay-off ma trix Y Strat egy Y So lu tion of a Game Y Maximin and minimax cri te rion of

optimality Y So lu tion of a rect an gu lar game with sad dle point Y So lu tion of a rect an gu lar game in terms of mixed strat e gies Y Im por tant prop er ties

of op ti mal mixed strat e gies Y So lu tion of 2 2× games with out sad dle point Y Dom i nance prop erty Y Graph i cal method for so lu tion of 2 × n and 2 × m

games Y Al ge braic method for Ap prox i mate So lu tion Y Equiv a lence of the rect an gu lar ma trix game and lin ear Pro gram ming Y Fun da men tal

theorem of Game theory (Minimax Theorem) Y Solution of a rectangular game by simplex method Y Summary of methods for solving the

rectangular games Y Minimax and Maximin of a function of several variables Y Saddle points of a function of several variables Y Necessary and

sufficient condition for a function [ ]E x y, to possess a saddle point.

Y Elements of Set Theory Y Sets and Sub sets Y Ba sic op er a tions on sets Y Re la tions Y Func tions Y Or der Y Some prop er ties of real num bers Y

Car di nal ity and Denumerability Y Closed and Open Sets in R Y Neigh bour hoods Y Open sets Y Struc ture of open sets in R Y Closed Sets Y

Ac cu mu la tion points Y Closed sets and ac cu mu la tion points Y Clo sure Y In te rior, Ex te rior and Bound ary of a set Y Dense, non-dense, per fect and iso lated

sets Y Cov er ing The o rem : Com pact ness Y Struc ture of closed sets on the real line; Can tor's Ter nary set Y Met ric Spaces Y Met ric Y Eu clid ean spaces Y

Some im por tant in equal i ties Y Bounded and Un bounded Met ric spaces Y Open and Closed Sets in a Met ric Space Y Spheres (or Balls) Y Open sets

Y Equiv a lent Metrices Y Closed sets Y Neigh bour hoods Y Ac cu mu la tion Points : Ad her ent Points Y Clo sure Y In te rior, Ex te rior, Fron tier and bound ary of

a set Y Bases Y Subspaces of a Met ric space Y Prod uct Spaces Y Top o log i cal Spaces Y Com plete Met ric Spaces Y Se quences and sub se quenc es in a

Met ric space Y Cauchy se quences Y Com plete Met ric spaces Y Baire Cat e gory The o rem Y Com plete ness and con tract ing Mappings Y Some com plete

Met ric spaces Y Connectedness Y Sep a rated sets Y Con nected and Dis con nected sets Y Connectedness on the real line Y Com po nents Y To tally

dis con nected spaces Y Lo cally con nected spaces Y Com pact ness Y Hausdroff ax iom Y Com pact spaces Y Countably com pact spaces Y Se quen tially

com pact spaces Y Lindel of spaces Y Lo cally com pact spaces Y Prod uct of two com pact spaces Y Con ti nu ity and Homeomorphism Y Some

Pre lim i nary re marks Y Lim its and con ti nu ity Y Homeomorphism Y Con ti nu ity and connectedness Y Continutiy and com pact ness Y Pro jec tion Mappings

Y Connectedness of the prod uct of two spaces Y Uniform Continuity Y Extension Theorems.

Y Cantor and Dedekind's Theories of Real Numbers Y Need for ex tend ing the sys tem of ra tio nal num bers Y Dedekind's the ory of real num bers

Y Re la tions of Equal ity and or der in cuts Y Ad di tion of Cuts Y Mul ti pli ca tion of Cuts Y Re cip ro cal of Cut Y Ra tio nal and Ir ra tio nal Cuts Y Dense ness

of Cuts Y Sec tions of Real Num bers: Dedekind's The o rem Y Can tor's The ory of Real Num bers Y Equiv a lence of Can tor and Dedekind's The o ries Y

El e ments of Set The ory Y Sets and their ba sic op er a tions Y Re la tions Y Func tions Y Or der Y Denumerable Sets Y Dec i mal, Ter nary and Bi nary

Rep re sen ta tions Y Car di nal Arith me tic Y Real and Com plex Num ber Sys tems Y Bi nary op er a tions or Bi nary Com po si tions in a Set Y Field

Ax i oms Y R as a Com plete Or dered Field Y Ex tended Real Num bers Y Com plex Num bers Y C as a Field Y Dif fer ence and di vi sion of two Com plex

Num bers Y Modulus and ar gu ment of a Com plex Num ber Y The geo met ri cal rep re sen ta tion of a Com plex Num bers Y Con ju gate Com plex Num bers

Y Prop er ties of Moduli Y Prop er ties of the Ar gu ments Y Im pos si bil ity of Or der ing Num bers Y Riemann Sphere and the Point at In fin ity Y

Se quences of Real Num bers Y Se quences and sub se quenc es Y Con ver gent Se quences Y Di ver gent Se quences Y Bounded Se quences Y

Mono tone Se quences Y Op er a tions on Con ver gent Se quences Y Cauchy's The o rems on Lim its Y Use of Cauchy's The o rems on Lim its Y Harder

Ex am ples on Lim its Y Nested In ter val The o rem Y Cauchy Se quences Y Limit Su pe rior and Limit In fe rior Y Se ries of Real Num bers Y Se ries of

Non-neg a tive terms Y All terms greater than some fixed pos i tive num ber Y Com par i son Test Y The aux il iary se ries Σ 12n Y Cauchy's Root Test Y D'

Alembert 's Ra tio test Y Cauchy's Con den sa tion test Y The Aux il iary Se ries Σ 1

npn(log )

Y Com par i son of Ra tios Y Raabe's Test Y Log a rith mic Test Y D'

24

–J.N. Sharma229-24 Mathematical Analysis-I (Metric Spaces)

–J.N. Sharma & A.R. Vasishtha231-29 Mathematical Analysis-II

Mor gan and Bertrand's Test Y An al ter na tive to Bertrand's Test Y Sum mary of Tests Y Gen eral Se ries Y Gen eral Prin ci ple of Con ver gence Y Al ter nat ing

se ries Y Ab so lute and non-ab so lute Con ver gence Y Re-ar range ment of Terms of an Ab so lutely Con ver gent Se ries Y Insersion of pa ren the ses Y

Re moval of Brack ets Y Re-ar range ment of terms of a Con di tion al ity Con ver gent Se ries Y Pringsheim' s Method Y Mul ti pli ca tion of In fi nite Se ries Y

Merten's The o rem Y Abel's The o rem Y Fail ure of Mul ti pli ca tion Rule Y Test for Ab so lute Con ver gence Y Abel’s In equal ity Y Dirichlet’s Test Y Abel's

Test Y Eu ler's Con stant Y In te gral Test for Se ries of Pos i tive Terms Y Open and Closed Sets of Real Num bers Y Neigh bour hoods Y Open Sets Y

The Struc ture of Open Sets in R Y Closed Sets Y Ac cu mu la tion Points, Ad her ent Points Y Closed Sets and Ac cu mu la tion Points Y Clo sure Y In te rior,

Ex te rior and Bound ary of a Set Y Dense, non -dense, per fect and iso lated Sets Y Cov er ing The o rems, Com pact ness Y Struc ture of Closed Sets on the real

line : Can tor's Ter nary Set Y Lim its and Con ti nu ity Y Def i ni tions Y Lim its Y Al ge bra of Lim its Y Con ti nu ity Y The four func tional lim its at a point Y

Kinds of discontinuties Y Saltus Y The o rems on con ti nu ity Y The o rems on dis con tin u ous func tions Y Pointwise dis con tin u ous func tions Y Uni form

con ti nu ity Y Ab so lute Con ti nu ity Y Con ti nu ity of the in verse func tion Y Some more ex am ples on con ti nu ity Y Differentiability Y De riv a tive at a point

Y Pro gres sive and re gres sive de riv a tives Y Differentiability in [ ]a b, Y De riv a tive of a func tion Y Mean ing of sign of a de riv a tive Y Geo met ri cal mean ing

of a de riv a tive Y A nec es sary con di tion for the ex is tence of a fi nite de riv a tive Y Al ge bra of de riv a tives Y The chain rule Y De riv a tive of an in verse

func tion Y Darboux Prop erty Y Rolle's The o rem Y Lagrange's Mean Value the o rem Y De duc tions from mean value the o rem Y Cauchy's mean value

the o rem Y Taylor's de vel op ment of a func tion in a fi nite form with Lagrange's form of re minder Y Tay lor's the o rem with Cauchy's form of re minder Y

The Riemann In te gral Y Sets of mea sure zero Y Par ti tions and Riemann Sums Y Up per and Lower R-integrals Y R-integrability Y Riemann's

nec es sary and suf fi cient con di tions for R-integrability Y Some classes of integrable Cal cu lus Y Mean Value The o rems Y In te gra tion by Sub sti tu tion

Y In te gra tion by Parts Y The in te gral as a limit Y The Riemann Stieltjes In te gral Y A gen er al iza tion of the Riemann In te gral Y Par ti tions Y Lower

and up per Riemann-Stietljes sums Y The lower and up per Riemann- Stieltjes Integrals Y The Riemann Stieltjes In te gral Y The QS-integrals as a limit

of sums Y Some classes of RS- integrable func tions Y A re la tion be tween R-in te gral and RS-in te gral Y In te gra tion of vec tor val ued func tion Y Some

more the o rems on In te gra tion Y Con ver gence of Im proper Integrals Y Im proper In te gral Y In te gral with in fi nite lim its Y Test for the con ver gence

of f x dxa

( )∞

∫ Y Com par i son Test Y To test the con ver gence of dx

x na

∞

∫ where a > 0 Y The µ- test Y Abel's test Y Dirichlet's test Y Ab so lute con ver gence

Y Test for the con ver gence of im proper in te gral f x dxa

b

( )∫ Y Com par i son test Y To test the con ver gence of dx

x a na

b

( )−∫ Y The µ- test Y Abel's test Y

Dirichlet's test Y Op er a tion with im proper integrals Y Met ric Spaces Y Eu clid ean spaces Y Met ric spaces Y Neigh bour hoods, limit points, open and

closed sets Y Connectedness Y Com pact ness Y Com plete ness and Can tor's In ter sec tion The o rem Y Baire cat e gory The o rem Y Com plete ness and

Con tract ing Map ping Y Lim its and Con ti nu ity Y Func tions of Sev eral Vari ables Y Con ti nu ity of Func tions of two Vari ables Y Par tial De riv a tives

Y In ter change of the Or der of Dif fer en ti a tion Y Differentiability of two vari ables Y Com pos ite Func tions Y Lin ear trans for ma tions Y Ma tri ces Y

Dif fer en ti a tion Y Par tial Dif fer en ti a tion Y The In verse Func tion The o rem Y The Im plicit Func tion The o rem Y Jacobians Y Def i ni tion Y Case of

Func tions of Func tions Y Jacobian of Im plicit Func tions Y Nec es sary and suf fi cient con di tion for a Jacobian to van ish Y Convariants and Invariants

Y Beta and Gamma Func tions Y Prin ci pal and gen eral value of an im proper in te gral Y In fi nite lim its Y To find the value of f x

F xdx

( )

( )−∞

∞

∫ Y To find

the value of the in te gral x

xdx

m

n

2

20 1 +

∞

∫ Y To find the value of x

xdx

m

n

2

20 1 −

∞

∫ Y De duc tions from x

xdx

m

n

2

20 1 +

∞

∫ and x

xdx

m

n

2

20 1 −

∞

∫ Y Method of

dif fer en ti a tion un der the in te gra tion sign Y Method of in te gra tion un der the in te gra tion sign Y Eu ler's Integrals− Beta and Gamma Func tions Y

El e men tary prop er ties of Gamma Func tions Y Trans for ma tions of Gamma Func tions Y An other form of Beta Func tion Y Re la tion be tween Beta and

Gamma func tions Y Other trans for ma tions Y To prove that Γ Γ Γm m mm

+

=−

1

2 22

2 1

π( ) Y To find the value of Γ Γ Γ Γ1 2 3 1

n n n

n

n

−

.... Y

Dou ble and Tri ple Integrals, Dirichlet's The o rem Y Dou ble Integrals Y Sec ond or der el e ment in po lar curves Y Mul ti ple Integrals Y Area of the

sur face Y Dirichlet's The o rem Y Liouville’s Ex ten sion of Dirichlet's The o rem Y Change of or der of in te gra tion Y Trans for ma tion of mul ti ple Integrals Y

Trans for ma tion for im plicit func tions Y Trans for ma tion of el e ment of sur face Y Vol umes and Sur faces Y Po lar Co or di nates Y Ex am ples on Sur faces Y

Uni form Con ver gence of Se quences and Se ries of Func tions Y Uni form Con ver gence Y Cauchy's gen eral prin ci ple of uni form con ver gence Y

Dini's Cri te rion for uni form con ver gence of a se quence of con tin u ous func tions Y Tests for uni form con ver gence Y Uni form con ver gence and con ti nu ity

Y Uni form con ver gence and in te gra tion Y Uni form con ver gence and dif fer en ti a tion Y Ev ery where con tin u ous but no where dif fer en tia ble func tions Y

Weierstrass's non- dif fer en tia ble func tion Y The Weierstrass’s Ap prox i ma tion The o rem Y Stone-Weier Strass The o rem Y Arzela's The o rem on

Equicontinous Fam i lies Y Power Se ries Y Def i ni tion Y Cauchy's the o rems on lim its Y Ra dius of con ver gence Y Uni form con ver gence of power se ries

Y Prop er ties of Power Se ries Y Abel's Summability Y Dif fer en ti a tion and In te gra tion of Vec tors Y Vec tor func tion Y Limit and con ti nu ity of a

vec tor func tion Y De riv a tive of a vec tor func tion with re spect to a sca lar Y Curves in space Y Ve loc ity and ac cel er a tion Y In te gra tion of vec tor func tions

Y Gra di ent, Di ver gence and Curl Y Par tial de riv a tives of vec tors Y The vec tor dif fer en tial op er a tor Del, ∇ Y Gra di ent of a sca lar field Y Level

Sur faces Y Di rec tional de riv a tive of a sca lar point func tion Y Tan gent plane and nor mal to a level sur face Y Di ver gence of a vec tor point func tion Y Curl

of a vec tor point func tion Y The Laplacian op er a tor ∇2 Y Im por tant vec tor iden ti ties Y Invariance Y Green's, Gauss's and Stoke's Theorems Y

Some preliminary concepts Y Line integrals Y Circulation Y Surface integrals Y Volume integrals Y Green's theorem in the plane Y The divergence

theorem of Gauss Y Green's theorem Y Stoke's theorem Y Line integrals independent of path Y Physical interpretation of div. and curl.

25

Y Basic Concepts of Set and Basic Set Operations Y Concepts of Set Y Notation Y Set of Sets Y Subset Y Super set Y Equality of Sets Y Proper

Subset Y Finite Set Y Infinite Set Y Null Set Y Power Set Y Universal Set Y Indexed Set and Index Set Y Heriditary Property Y Pairwise Disjoint Y Set

Operations Y Union Y Intersection Y Disjoint Sets Y Difference of Sets Y Complement of a Set Y Symmetric Difference of Sets Y Distributive Law Y

De-Morgan's Law Y Ordered Pair Y Equality of Ordered Pairs Y Product of Sets Y Product Sets in General Y Functions and Sequences Y Function Y

Onto and Into Mappings Y One-one and Many-one Mappings Y Real Valued Map Y Set Function Y Real Valued Set Function Y

Extended Real Valued Set Function Y Sequence Y Convergent Sequence Y Bounded Sequence Y Metric Space Y Monotonic (Increasing & Decreasing)

Y Axiom of Choice Y Choice Function Y Axiom of Choice Y Zermelo's Postulate Y Chain Y Finite Character Y Hausdorff Maximal Principle

Y Tukey's Lemma Y Zorn's Lemma Y Well-ordering Theorem Y Kuratowski Lemma Y Hausdorff Maximal Principle Y Ordered Sets Y Partially Ordered

Set Y Comparable and Uncomparable Y The Totally Ordered Set Y Subset of an Ordered Set Y Order Complete Y Theorem Y Initial Segment

Y First and Last Element Y Well Ordered Set Y Principle of Transfinite Induction Y Ordinal Number Y Ordinals Y Bounded Sets, Derived Sets,

Open Sets and Closed Sets on the Real Line Y Real Line Y Open and Closed Intervals Y Open Set Y Continuity Y Bounded Linear Set Y Limit

Point and Derived Set Y Condensation Point Y Closed Set Y Open Set Y Countability of Sets Y Cardinally Equivalent Y Cardinal Numbers

Y Sum of Cardinal Numbers Y Product of Cardinal Numbers Y Infinite Set Y Finite Set Y A Set A is Called a Denumerable Set if A ~N

Y Countable Sets Y Uncountable Set Y Power of Continum Y Power Set Y Cardinal Number of Isolated Set Y Continum Hypothesis Y Schroeder-

Bernstein Y Measure and Outer Measure Y Boolean Ring (or Ring of Sets) Y σ-Ring of Sets Y Algebra of Sets (or Boolean Algebra or Field)

Y σ-Algebra of Sets (or σ - Field) Y Semi-ring Y Monotone Class Y Complete Lattice Y Set Function Y Extended Real Valued Set Function Y

Finite Function Y Postulates for an Ideal Measure Function Y Measurable Space Y Measure Function Y Heriditary Property Y Caratheodory's Postulate for

Outer Measure Y Measurable Set Y Elementary Set Y Lebesgue Measure of a Set Y Measure of an Open Set and a Closed Set Y Measure of an Open

Interval Y Measure of a Closed Interval Y Measure of Rectangle Y Measure of a Parallelopiped Y Exterior and Interior Measure (Lebesgue Measurable

Set) Y Almost Everywhere Y Cantor's Tennary Set Y Limiting Sets Y Covering in the Sense of Vitali Y Set of the Type Fσ Y Set of the Type Gδ Y Borel Set

Y Borel Measurable Y Measurable Functions Y Almost Everywhere Y Equivalent Functions Y Characteristic Function Y

Simple Function Y Step Function Y Limit Superior and Limit Inferior Y Lebesgue Measurable Functions Y Borel Measurability Functions Y Little

Wood's Three Principles Y The Lebesgue Integral of a Function Y General Def. of Lebesgue Integral of a Function Y To Define Lebesgue Integral Y

To Define Lebesgue Integral of an Unbounded Function f(x) Defined Over a Measurable Set E, (Integral of a Non-negative Function) Y Theorems on

Convergence of Sequences of Measurable Functions Y Convergence in Mean Y Convergence in Measure Y Pointwise Convergence Y

Convergence Almost Everywhere Y Uniform Convergence Y Absolute Continuous Functions, Indefinite Integral and Differentiation Y

Continuous Function Y Absolute Continuous Function Y Indefinite Integral Y Differentiable Y Monotonic Functions Y Function of Bounded Variation Y

Lipschitz Condition Y Lebesgue Point Y Fundamental Theorem of Integral Calculus Y Problems Related to Functions of Bounded Variation Y Variation

Function Y Problems Related to Absolute Continuous Functions Y Problems Related to Indefinite Integral Y Problems Related to Lebesgue Point of a Function

Y Some Miscellaneous Problems on Absolute Continuous Functions Y Lp -Spaces Y Conjugate Number Y LP-Space Y Norm of an element of Lp -Space

Y Distance Function or Metric Y Convergent Sequence Y Cauchy Sequence Y Completeness of Lp-Space Y Approximation by Continuos Functions Y

Convex Function Y Examples on Convex Functions Y Definition Y Jensen Inequality Y Fur ther The o rems on Lebesgue In te gra tion Y Integration

by Parts Y Introduction : Stieltjes Integral Y Cumulative Distribution Function Y Lebesgue-Stieltjes Integral Y The Weierstrass Ap prox i ma tion

The o rems and Semi-Con tin u ous Func tions Y Polynomial Function Y Bernstein Polynomial Y Article Y Theory and Problems Related to

Semi-continuous Functions Y Signed Mea sure Y Positive and Negative Sets Y Distinction between a Set of Measure Zero and a Null Set Y Singular

Measures Y Jordan Decomposition Y Absolutely Continuous Measure Function Y Prod uct Mea sure Y Rectangle Y Section Y X-Section Y Product

Measure Y Double Integral Y Fou rier Se ries Y Periodic Function Y Finite Discontinuity Y Even and Odd Functions Y Orthogonal Functions Y

Trigonometric Polynomial Y Fourier Series Y The L2-Theory of Fourier Series Y Summation of Series by Arithmetic Means Y Summability of Fourier

Series Y Banach Space Y Vector Space or (Linear Space) Y Subspace Y Linear Sum or Sum Y Direct Sum Y Quotient Space Y Basis or Hamel Basis Y

Infinite Dimension Y Linear Transformation Y A Linear Map Y Partially Ordered Set Y Minimal and Maximal Elements Y Supremum : Infimum Y Zorn's

Lemma Y Normed Linear Space Y Banach Space Y Continuity Y Bounded Map Y Isometric Isomorphism Y Graph of a Function Y Closed Linear Map Y

Some Elementary Definitions Y Functional Conjugate Space Y Convergence Y Projection Y Algebra Y Banach Algebra Y Conjugate of an Operator Y

Weak and Strong Convergence Y Uniform Convergence Implies Strong Convergence Y Dense Subset Y Separable Space Y Definition : Summable Y

Hilbert Space Y Inner Product Space Y Hilbert Space Y Adjoint Operator (Conjugate Operator) Y Different Types of Operators Y Conjugate Space H* Y

Continuity Y Orthogonality Y Orthogonal Set Y Complete Orthonormal Set Y Perpendicular Projection Y Parallelogram Law Y Invariant Y Uniformly

Convex Y Problems Related to Operators Y Projection Y Some Numerical Problems on Orthogonalization Y Fi nite Di men sional Spec tral The ory

Y Spectral Solution Y Invariant Y Reducibility Y Banach Al ge bra Y Algebra Y Division Algebra Y Banach Algebra Y Regular Elements Y Singular

Element Y Spectrum Radius Y Spectral Radius Y Topological Divisors of Zero.

26

–K.P. Gupta & Ashutosh Shanker Gupta232-26Measure & Integration (Measure Theory & Functional Analysis)

Y Cantor and Dedekind's Theories of Real Numbers Y Need for extending the system of rational numbers Y Dedekind’s theory of real numbers Y

Relations of Equality and order in cuts Y Addition of Cuts Y Multiplication of Cuts Y Reciprocal of a Cut Y Rational and Irrational Cuts Y Denseness of

Cuts Y Sections of Real Numbers: Dedekind’s Theorem Y Cantor’s Theory of Real Numbers Y Equivalence of Cantor and Dedekind’s Theories Y

Elements of Set Theory Y Sets and their basic operations Y Relations Y Functions Y Order Y Denumerable Sets Y Decimal, Ternary and Binary

Representations Y Cardinal Arithmetic Y Real and Complex Number Systems Y Binary operations or Binary Composition in a Set Y Field Axioms Y

R as a Complete Ordered Field Y Extended Real Numbers Y Complex Numbers Y C as a Field Y Difference and division of two Complex Numbers Y

Modulus and argument of a Complex Number Y The geometrical representation of a Complex Numbers Y Conjugate Complex Numbers Y Properties of

Moduli Y Properties of Arguments Y Impossibility of Ordering Complex Numbers Y Riemann Sphere and the Point at Infinity Y Sequences of Real

Numbers Y Sequences and subsequences Y Convergent Sequences Y Divergent Sequences Y Bounded Sequences Y Monotone Sequences Y

Operations on Convergent Sequences Y Cauchy’s Theorems on Limits Y Use of Cauchy’s Theorems on Limits Y Harder Examples on Limits Y Nested

Interval Theorem Y Cauchy Sequences Y Limit Superior and Limit Inferior Y Series of Real Numbers Y Series of Non-negative terms Y All terms

greater than some fixed positive number Y Comparison Tests Y The Auxiliary Series Σ 1

np Y Cauchy’s Root Test Y D’ Alembert’s Ratio Test Y Cauchy’s

Condensation Test Y The Auxiliary Series Σ 1

n n p(log ) Y Comparison of Ratios Y Raabe’s Test Y Logarithmic Test Y D’ Morgan and Bertrand’s Test Y An

alternative to Bertrand’s Test Y Summary of Test Y General Series Y General Principle of Convergence Y Alternating series Y Absolute and non-absolute

Convergence Y Re-arrangement of Terms of an Absolutely Convergent Series Y Insersion of parentheses Y Removal of Brackets Y Re-arrangement of

terms of a Conditionally Convergent Series Y Pringsheim’s Method Y Multiplication of Infinite Series Y Merten’s Theorem Y Abel’s Theorem Y Failure of

Multiplication Rule Y Test for Absolute Convergence Y Abel’s Inequality Y Dirichlet’s Test Y Abel’sTest Y Euler’s Constant Y The Integral Test for Series

of Positive Terms Y Open and Closed Sets of Real numbers Y Neighbourhoods Y Open Sets Y The Structure of Open Sets in R Y Closed Sets Y

Accumulation Points, Adherent Points Y Closed Sets and Accumulation Points Y Closure Y Interior, Exterior and Boundary of a Set Y Dense, non-dense,

perfect and isolated Sets Y Covering Theorems, Compactness Y Structure of Closed Sets on the real line: Cantor’s Ternary Set Y Limits and Continuity

Y Definitions Y Limits Y Algebra of Limits Y Continuity Y The four functional limits at a point Y Kinds of discontinuities Y Saltus Y Theorems on

continuity Y Theorems on discontinuous functions Y Pointwise, discontinuous functions Y Uniform continuity Y Absolute continuity Y Continuity of the

inverse function Y Some more examples on continuity Y Differentiability Y Derivative at a point Y Progressive and regressive derivatives Y

Differentiability in [ , ]a b Y Derivative of a function Y Meaning of the sign of derivative Y Geometrical meaning of a derivative Y A necessary condition for

the existence of a finite derivative Y Algebra of derivatives Y The chain rule Y Derivative of an inverse function Y Darboux Property Y Rolle’s Theorem Y

Lagrange’s Mean Value theorem Y Deductions from mean value theorem Y Cauchy’s mean value theorem Y Taylor’s development of a function in a

finite form with Lagrange’s form of remainder Y Taylor’s theorem with Cauchy’s form of remainder Y The Riemann Integral Y Sets of measure zero Y

Partitions and Riemann Sums Y Upper and Lower R-integrals Y R-integrability Y Riemann’s necessary and sufficient conditions for R-integrability Y Some

classes of integrable functions Y Algebra of integrable functions Y Fundamental theórem of Integral Calculus Y Mean Value Theorems Y Integration by

Substitution Y Integration by Parts Y The integral as a limit Y The Riemann Stieltjes Integral Y A generalization of the Riemann Integral Y Partitions Y

Lower and upper Riemann-Stieltjes sums Y The lower and upper Riemann-Stieltjes Integrals YThe Riemann Stieltjes Integral Y The RS-integral as a limit

of sums Y Some classes of RS-integrable functions Y Algebra of RS-integrable functions Y A relation between R-integral and RS-integral Y Integration of

vector valued functions Y Function of bounded variation Y Some more theorems on Integration Y Convergence of Improper Integrals Y Improper

Integrals Y Integral with infinite limits Y Tests for the convergence of a

f x dx∞

∫ ( ) Y Comparison Test Y To test the convergence of a

n

dx

x

∞

∫ where a > 0 Y The

µ-test Y Abel’s test Y Dirichlet’s test Y Absolute convergence Y Test for the convergence of improper integral a

f x dx∞

∫ ( ) Y Comparison test Y To test the

convergence of a

b

n

dx

x a∫ −( ) Y The µ-test Y Abel’s test Y Dirichlet’s test Y Operations with Improper integrals Y Metric Spaces Y Euclidean spaces Y Metric

spaces Y Neighbourhoods, limit points, open and closed sets Y Connectedness Y Compactness Y Completeness and Cantor’s Intersection Theorem Y

Baire category Theorem Y Completeness and Contracting mappings Y Limits and Continuity Y Functions of Several Variables Y Continuity of

Functions of two Variables Y Partial Derivatives Y Interchange of the Order of Differentiation Y Differentiability of two variables Y Composite Functions Y

Linear transformations Y Matrices Y Differentiation Y Partial Differentiation Y The lnverse Function Theorem Y The Implicit Function Theorem Y

Jacobians Y Definition Y Case of Functions of Functions Y Jacobian of Implicit Functions Y Necessary and sufficient condition for a Jacobian to vanish Y

Convariant and Invariants Y Beta and Gamma Functions Y Principal and general values of an improper integral Y Infinite limits Y To find the value

of −∞

∞

∫ f x

F xdx

( )

( ) Y To find the value of the integral

0

2

21

∞

∫ +x

xdx

m

n Y To find the value of

0

2

21

∞

∫ −x

xdx

m

n Y Deductions from

0

2

21

∞

∫ +x

xdx

m

n and

27

–J.N. Sharma & A.R. Vasishtha233-43 Real Analysis (General)

0

2

21

∞

∫ −x

xdx

m

n Y Method of differentiation under the integration sign Y Method of integration under the sign of integration Y Euler’s Integrals–Beta and

Gamma Functions Y Elementary properties of Gamma Function Y Transformation of Gamma Function Y Another form of Beta Function Y Relation

between Beta and Gamma functions Y Other transformations Y To prove that Γ Γ Γm m mm

+

=−

1

2 22

2 1

π( ) Y To find the value of

Γ Γ Γ Γ1 2 3 1

n n n

n

n

−

... Y Double and Triple Integrals, Dirichlet's Theorem Y Double Integrals Y Second order element in polar curves Y

Multiple Integrals Y Area of the surface Y Dirichlet’s Theorem Y Liouville’s Extension of Dirichlet’s Theorem Y Change of order of integration Y

Transformation of Multiple Integrals Y Transformation for implicit functions Y Transformation of element of a surface Y Volumes and Surfaces Y Polar

Coordinates Y Examples on Surfaces Y Uniform Convergence of Sequences and Series of Functions Y Some definitions Y Uniform Convergence

Y Cauchy’s general principle of uniform convergence Y Dini’s Criterion for uniform convergence of a sequence of continuous functions Y Tests for uniform

convergence Y Uniform convergence and continuity Y Uniform convergence and integration Y Uniform convergence and differentiation Y Everywhere

continuous but nowhere differentiable functions Y Weierstrass’s non-differentiable function Y The Weierstrass’s Approximation Theorem Y The

Stone-Weierstrass Theorem Y Arzela’s Theorem on Equicontinuous Families Y Power Series Y Definition Y Cauchy’s theorems on limits Y Radius of

convergence Y Uniform convergence of power series Y Properties of Power Series Y Abel’s Summability.

Y Differentiation and Integration of Vectors Y Vector function Y Limits and continuity of a vector function Y Derivative of a vector function with

respect to a scalar Y Curves in space Y Velocity and acceleration Y Integration of vector functions Y Gradient, Divergence and Curl Y Partial

derivatives of vectors Y The vector differential operator Del. V Y Gradient of a scalar field Y Level Surfaces Y Directional derivative of a scalar point

function Y Tangent plane and normal to a level surface Y Divergence of a vector point function Y Curl of a vector point function Y The Laplacian operator

∇2 Y Important vector identities Y Invariance Y Green's, Gauss’s and Stoke's Theorems Y Some preliminary concepts Y Line Integrals Y Circulation

Y Surface integrals Y Volume integrals Y Green’s theorem in the plane Y The divergence theorem of Gauss Y Geeen’s theorem Y Stoke’s theorem Y Line

integrals independent of path Y Physical interpretation of divergence and curl.

Y Some Basic Set Theoretic Concepts Y Math e mat i cal logic Y Tau tol o gies Y Set Y Sub sets of a set Y Un ion of Sets Y In ter sec tion of sets Y Car te sian

prod uct of two sets Y Func tions or mappings Y Bi nary op er a tion Y Re la tions Y Equiv a lence re la tions Y Equiv a lence classes Y Par ti tions Y Par tial or der

re la tions Y Groups Y Bi nary op er a tion on a set Y Al ge braic struc ture Y Group, Def i ni tion Y Abelian Group Y Fi nite and in fi nite groups Y Or der of a fi nite

group Y Gen eral prop er ties of groups Y Def i ni tion of a group based upon left axions Y Com po si tion ta bles for fi nite sets Y Ad di tion modulo m

Mul ti pli ca tion modulo p Y Res i due classes of the set of in te gers Y An al ter na tive set of pos tu lates for a group Y Per mu ta tions Y Group of per mu ta tions Y

Cy clic per mu ta tions Y Even and odd per mu ta tions Y In te gral power of an el e ment of a group Y Or der of an el e ment of a group Y Isomorphism of groups

Y The re la tion of isomorphism in the set of all groups Y Com plexes and subgroups of a group Y In ter sec tion of sub groups Y Cosets Y Re la tion of

con gru ence modulo a subgroup H in a group a G Y Lagrange’s the o rem Y Eu ler’s the o rem Y Fermat’s the o rem Y Or der of the prod uct of two sub groups of

fi nite or der Y Cayley’s the o rem Y Cyclic groups Y Subgroup generated by a subset of a group Y Generating system of a group Y Groups (Continued) Y

Normal subgroups Y Conjugate Y elements Y Normalizer of an element of a group Y Class equation of a group Y Centre of a group Y Conjugate

subgroups Y Invariant subgroups Y Quotient Groups Y Homomorphism of Groups Y Kernel of a homomorphism Y Fundamental theorem on

homomorphism of groups Y Automorphisms of a group Y Inner automorphisms Y More results on group homomorphism Y Maximal subgroups Y

Composition series of a group and the Jordan-Holder theorem Y Solvable groups Y Commutator subgroup of a group Y Direct products Y External direct

products Y Internal direct products Y Cauchy's theorem on abelian groups Y Cauchy’s theorem Y Sylow’s theorem Y Rings Y Elementary properties of a

ring Y Rings with or without zero divisors Y Internal domain Y Field Y Division ring or skew field Y Isomorphism of rings Y Subrings Y Subfields Y

Characteristic of a ring Y Ordered internal domains Y Imbedding of a ring into another ring Y The field of quotients Y Ideals Y Principal ideal Y Principal

ideal ring Y Divisibilty in an integral domain Y Units Y Associates Y Prime elements Y Greatest common divisior Y Polynomial rings Y Polynomials over an

internal domain Y Division algorithm for polynomials over a field Y Euclidean algorithm for polynomials over a field Y Unique factorization domain Y

Unique factorization theorem for polynomials over a field Y Remainder theorem Y Prime fields Y Rings of endomorphisms of an Abelian group Y Rings

(Continued) Y Quotient rings or residue class rings Y Homomorphism of rings Y Kernel of a ring homomorphism Y Maximal ideals Y Prime ideals Y

Euclidean rings or Euclidean domains Y Polynomial rings over unique factorization domains Y Vector Spaces Y Definition Y General properties of

28

–J.N. Sharma & A.R. Vasishtha234-38 Vector Calculus

–A.R. Vasishtha & A.K. Vasishtha235-68 Modern Algebra (Abstract Algebra)

vector spaces Y Vector subspaces Y Linear combination of factors Y Linear span Y Linear sum of two subspaces Y Linear dependence and linear

independence of vectors Y Basis of a vector space Y Finite dimensional vector spaces Y Dimension of a finitely generated vector space Y Homomorphism

of vector spaces or linear transformations Y Isomorphism of vector spaces Y Quotient space Y Direct sum of space Y Complementary subspaces Y

Coordinates Y Vector Spaces (Continued) Y Linear transformations as vectors Y Dual space Y Dual basis Y Reflexivity Y Annihilators Y Modules Y

Definition Y Submodules Y Direct sum of submodules Y Homomorphism of modules or linear transformations Y Quotient modules Y Cyclic modules Y

Straight-Edge Y Finitely Generated Modules Y Extension Fields and Galois Theory Y Field extensions Y Finite field extension Y Field adjunctions Y

Simple field extension Y Algebraic field extension Y Transcendental element Y Roots of polynomials Y Multiple root Y Splitting field of decomposition field

Y Uniqueness of the splitting field Y Derivative of a polynomial Y Separable extension Y Perfect field Y The elements of Galois theory Y Fixed field Y

Normal extension Y Galois group Y Fundamental theorem of Galois theory Y Construction with Straight-Edge rules and compass Y Solvability by radicals

Y Finite fields Y Number Theory Y Two basic binary operations on the set of integers Y Order relation Y Well ordering principle Y Absolute value or

modulus of an integer Y Divisibility in the set of integers Y The Division Algorithm Y Greatest Common Divisor Y Euclidean Algorithm Y Relatively prime

integers Y Least command multiple Y Primes and composite integers Y Euclid’s lemma Y The Fundamental theorem of Arithmetic Y The number of

divisors of a positive integer Y Mersenne Numbers Y Congruence of integers Y Residue class Y Complete set of residues modulo m Y Linear congruences Y

Euler’s φ-function Y Fermat’s theorem Y Euler’s theorem Y Wilson’s theorem Y Lagrange’s theorem.

Y Algebra of Matrices Y Ba sic Con cepts Y Ma trix Y Square ma trix Y Unit ma trix or Iden tity Ma trix Y Null or zero ma trix Y Submatrices of a

ma trix Y Equal ity of two ma tri ces Y Ad di tion of ma tri ces Y Mul ti pli ca tion of a ma trix by a sca lar Y Mul ti pli ca tion of two ma tri ces Y

Tri an gu lar, Di ag o nal and Sca lar Ma tri ces Y Trace of a Ma tri x Y Trans pose of a Ma trix Y Con ju gate of a Ma trix Y Tansposed con ju gate of a

Ma trix Y Sym met ric and skew-sym met ric ma tri ces Y Hermitian and Skew-Hermitian Ma tri ces Y De ter mi nants Y De ter mi nants of or der 2 Y

De ter mi nants of or der 3 Y Mi nors and co factors Y De ter mi nants of or der n Y De ter mi nant of a square ma trix Y Prop er ties of De ter mi nants Y

Prod uct of two de ter mi nants of the same or der Y Sys tem of non-ho mo ge neous lin ear equa tions (Cramer’s Rule) Y In verse of a Ma trix Y

Adjoint of a square ma trix Y In verse or Re cip ro cal of a Ma trix Y Sin gu lar and non-sin gu lar ma tri ces Y Re ver sal law for the in verse of a

prod uct of two ma tri ces Y Use of the in verse of a ma trix to find the so lu tion of a sys tem of lin ear equa tions Y Or thogo nal and uni tary

ma tri ces Y Par ti tion ing of ma tri ces Y Rank of a Ma trix Y Sub-ma trix of a Matrix Y Mi nors of a Ma trix Y Rank of a ma trix Y Ech e lon form of a

matrix Y El e men tary trans for ma tion of a ma trix Y El e men tary Ma tri ces Y Invariance of rank un der el e men tary trans for ma tions Y Re duc tion

to normal form Y Equiv a lence of ma tri ces Y Row and Col umn equiv a lence of ma tri ces Y Rank of a prod uct of two ma tri ces Y Com pu ta tion of

the in verse of a non-sin gu lar ma trix by el e men tary trans for ma tions Y Vec tor Space of n-tuples Y Vec tors Y Lin ear de pend ence and lin ear

in de pend ence of vec tors Y The n -vec tor space Y Sub-space of an n-vec tor space Vn Y Ba sis and di men sion of a subspace Y Row rank of a

ma trix Y Left nul lity of a ma trix Y Col umn rank of a ma trix Y Right nul lity of a ma trix Y Equal ity of row rank, col umn rank and rank Y Rank of

a sum Y Lin ear Equa tions Y Homogeneous linear equations Y Fundamentals set of solutions of the Equation Y System of linear

non-homogeneous equations Y Condition for consistency Y Eigen values and Eigen vectors Y Matrix polynomials Y Characteristic

values and characteristic vectors of a matrix Y Characteristic roots and characteristic vectors of a matrix Y Cayley-Hamiltion theorem Y

Eigen Values and Eigen Vectors (Continued) Y Characteristic subspaces of a matrix Y Rank multiplicity Theorem Y Minimal

polynomial and minimal equation of a matrix Y Orthogonal Vectors Y Inner product of two vector Y Orthogonal vectors Y Unitary and

orthogonal matrices Y Orthogonal group Y Similarity of Matrices Y Similarity of matrices Y Diagonalizable matrix Y Orthogonally similar

matrices Y Unitarily similar matrices Y Normal matrices Y Quadratic Forms Y Quadratic Forms Y Linear transformations Y Congruence of

matrices Y Reduction of a real quadratic form Y Canonical or Normal form of a real quadratic form Y Signature and index of a real quadratic

form Y Sylvester’s law of inertia Y Definite, semi-definite and indefinite real quadratic forms Y Hermitian forms.

Y Spherical Harmonics Y Kel vin's The o rem Y Legendre's Equa tion from Laplace's Equa tion Y Bessel's Equa tion from Laplace's Equa tion Y

Legendre's Equa tion Y So lu tion of Legendre's Equa tion Y Def i ni tion of P xn( ) and Q xn( ) Y Gen eral so lu tion of Legendre's Equa tion Y To show

that p xn( ) is the co ef fi cient of hn in the ex pan sion of ( – )– /1 2 2 1 2xh h+ Y Laplace's Def i nite integrals for p xn( ) Y Or thogo nal prop er ties of

Legendre's Poly no mi als Y Re cur rence for mu lae Y Beltrami's Re sults Y Christoffel's Ex pan sion Y Christoffel's Sum ma tion For mula Y Rodrigue's

For mula Y Even and odd func tions Y Ex pan sion of x n in Legendre's Poly no mi als Y Gen eral Re sult Y An im por tant case Y As so ci ated Legendre's Equa tion

29

–A.R. Vasishtha & A.K. Vasishtha236-47 Matrices

–J.N. Sharma & R.K. Gupta237-20Mathematical Methods (Special Functions & Boundar y Value Problems)

Y If v is a so lu tion of Legendre's Equa tion then ( – ) /1 2 2xd v

dx

mm

m is a so lu tion of as so ci ated Legendre's Equa tion Y As so ci ated Legendre's Func tion Y

Prop er ties of the as so ci ated Legendre's Func tion Y Or thogo nal prop er ties of as so ci ated Legendre's Func tions Y Re cur rence for mu lae for as so ci ated

Legendre's Func tions Y Trig o no met ri cal Se ries for p xn( ) Y Legendre's Func tions of the Sec ond Kind Q (x)n Y Legendre's Func tions of the

sec ond kind Y Neumann's In te gral Y Re cur rence for mu lae for Q xn( ) Y Re la tion be tween P xn( ) and Q xn( ) Y Christoffel's Sec ond Sum ma tion

for mula Y As sum ing P xn( ) as a so lu tion of Legendre's Equa tion, show that the com plete so lu tion of this Legendre's Equa tion is AP x BQ xn n( ) ( )+Y Hypergeometric Func tions Y Gauss's Hypergeometric Equa tion Y The Hypergeometric Se ries Y Par tic u lar cases of Hypergeometric Se ries Y

Dif fer ent forms of Hypergeometric Func tion Y So lu tion of Hypergeometric Equa tion Y Lin ear re la tions be tween the so lu tion of the Hypergeometric

Equa tion Y Sym met ric prop erty of Hypergeometric Func tion Y In te gral for mula for the Hypergeometric Func tion Y Kummer's The o rem Y Gauss's The o rem Y

Vandermonde's The o rem Y Dif fer en ti a tion of Hypergeometric Func tion Y The con flu ent Hypergeometric Func tions Y The o rem Y Whitakar's con flu ent

Hypergeometric Func tion Y In te gral rep re sen ta tion of con flu ent Hypergeometric func tion 1 1F x( ,α, γ ) Y Dif fer en ti a tion of con flu ent Hypergeometric Func tion

Y Con tin u ous Hypergeometric Func tion Y The o rem Y Dixon's The o rem Y Bessel's Equa tion Y So lu tion of Bessel's Gen eral Dif fer en tial Equa tion Y

Gen eral So lu tion of Bessel's Equa tion Y In te gra tion of Bessel's Equa tion in se ries for n = 0 Y Def i ni tion of J xn( ) Y Re cur rence For mu lae for J xn( ) Y

Gen er at ing func tion for J xn( ) Y A sec ond so lu tion of Bessel's Equa tion Y Hermite Poly no mi als Y Hermite Dif fer en tial Equa tion Y So lu tion of

Hermite Equa tion Y Hermite's Poly no mi als Y Gen er at ing Func tion Y Other forms for the Hermite Poly no mi als Y To find first few Hermite

Poly no mi als Y Or thogo nal prop er ties of Hermite Poly no mi als Y Re cur rence for mu lae for Hermite Poly no mi als Y Laguerre Poly no mi als Y

Laguerre Dif fer en tial Equa tion Y So lu tion of Laguerre Equa tion Y Laguerre's Poly no mi als Y Gen er at ing Func tion Y Other form for the

Laguerre Poly no mi als Y To find first few Laguerre Poly no mi als Y Or thogo nal prop erty of the Laguerre Poly no mi als Y Re cur rence for mu lae for

Laguerre Poly no mi als Y As so ci ated Laguerre's Equa tion Y If v is a so lu tion of Laguerre's Equa tion of or der n + α then d v

dx

α

α sat is fies Laguerre's

As so ci ated Equa tion Y As so ci ated Laguerre's Poly no mi als (Def.) Y As so ci ated Laguerre's Poly no mi als L xnα ( ) Y Gen er at ing func tion Y Other

form for as so ci ated Laguerre Poly no mial Y Or thogo nal prop erty of the as so ci ated Laguerre Poly no mi als Y Re cur rence formulae for the

associated Laguerre Polynomials Y Chebyshev Polynomials Y Chebyshev's Diff. Equa tion Y Chebyshev Poly no mi als Y To prove that T xn( )

and U xn( ) are in de pend ent so lu tions of Chebyshev's Equa tion Y Re la tions for T xn( ) and U xn( ) Y To find first few Chebyshev Polys. Y Gen er at ing Func tion Y

Or thogo nal prop er ties of Chebyshev Poly no mi als Y Re cur rence for mu lae for T xn( ) and U xn( ) Y Orthogonal Set of Functions Y Definitions Y

Gen er al ized Fou rier Se ries Y Other Type of orthogonality Y Strum-Liouville Equa tion Y The o rem Y The o rem: Eigen Val ues of the

Strum-Liouville prob. are all real Y The o rem Y Orthogonality of Legendre Poly no mi als Y Orthogonality of Bessel Func tion Y Orthogonality of

Hermite Poly no mi als Y Orthogonality of Laguerre Poly no mi als Y Orthogonality of Chebyshev Poly no mi als Y Orthogonality of Jacobi

Polynomials Y Bessel's In equal ity and com plete ness re la tion Y Def i ni tions Y The o rem: If an orthonormal set {φn x( )} is closed,it is com plete Y

Wave, Heat and Laplace's Equations Y One-Di men sional wave-equa tion Y Two-Di men sional wave-equa tion Y Heat Equa tion Y

Laplace's Equa tion Y Laplace's Equa tion in terms of spher i cal co-or di nates Y Laplace's Equa tion in terms of cy lin dri cal co-or di nates Y

Dif fu sion Equation Y Boundary Value Problems Y Cylindrical Y So lu tion of sep a ra tion of Vari ables Y So lu tion of one-di men sional wave

equa tion Y So lu tion of two-di men sional wave equa tion Y Vari a tion of a cir cu lar mem brane Y So lu tion of one-di men sional heat equa tion Y So lu tion of

two-di men sional Laplace's Equa tion Y So lu tion of two-di men sional heat equa tion Y So lu tion of two-di men sional Laplace's Equa tion given in the

cy lin dri cal co-or di nates Y So lu tion of the Laplace's Equa tion given in co-or di nates Y So lu tion of Laplace's Equa tion given in spher i cal co-or di nates Y

So lu tion of three di men sional wave equa tion in spher i cal po lar co-or di nates Y So lu tion of Dif fu sion Equa tion (By the method of the sep a ra tion of vari ablesY

So lu tion of Dif fu sion Equa tion in cy lin dri cal co-or di nates Y So lu tion of Dif fu sion Equa tion in cy lin dri cal po lar co-or di nates.

Y Spher i cal Har mon ics Y Kel vin’s The o rem Y Legendre’s equa tion from Laplace’s equa tion Y Bessel’s equa tion from Laplace’s equa tion Y

Legendre's Equa tion Y So lu tion of Legendre’s Equa tion Y Def i ni tion of P xn( ) and Q xn( ) Y Gen eral so lu tion of Legendre’s equa tion Y To show that P xn( )

is the co ef fi cient of hn in the ex pan sion of ( ) /1 2 2 1 2− + −xh h Y Laplace’s Def i nite integrals for P xn( ) Y Or thogo nal prop er ties of Legendre’s Poly no mi als Y

Re cur rence for mula Y Beltrami’s Re sult Y Christoffel’ Ex pan sion Y Christoffel’s Sum ma tion For mula Y Rodrigues for mula Y Even and odd func tions Y

Ex pan sion of x n in Legendre’s Poly no mial’s Y Gen eral Re sults Y An im por tant case Y As so ci ated Legendre’s Equa tion Y If v is a so lu tion of Legendre’s

equa tion then ( ) /1 2 2− xdm

dx

m vm

is a so lu tion of as so ci ated Legendre’s equa tion Y As so ci ated Legendre’s Func tion Y Prop er ties of the as so ci ated

Legendre’s Func tion Y Or thogo nal Prop er ties of as so ci ated Legendre’s Func tions Y Re cur rence for mulae for as so ci ated Legendre’s func tions Y

Trig o no met ri cal Series for P xn( ) Y Legendre's Func tions of the Sec ond Kind Q (x)n Y Legendre’s func tions of the Sec ond Kind Y Neumann’s

30

–J.N. Sharma & R.K. Gupta238-30 Special Functions (Spherical Harmonics)

In te gral Y Re cur rence for mulae for Q xn( ) Y Re la tion be tween P xn( ) and Q xn( ) Y Christoffel’s Sec ond Sum ma tion for mula Y As sum ing Pn as a so lu tion of

Legendre’s equa tion show that the com plete so lu tion of Legendre equa tion is AP x BQ xn n( ) ( )+ Y Hypergeometric Func tions Y Gauss’s

hypergeometric equa tion Y The hypergeometric se ries Y Dif fer ent forms of hypergeometric func tion Y So lu tion of hypergeometric equa tion Y Lin ear

re la tions between the so lu tion of the hypergeometric equa tions Y Symmetric Prop erty of hypergeometric function Y In te gral for mula for the

hypergeometric function Y Kummer’s Theorem Y Gauss’s The o rem Y Vandermonde’s Theorem Y Dif fer en ti a tion of hypergeometric function Y The

con flu ent hypergeometric function Y Theorem Y Whtitakar’s hypergeometric func tion Y In te gral rep re sen ta tion of con flu ent hypergeometric function

1 1F x( , , )α γ Y Differentiation of con flu ent hypergeometric func tion Y Con tin u ous hypergeometric function Y The o rem Y Dixon’ The o rem Y Bessel's

Equations Y Solution of Bessel’s General Differential Equations Y General solution of Bessel’s Equation Y Integration of Bessel’s equation in series for

n = 0 Y Definition of J xn( ) Y Recurrence formula for J xn( ) Y Generating function for J xn( ) Y A second solution of Bessel’s Equation Y Hermite

Polynomials Y Hermite Differential Equation Y Solution of Hermite Equation Y Hermite's Polynomials Y Generating Function Y Other forms for

Hermite Poly Y To find first few Hermite Polys Y Orthogonal properties of Hermite Polynomials Y Recurrence formulae for Hermite Polynomials Y

Laguerre Polynomials Y Laguerre’s Differential equation Y Solution of Laguerre’s equation Y Laguerre Polynomials Y Generating function Y Other

forms for the Laguerre Polys. Y To find first few Laguerre Polys. Y Orthogonal Property of the Laguerre Polys. Y Recurrence formula for Laguerre

Polynomials Y Associated Laguerre’s Equation Y If v is a solution of Laguerre’s equation of order n + α then d v

dx

α

α satisfies Laguerre’s associated equation

Y Associated Laguerre’s Polynomials (Def.) Y Associated Laguerre’s Polynomials Lnα ( )x Y Generating function Y Other forms for associated Laguerre

Polynomial Y Orthogonal property of associated Laguerre Polynomials Y Recurrence formulae for the associated Laguerre Polynomials Y Chebyshev

Polynomials Y Chebyshev’s Diff. Equations Y Chebyshev Polynomials Y To prove that T xn( ) and U xn( ) are independent solution of Chebyshev’s

Equation Y Relations for T xn( ) and U xn( ) Y To find first few Chebyshev Polys. Y Generating function Y Orthogonal properties of Chebyshev polys. Y

Recurrence formulae for T xn( ) and U xn* ( ) Y Orthogonal Set of Functions Y Generalized Fourier Series Y Other Type of orthogonality Y Strum-Liouville

Equation Y Theorem Y Theorem: Eigen Values of the Strum-Liouville Problem are all Real Y Theorem Y Orthogonality of Legendre Polys. Y

Orthogonality of Bessel Functions Y Orthogonality of Hermite Polys. Y Orthogonality of Laguerre Polys. Y Orthogonality of Chebyshev Polys. Y

Orthogonality of Jacobi Polys. Y Bessel’ inequality and Completeness Relation Y Definitions Y Theorem: If an orthonormal set is closed, it is complete Y

Elliptic Functions Y Periodic Functions Y Elliptic Functions Y Order of an Elliptic Function Y Properties of an Elliptic Function Y Weierstrass’s function

Y Properties of Weierstrass’s Sigma function Y Properties of ξ( )z Y Weierstrass’s Elliptic Function Y Properties of p z( ) Y An algebraic relation connecting

two elliptic functions Y The differential equation satisfied by two Weiestrass’s elliptic function p z( ) Y The three roots e e e1 2 3, , of eqn. 4 22 3ω ω− −g g = 0

are all distinct Y The pseudo periodicity of ζ ( )z and σ ( )z Y Jacobi’s Elliptic Functions Y Construction of Jacobi’s elliptic functions Y Relation between

Jacobi’s elliptic functions and their derivatives Y The complementary modulus Y Few important Results Y Elliptic Integrals Y Derivatives of s z c zn n( ), ( ) and

d zn ( ) Y Periods of s z c z d zn n n( ), ( ), ( ) Y The addition theorem Y Beta and Gamma Functions Y Principle and general values of an improper integral Y

Infinite limits Y To find value of − ∞

∞

∫ f x

F xdx

( )

( ) Y To find value of

− ∞

∞

∫ +x

xdx

m

n

2

21 Y To find value of

− ∞

∞

∫ −x

xdx

m

n

2

21 Y Deduction from

0

2

21

∞

∫ +x

xdx

m

n and

0

2

21

∞

∫ −x

xdx

m

n Y Method of differentiation under the integration sign Y Method of integration under the sign of integration Y Euler’s Integrals — Beta and

Gamma Functions Y Elementary properties of Gamma Functions Y Transformation of Gamma Functions Y Another form of Beta Function Y Relation

between Beta and Gamma Functions Y Other Transformations Y To find the value of Γ Γ Γ1 2 1

n n

n

n

−

...... Y The Dirac Delta Function Y Delta

Function Y Properties of Dirac delta function Y Derivatives of δ ( )x Y The Heaviside Unit Function Y The Dirac delta function is the derivative of the

Heaviside unit function H x( ) .

Y Vectors-Addition and Subtraction Y Vec tor and Sca lar Quan ti ties Y Ad di tion of Vec tors Y Mul ti pli ca tion of a Vec tor by a Sca lar Y Res o lu tion of a

Vec tor in Terms of Coplanar Vec tors Y Non-coplanar Vec tors Y Orthonormal Sys tem of Unit Vec tors i, j, k Y Po si tion Vec tor Y Collinearity of Three Points

Y Prod ucts of two Vec tors Y The Sca lar or Dot Prod uct of Two Vec tors Y Con di tion for Per pen dic u lar ity of Two Vec tors Y Dis trib u tive Law for Sca lar

Prod uct Y The Vec tor or Cross-prod uct of Two Vec tors Y Ex pres sion for Vec tor Prod uct in Terms of Rect an gu lar Com po nents of the Vec tors Y Vec tor

Area Y Mul ti ple Prod ucts Y Sca lar Tri ple Prod ucts Y Dis trib u tive Law for Vec tor Prod uct of Two Y Con di tion for Three Vec tors to be Complanar Y To

Ex press the Value of the sca lar Tri ple Prod uct in Terms of Rect an gu lar Com po nents of Vec tors Y Vec tor Tri ple Prod uct Y To Prove that

a (b c)=(a c) b (a b)c× × • − • Y Sca lar Prod uct of Four Vec tors Y Vec tor Prod uct of Four Vec tors Y Re cip ro cal Sys tem Vec tors Y Vec tor Equa tions

31

–A.R. Vasishtha239-23 Vector Algebra

of a Line and a Plane Y Vec tor Equa tion of a Straight Line Y Con di tion for Three Points to be Col lin ear Y Bi sec tor of the An gle be tween Two Straight Lines

Y Vec tor Equa tion of a Plane Y Con di tion for Four Points to be Coplanar Y Cen troids Y Fur ther Ap pli ca tion of Vec tors to Ge om e try and

Me chan ics- Ge om e try of Plane Y Vec tor Equa tion of a Plane Y An gle be tween Two Planes Y The Two Sides of a Plane Y Per pen dic u lar Dis tance of a

Point From a Plane Y Planes Through the In ter sec tion of Two Planes Y Ge om e try of the Straight Line Y Line of in ter sec tion of Two Plane Y Con di tion

for Two Lines to be Coplanar Y Short est Dis tance be tween Two Non-in ter sect ing Lines Y Vol ume of a Tet ra he dron Y Ge om e try of the Sphere Y The

Vec tor Equa tion of a Sphere Y Tan gent Plane at a Given Point Y Di am et ral Plane Y Rad i cal Plane Y Ap pli ca tion to Me chan ics Y Lami’s The o rem Y

Work Y Vec tor Mo ment or Torque Y Rel a tive Mo tion Y Rel a tive Ve loc ity.

Y Meaning and Purpose of Statistics Y Or i gin of Sta tis tics Y Def i ni tion of Sta tis tics Y Scope and Lim i ta tions of Sta tis tics Y Pop u la tion and sam ple Y

Main Stages of a Sta tistical En quiry Y Dis trust of Sta tis tics Y Fre quency Dis tri bu tions and Mea sures of Cen tral Ten dency Y Clas si fi ca tion and

Tab u la tion Y Fre quency Poly gon, His to gram and Ogive Y Var i ous forms of Fre quency Curves Y Mea sures of Cen tral Ten dency Y Arith me tic Mean Y

Proof of the for mula on Means NM N M N M N Mr r= + + +1 1 2 2 ... Y Me dian Y Mode Y Mode by Group ing Y Geo met ric Mean Y Har monic Mean Y

Quartiles and Par ti tion Val ues Y De sid er ata for a Sat is fac tory Av er age Y Em pir i cal Re la tion be tween Mean, Me dian and Mode Y Mea sures of

Dis per sion and Skew ness Y Range Y Quartile De vi a tion or Semi-Inter Quartile Range Y Stan dard De vi a tion or the Root Mean Square De vi a tion from

the Mean Y Mean De vi a tion or the Av er age De vi a tion Y Co ef fi cient of Vari a tion Y Mo ments Y Prob a bil ity Y Def i ni tion Y Sta tis ti cal or Em pir i cal

Def i ni tion Y Some Def i ni tions Y Ad di tion The o rem of Prob a bil ity Y Mul ti pli ca tion the ory of Prob a bil ity Y De pend ent and In de pend ent Events Y

Prob a bil ity of atleast One Event Y Bi no mial and Multinomial The o rems Y Multinomial The o rem Y Prob a bil i ties : Ax i om atic Ap proach Y Ad di tion

The o rem of Prob a bil ity Y Con di tional Prob a bil ity Y Variate or a Ran dom Vari able Y Rel a tive fre quen cies and prob a bil i ties Y The o rem of To tal Prob a bil ity

for Com pound Events Y Baye’s The o rem Y Ran dom Vari able Dis tri bu tion Func tion Y Ra ndom Vari able or Variate Y Type of Ran dom Vari ables Y

Dis tri bu tion Func tion Y Prob a bil ity Mass Func tion Y Dis crete Dis tri bu tion Y Function Y Probability Density Function Y Con tin u ous Fre quency

Dis tri bu tions Y Dis crete and Con tin u ous Vari ables Y Con tin u ous Dis tri bu tions Y Cu mu la tive Dis tri bu tions or Prob a bil ity Dis tri bu tion Func tion Y Any

Func tions Y Mo ments Y Geo met ric Mean G Y Sum of Ran dom Vari ables, Con vo lu tions Y Mar ginal and Con di tional Prob a bil i ties Y Chebyshev’s and

Markov’s In equal i ties Y Prob a bil ity Func tion of a Quo tient Y Change of vari able Y Bivariate Dis tri bu tions Y Mar ginal Dis tri bu tions Y Con di tional

Prob a bil ity Den sity Y Sto chas tic In de pend ence Y Im por tant The o ret i cal Dis tri bu tions Y The o ret i cal Dis tri bu tions Y Binomial dis tri bu tion Y Pascal’s

Tri an gle Y Mo ments of the Binomial Dis tri bu tion Y Proof of the for mula M pp

p qk

kn=

+∂∂

( ) Y Mode of Bi no mial Dis tri bu tion Y Re cursion for mula for

bi no mial dis tri bu tion µ µ µr r

rpq nrd

dp+ −= +

1 1 Y Pois son Dis tri bu tion Y Pois son Pro cess Y Mode of the Pois son Dis tri bu tion Y Con stan ts of the

Pois son Dis tri bu tion Y Multinomial Dis tri bu tion Y Hy per-geo met ric Dis tri bu tion Y Mean and vari ance of the Hy per-geo met ric Dis tri bu tion Y Nor mal

Dis tri bu tion Y Der i va tion of Nor mal Dis tri bu tion Y Prop er ties of Nor mal Dis tri bu tion Y Con stants of the Nor mal Dis tri bu tion Y Some fur ther prop er ties

of the Nor mal Dis tri bu tion Y Prob a ble Er ror Y Im por tance of the Nor mal Dis tri bu tion Y Fit ting a Nor mal Dis tri bu tion Y Cen tral Limit The o rem Y The Law

of Large Num bers Y Weak law of Large Num bers Y Mo ment Gen er at ing Func tions and Cumulants Y Ex pec ta tion of a Ran dom Vari able Y

Ex pec ta tion of Func tions of Ran dom Vari able Y Ex pec ta tion of Func tions of two Ran dom Vari ables Y Mo ment Gen er at ing Func tion Y Change of or i gin

and scale in Mo ment Gen er at ing Func tion Y M.G.F. of a sum Y Bi no mial Dis tri bu tion Y Pois son Dis tri bu tion Y Neg a tive Bi no mial Dis tri bu tion Y Nor mal

Dis tri bu tion Y Sum of In de pend ent Nor mal vari ates Y Cumulants Y Ad di tive Prop erty of Cumulants Y Fac to rial Mo ments Y Sum of Pois son vari ates Y

Char ac ter is tic Func tion Y In ver sion Y Cauchy’s Dis tri bu tion Y Prob a bil ity Gen er at ing Func tion Y Prop er ties of Char ac ter is tic Func tions Y Some

Char ac ter is tic Func tions Y Prob a bil ity Gen er at ing Func tions Y Re la tion be tween Prob a bil ity Gen er at ing Func tion and Char ac ter is tic Func tion Y Fac to rial

Mo ment Gen er at ing Func tions Y Method of Least Squares and Curve Fit ting Y Method of Least squares Y Some Spe cial Curves Y Bivariate

Dis tri bu tion, Re gres sion and Cor re la tion Y Scat ter or Dot Di a gram Y r in de pend ent of Or i gin and Scale Y Sterograms and Col lec tion Sur face Y

Prob a ble er ror of co ef fi cient of Cor re la tion Y Rank Cor re la tion Y r lies be tween −1 and 1 Y Vari ance of a sum or dif fer ence Y Re gres sion Y Range of r Y

Cor re la tion ra tio Y Lin ear Re la tion ship Y Cau sa tion and ef fect Y Re gres sion and Correla tion Y Mul ti ple and Par tial Cor re la tion Y Mul ti ple

Cor re la tion and Par tial Cor re la tion Y Equa tion of the Re gres sion Plane Y Mul ti ple Cor re la tion Co ef fi cient Y Par tial Cor re la tion Co ef fi cient Y

Con sis tence of Data and As so ci a tion of At trib utes Y At trib utes Y Clas si fi ca tion with ref er ence to at trib utes Y Class fre quen cies Y Re la tion be tween

class fre quen cies Y Con sis tence of data Y In de pend ence and As so ci a tion of at trib utes Y Yule’s co ef fi cient of As so ci a tion Y Fi nite Dif fer ences and

In ter po la tion Y Dif fer ence Ta ble Y Some No men cla tures Y E and ∆ no ta tion Y Fac torial No ta tion Y In ter po la tion Y Al ge braic Meth ods of In ter po la tion Y

New ton’s For mula for equal In ter vals Y Lagrange’s For mula Y Cen tral Dif fer ences Y Gauss’s Back ward For mula Y Gauss’s For ward For mula Y Bessel’s

For mula Y Stirling’s For mula Y Dis tinc tion be tween In ter po la tion and Ex trap o la tion Y Di vided Dif fer ences For mula Y New ton’s Di vided Dif fer ences

For mula Y Pre lim i nary Con cepts on Sam pling Y Uni verse, def i ni tion Y Sam pling : Types of sam ples Y Sim ple Sam pling Y De vices for Ran dom

Sam pling Y Tippet’s Num bers Y Strat i fied Sam pling Y Sim ple Sam pling of At trib utes-Large Sam ples Y Pop u la tion and Sam ples Y Sim ple

32

–J.N. Sharma & J.K. Goyal240-30 Mathematical Statistics

Sam pling of At trib utes Y Mean and Stan dard De vi a tion in Sim ple Sam pling of At trib utes Y Test of sig nif i cance for large sam ples Y Stan dard Er ror Y

Pre ci sion Y Con di tions for Sim ple Sam pling Y Com par i sons of Large Sam ples Y The Sam pling of Vari ables-Large Sam ples Y Sam pling Dis tri bu tion

Y Util ity of Sam pling Dis tri bu tion Y Stan dard Er ror Y Mean and Stan dard Er ror of the Sam pling Dis tri bu tion of means of sam ples Y Dis tri bu tion of the

dif fer ence be tween two sam ple means Y Lev els of Sig nif i cance Y Means of the sam ples Y Fiducial or Con fi dence Lim its Y Test of Sig nif i cance of the

means of two Large Sam ples Y Sam pling Dis tri bu tion Y Stan dard Er ror of other Pa ram e ters Y χ2 Dis tri bu tion Y Def i ni tion of χ2 Y De grees of free dom

and Con straints Y Der i va tion of χ2-dis tri bu tion Y Con di tions for the ap pli ca tion of χ2 test Y Prop er ties of χ2-dis tri bu tion Y Lev els of Sig nif i cance Y Test of

good ness of fit Y Test of in de pend ence Y Yate’s Cor rec tions for Con ti nu ity Y Ad di tive prop erty of χ2 Y Dis tri bu tion of Quo tient of two χ2 vari ates Y χ2

de ter mi na tion of con fi dence lim its Y χ2 test for Pop u la tion vari ance Y The Sam pling of Vari ables-Small Sam ples Y Es ti mates of the arith me tic mean

Y Es ti mate of vari ance Y t-dis tri bu tion Y Prob a bil ity ta bles of t-dis tri bu tion Y Use of t-dis tri bu tion Y Test of sig nif i cance of the mean of ran dom sam ples

from nor mal pop u la tion Y Test of sig nif i cance of the dif fer ence be tween the sam ple means Y To test the sig nif i cance of the ra tio of two in de pend ent

es ti mates of the pop u la tion vari ance the z-test Y Fisher’s z-ta bles of points and the sig nif i cance test Y F-dis tri bu tion Y Snedecor’s F-ta bles and sig nif i cance

test Y Mo ments of the F-dis tri bu tion Y Re la tion ship be tween t, F and χ2 dis tri bu tions Y To test the sig nif i cance of Cor re la tion Co ef fi cient (small sam ples) Y

Test of sig nif i cance of cor re la tion co ef fi cient based on Fisher’s z-trans for ma tion (large sam ples) Y Sig nif i cance of the dif fer ence be tween two in de pend ent

cor re la tion co ef fi cients Y Anal y sis of Vari ance Y Mean ing and Def i ni tion Y Vari ance within and be tween classes Y One cri te rion of clas si fi ca tion Y

Cal cu la tions of ANOVA Y Two cri te ria of clas si fi ca tion Y As sump tions for Anal y sis of vari ance test Y De sign of Ex per i ments (Ran dom iza tion and

Latin Squares) Y Ba sic Prin ci ples of Ex per i men tal De sign Y The Latin squares Y Ran dom ised Block De sign (R.B.D.) Y Com par i son of Latin Square

De sign Y In fer ence-1 (Es ti ma tion ) Y Point Es ti ma tion Y Prop er ties of Good Es ti ma tor Y Un bi ased Es ti ma tor Y Con sis tent Es ti ma tor Y Ef fi cient

Es ti ma tor Y Suf fi cient Es ti ma tor Y Max i mum Like li hood pa ram e ter Y Prop er ties of Max i mum Like li hood pa ram e ter Y Prop er ties of Max i mum Like li hood

Es ti ma tion Y Meth ods of Mo ments Y In fer ence-2 (Test ing of Hy poth e sis) Y Sta tis ti cal Hy poth e ses Y Null Hy poth e sis Y Type I and Type II Er rors Y

Crit i cal Re gion and Ac cep tance Re gion Y The best Test for a Sim ple Hy poth e sis Y The Neymen Pearson Lemma Y The Like li hood Ra tio Test Y Test for

sin gle Mean Y In ter val Es ti ma tion Y Def i ni tion Y Con fi dence In ter vals Y Con fi dence In ter val for mean of Nor mal pop u la tion Y Con fi dence In ter val

when σ is un known Y Con fi dence In ter val for Dif fer ence be tween Means Y Con fi dence In ter val for pro por tions Y Con fi dence In ter val for Vari ance

Y In dex Num bers Y Uses Y Con struc tion of In dex num bers Y Fixed and Chain Bases Y Av er age Y In dex Num bers based on Arith me tic Mean Y

Re vers ibil ity Test Y Fisher’s Ideal In dex Num ber Y Cir cu lar Test Y Anal y sis of Time Se ries Y Move ments Y Method of de ter min ing trend Y Es ti ma tion

of Sea sonal Trend Y Re mov ing the trend Y Util ity of time se ries Y Ap pen dix Y Inter Re la tions among cer tain dis crete dis tri bu tions Y Re pro duc tive

(Ad di tive) Property of some dis crete dis tri bu tions Y Re la tions be tween con tin u ous dis tri bu tions Y Rao Cramer In equal ity Y Non Parametric Tests Y

Me dian Test Y Sign Test Y Run Test Y Wilcoxon-Mann Whit ney Test Y Bivariate Nor mal Dis tri bu tion Y Lo rentz curve Y Lo gis tic curve Y Mo ment

Gen er at ing Func tions of some im por tant distributions M t0 ( ) Y Dis cover ers of Some Dis tri bu tions Y Tables.

Y Introduction to Operations Research Y In tro duc tion (The Or i gin and the De vel op ment of OR) Y Na ture and Def i ni tion of OR Y Ob jec tive of

OR Y Phases of OR Method Y Ar eas of Ap pli ca tions (Scope) of OR Y Op er a tions Re search and De ci sion-Mak ing Y Sci en tific Method in OR Y

Char ac ter is tics of Op er a tions Re search Y Mod el ing in OR Y Types of Mod els Y Gen eral Meth ods of So lu tion for OR Mod els Y Mathematical

Preliminaries Y El e men tary Prob a bil ity The ory Y Sam ple Space Y Events Y Al ge bra of Events Y Clas si cal Def i ni tion of Prob a bil ity Y Odds in

Fa vour and Odds Against Y The Sta tis ti cal (or Em pir i cal) Def i ni tion of Prob a bil ity Y Ax i om atic Def i ni tion of Prob a bil ity Y Nat u ral As sign ment of

Prob a bil i ties Y The o rem of To tal Prob a bil ity or Ad di tional The o rem of Prob a bil ity Y Com pound Events Y In de pend ent and De pend ent Events Y

Con di tional Prob a bil ity Y Mul ti pli ca tion The o rem of Prob a bil ity Y Ran dom Vari able Y Dis crete Prob a bil ity Dis tri bu tions Y Ex pec ta tion of a Ran dom

Vari able Y Spe cial Dis crete Prob a bil ity Dis tri bu tions Y Con tin u ous Prob a bil ity Dis tri bu tions Y Spe cial Con tin u ous Prob a bil ity Dis tri bu tions Y Ma tri ces

and De ter mi nants Y Def i ni tions Y Op er a tions of Ma trix Ad di tion and Mul ti pli ca tion Y Sub-Ma trix Y Mi nor of Or der k Y De ter mi nant Y Im por tant

Prop er ties of De ter mi nants Y Mi nors Y Co factors Y Rank of a Ma trix Y Adjoint of a Ma trix Y Sin gu lar and Non-Sin gu lar Ma tri ces Y In verse of a Ma trix Y

Vec tors and Vec tor Spaces Y Def i ni tions Y Eu clid ean Space Y Lin ear De pend ence and In de pend ence of Vec tors Y Lin ear Com bi na tion (L.C.) of

Vec tors Y Span ning Set Y Ba sis Set Y Some Use ful The o rems of Lin ear Al ge bra Y Si mul ta neous Lin ear Equa tion Y Fi nite Dif fer ence Y First Dif fer ence

of f (x) Y Sec ond Dif fer ence of f(x) Y Con di tions for a Max i mum or Min i mum of f(x) Y Dif fer en ti a tion of Integrals Y Generating Func tions Y

Inventory Theory Y In ven tory Y Vari ables in In ven tory Prob lems Y Need of In ven tory Y In ven tory Prob lems Y Ad van tages and Dis ad van tages of

In ven tory Y Clas si fi ca tion or Cat e go ries of In ven tory Mod els Y Some Gen eral No ta tions Used in In ven tory Mod els Y De ter min is tic Mod els Y Eco nomic

Lot Size Mod els Y Model I : Eco nomic Lot Size Model with Uni form Rate of De mand Y In fi nite Pro duc tion Rate and hav ing no Short ages Y An other form

of Model I Y Model II : Eco nomic Lot-size Model with Dif fer ent Rates of De mand in Dif fer ent Pro duc tion Cy cles, In fi nite Pro duc tion Rate and hav ing no

Short ages Y Model III : Eco nomic Lot-size Model with Uni form Rate of De mand, Fi nite Rate of Re plen ish ment hav ing no Short ages

De ter min is tic Mod els With Short ages Y Model IV : Fixed Time Model Y Model V : Eco nomic Lot-size Model with Uni form Rate of De mand, In fi nite Rate of

33

–R.K. Gupta241-34 Operations Research

Pro duc tion and Hav ing Short ages which are to be Ful filled Y Model VI : Eco nomic Lot-size Model with Uni form Rate of De mand, Fi nite Rate of

Pro duc tion and hav ing Short ages which are to be Ful filled Y Multi Item, De ter min is tic Mod els with One Con stant Y Proba bil is tic Mod els Y Model VII :

Sin gle Pe riod Model with Dis con tin u ous or In stan ta neous De mand and Time In de pend ent Costs (No Set up Cost Model) Y Model VIII : Sin gle Pe riod

Model with Uni form De mand (No Set up Cost Model) Y Model IX : The Gen eral Sin gle Pe riod Model of Profit Max i mi za tion with Time In de pend ent Cost

Y Model X : Proba bil is tic Or der Level Sys tem with Lead-Time Y Pur chase In ven tory Mod els with Price Breaks Y Model XI : Pur chase In ven tory Model Y

Model XII : Pur chase In ven tory Model with One Price Break Y Model XIII : Pur chase In ven tory Model with Two Price Breaks Y Model XIV : Pur chase

In ven tory Model with Mul ti ple Price Breaks Y Re place ment Prob lems Y Re place ment of Ma jor or Cap i tal Item (Equip ment) that de te ri o rates with

Time Y To find the Best Re place ment Age (Time) of a Ma chine Y Few Im por tant Terms Y To De ter mine the Best Re place ment Age of Items

whose Main te nance Costs In crease with Time and the Value of Money also Changes with Time Y A Dis counted Cost P n( ) is In vested by tak ing Loan at

the In ter est Rate r; and the Loan is Re paid by Fixed An nual Pay ments say x, through out the Life of the Ma chine. To Find the Min i mum Value of x for

Op ti mum Pe riod n at which to Re place the Ma chine Y Re place ment of Items in An tic i pa tion of Com plete Fail ure the prob a bil ity of which in creases with

Time Y To de ter mine the In ter val of Op ti mum Re place ment Y Prob lems in Mor tal ity Y Staff ing Prob lem Y Mor tal ity Ta bles Y Wait ing Line or

Queu ing The ory Y Ba sic Queu ing Pro cess (sys tem) and Its Char ac ter is tics Y Cus tom ers Be hav iour in a Queue Y Im por tant Def i ni tions in

Queu ing Prob lem Y The State of the Sys tem Y Pois son Pro cess Y Pois son Ar riv als Y The o rem Y Some Dis tri bu tions Y An Im por tant The o rem Y

No ta tions Y Clas si fi ca tion of Queu ing Mod els Y So lu tion of Queue Mod els Y Model I (M/M/1) : ( / )∞ FCFS (Birth and Death Model) Y Re la tion ship

be tween Y Model II (Gen eral Er lang Queu ing Model) Y Model III : (M/M/1) : (N/FCFS) Y Model IV (M/M/S) : ( / )∞ FCFS Y Model V :

( / ) : ( / )M/E FCFSk 1 ∞ Y Model VI ( / ) : ( / )M/E FCFSk 1 1 Y Ma chine Re pair Prob lem Y Model VII ( ) : ( ),M / M / R k /GD k <R Y Model VIII Power Sup ply

Model Y Al lo ca tion (Gen eral Lin ear Pro gram ming Prob lems) Y Gen eral Lin ear Pro gram ming Prob lems Y Math e mat i cal For mu la tion of a

L.P.P. Y Ba sic So lu tion (B.S.) Y An Im por tant The o rem Y Some Im por tant Def i ni tions Y So lu tion of a Lin ear Pro gram ming Prob lem Y

Geo met ri cal (or Graph i cal) Method for the So lu tion of a Lin ear Pro gram ming Prob lem Y An a lytic Method (Trial and Er ror Method) Y Slack and

Sur plus Vari ables Y Ap pli ca tions of Lin ear Pro gram ming Tech niques Y Ad van tages of Lin ear Pro gram ming Tech niques Y Lim i ta tions of Lin ear

Pro gram ming Y Con vex Sets and their Prop er ties Y Def i ni tions Y Some Im por tant The o rems Y Sim plex Method Y Some Def i ni tions and

No ta tions Y Fun da men tal The o rem of Lin ear Pro gram ming Y To ob tain B.F.S. from a F.S. Y To De ter mine Im proved B.F.S. Y Un bounded

So lu tions Y Optimality Con di tions Y Al ter na tive Op ti mal So lu tions Y In con sis tency and Re dun dancy in Con straint Equa tions Y To De ter mine

Start ing B.F.S. Y Com pu ta tional Pro ce dure of the Sim plex Method for the So lu tion of a Max i mi za tion L.P. Y Ar ti fi cial Vari ables Tech nique Y

De gen er acy in Sim plex Method Y Con di tions for the Oc cur rence of De gen er acy in a L.P.P. Y Com pu ta tional Pro ce dure to Re solve De gen er acy by

Carner's Per tur ba tion Method Y Some Im por tant Tips for Sim plex Method Y Lin ear Pro gram ming Prob lem (Spe cial Cases) Y So lu tion of Sys tem of

Si mul ta neous Lin ear Equa tions by Sim plex Method Y In verse of a Ma trix by Sim plex Method Y Du al ity in Lin ear Pro gram ming Y Stan dard Form

of the Pri mal Y Sym met ric Dual Prob lem Y Unsymmetric Dual Prob lem Y The Dual of a Mixed Sys tem Y Du al ity The o rems Y Cor re spon dence

be tween Pri mal and Dual Y To Read the So lu tion to the Dual from the Fi nal Sim plex Ta ble of the Pri mal and Vice-Versa Y Dual Sim plex Method

Y Ad van tage of Dual Sim plex Al go rithm Y Com pu ta tional Pro ce dure of the Dual Sim plex Al go rithm Y Sen si tiv ity Anal y sis Y Vari a tion in the

Price Vec tor c Y Vari a tion in the Re quire ment Vec tor b Y Vari a tion in the El e ment aij of the Co ef fi cient Ma trix A Y Ad di tion of a New Vari able to

the Prob lem Y Ad di tion of a New Con straint to the Prob lem Y Para met ric Lin ear Pro gram ming Y Lin ear Vari a tions in c Y Lin ear Vari a tion in

b Y Integer Programming Y Im por tance (or need) of I.P.P. Y So lu tion of I.P.P. Y Gomory’s all I.P. Method Y Con struc tion of Gomory’s Con straint

and Gomory’s Cut ting Plane Y All-In te ger Cut ting Plane Al go rithm Y Mixed-In te ger Cut ting Plane Al go rithm Y The Branch-and-Bound Tech nique Y

Branch-and-Bound Al go rithm Y As sign ment Prob lem Y Im por tant The o rem Y Hun gar ian Method (Re duced Ma trix Method) Y Un bal anced

As sign ment Prob lems Y Max i mi za tion As sign ment Prob lem Y Re stric tions on Assignment Y Trans por ta tion Prob lem Y Dif fer ence be tween a

Trans por ta tion and an As sign ment Prob lem Y So lu tion of a Trans por ta tion Prob lem Y To Find an Ini tial Fea si ble So lu tion Y Optimality Test Y

The o rem Y Com pu ta tional Pro ce dure of Optimality Test Y Trans por ta tion Al go rithm or MODI (Mod i fied Dis tri bu tion) Method Y De gen er acy in

Trans por ta tion Prob lems Y Un bal anced Trans por ta tion Prob lems Y Profit Max i mi za tion Prob lems Y Pro hib ited Trans por ta tion Route Y Se quenc ing

(In clud ing Trav el ling Sales man Prob lem) Y A Se quenc ing Prob lem Y Gen eral As sump tions Y Se quenc ing De ci sion Prob lem for n-jobs on

two Ma chines Y Se quenc ing De ci sion Prob lem for n-Jobs on Three Ma chines Y Se quenc ing De ci sion Prob lem for n Jobs on m Ma chines Y Pro cess ing Two

Jobs Through m Ma chines Y Graph i cal Method Y Trav el ling Sales man (or Rout ing) Prob lem Y Dynamic Programming Y Bell man’s Prin ci ple of

Optimality in Dy namic Pro gram ming Y Mul ti stage De ci sion Prob lem Y Char ac ter is tics of Dy namic Pro gram ming Prob lems Y So lu tion of a

Multi-stage Prob lem by Dy namic Pro gram ming with Fi nite Num ber of Stages Y So lu tion of Lin ear Pro gram ming Prob lem as a Dy namic

Pro gram ming Prob lem Y So lu tion of an In ven tory Prob lem as a Dy namic Pro gram ming Prob lem Y Game Theory (Competitive Strategies) Y

Com pet i tive Games Y Fi nite and In fi nite Games Y Zero Sum Game Y Two Per son Zero Sum (or Rect an gu lar) Games Y Pay-off Ma trix Y Strat egy

Y So lu tion of a Game Y Maximin and Minimax Cri te rion of Optimality Y Sad dle Point Y So lu tion of a Rect an gu lar Game with Sad dle Point Y So lu tion

of a Rect an gu lar Game in terms of Mixed Strat e gies Y Im por tant Prop er ties of Op ti mal Mixed Strat e gies Y So lu tion of 2 × 2 Games With out Sad dle Point

Y Dom i nance Prop erty Y Arith me tic Method (or the Method of Odd ments or the Short Cut Y Method) for the So lu tion of 2 × 2 Game with out Sad dle Y

Graph i cal Method for the So lu tion of (2 × n ) and (m × 2) Games Y Al ge braic Method for the So lu tion of a Gen eral Game Y It er a tive Method for Ap prox i mate

So lu tion Y Equiv a lence of the Rect an gu lar Ma trix Game and Lin ear Pro gram ming Y Fun da men tal The o rem of Game The ory (Minimax The o rem) Y So lu tion of a

Rect an gu lar Game by Sim plex Method Y Ma trix Method for n×n (i.e, Square) Games Y Sum mary of Meth ods for Solv ing the Rect an gu lar (Two Per son Zero

Sum) Games Y Minimax and Maximin of a Func tion of Sev eral Vari ables Y Sad dle Point of a Func tion of Sev eral Vari ables Y Nec es sary and Suf fi cient

34

Con di tion for the Func tion E( )x, y to Pos sess a Sad dle Point (Ex is tence of Sad dle Point) Y Goal Pro gram ming Y Con cept of Goal Pro gram ming Y

For mu la tion of a G.P. Prob lem as a L.P. Prob lem Y Mul ti ple Goals with Pri or i ties and Weights Y So lu tion of a Lin ear G.P. Prob lem Y Graph i cal

Method for the So lu tion of a Lin ear G.P. Prob lem Y Sim plex Method (Mod i fied) for the So lu tion of a Lin ear G.P. Prob lem Y Net work Anal y sis

(PERT/CPM)Y The ory of Graphs Y Few Im por tant Def i ni tions Y Net work Y Sched ule Chart (Gantt Bar Chart) Y Dif fer ence be tween CPM and

PERT Y Net work Com po nents Y Con struc tion of the Net work Di a gram Y Dummy Ac tiv ity Y Er rors in Draw ing a Net work Y Gen eral Pro ce dure for

the Con struc tion of a Net work Di a gram Y Num ber ing the Events (Fulkeron's Rule) Y CPM Com pu ta tion (or Anal y sis) Y The Float and Slack Y

In equal ity re la tion be tween Floats of an Ac tiv ity Y Event Slacks Y Crit i cal, Event, Ac tiv ity and Path Y Pro ce dure of de ter mi na tion of Crit i cal Path

Y PERT Y Es ti mate of Prob a bil ity of Com plet ing the Pro ject by Sched uled Time Y In for ma tion The ory Y Com mu ni ca tion Pro cess Y De scrip tion

of a Com mu ni ca tion Sys tem Y A Quan ti ta tive Mea sure of In for ma tion Y A Bi nary Unit of In for ma tion Y Mea sure of un cer tainty or En tropy Y

Prop er ties of Av er age Mea sure of Un cer tainty or En tropy Y Im por tant Re la tions for Var i ous Entropies Y En cod ing Y Unique Decipherability Y Uses

of En cod ing; Ef fi ciency and Re dun dancy Y Shan non-Fano En cod ing Pro ce dure Y Nec es sary and Suf fi cient Con di tion for Noise less Cod ing Y

Non-lin ear Pro gram ming Y Gen eral Non-lin ear Pro gram ming Prob lem Y Math e mat i cal For mu la tion of GNLPP Y So lu tion of NLPP with all

Equally Con straints Y Suf fi cient Con di tions for Max i mum or Min i mum of the Ob jec tive Func tion Y So lu tion of NLPP when Con straints are Not

all Equal ity Con straints Y Kuhn-Tucker Nec es sary Con di tions for the Optimality of the Ob jec tive Func tion in a GNLP Prob lem Y Kuhn-Tucker

Suf fi cient Con di tions for the Optimality of the Ob jec tive Func tion of a GNLPP with in equal ity Con straints Y Graph i cal So lu tion Y Tables.

Y Moment of Inertia Y Def i ni tions Y Mo ments of in er tia in some sim ple cases Y The o rem of par al lel axes Y Pappus the o rem (solid gen er ated); to find the

M.I. of the body Y Pappus the o rem (sur face gen er ated); to find the M.I. of the body Y Mo ment of in er tia about a line Y To find M.I. about any axis which

passes through the in ter sec tion of two per pen dic u lar axis in the plane pro vided M.I.’s and P.I.’s about these axes are known Y El e men tary the o rem on

mo ment of in er tia Y Method of dif fer en ti a tion Y M.I. of het er o ge neous bod ies Y Momental el lip soid Y Momental el lipse Y Bod ies Y Equipmomental

The o ries Y Prin ci pal axes Y Prin ci pal mo ments Y D' Alembert's Prin ci ple Y Mo tion of a par ti cle and a Rigid body Y Im pressed and Ef fec tive forces Y D’

Alembert’s Prin ci ple Y An gu lar Mo men tum Y Gen eral equa tions of Mo tion Y Lin ear mo men tum Y Mo tion of the Cen tre of in er tia Y Mo tion about Cen tre

of in er tia Y Im pul sive forces Y Im pul sive forces con tinued Y Ap pli ca tion of D’Alembert’s Prin ci ple to im pul sive forces and gen eral equa tion of mo tion Y

Mo tion about a Fixed Axis Y Mo ment of the ef fec tive forces about the axis of ro ta tion Y Ki netic En ergy Y Equa tion of Mo tion Y Com pound Pen du lum

Y Cen tre of Sus pen sion Y Cen tre of Sus pen sion and os cil la tions are Con vert ible Y Min i mum time of os cil la tion of com pound pen du lum Y Re ac tion of the

axis of rotation Y Mo tion about a fixed axis (im pul sive forces) Y Cen tre of Per cus sion Y Cen tre of Per cus sion of a rod Y Gen eral case of cen tre of

per cus sion Y Mo tion in Two Di men sions Y Equa tion of mo tion Y Ki netic en ergy Y Mo ment of Mo men tum Y Mo tion of a solid sphere down an in clined

plane Y Slip ping of rods Y A uni form straight rod sliding down in a ver ti cal plane, its ends be ing in con tact with two planes, one hor i zon tal and other

ver ti cal Y Mo tion of a solid sphere down an in clined plane when roll ing and sliding are com bined Y Mo tion of a cir cu lar disc Y When two bod ies are in

con tact, then to de ter mine wheather the rel a tive mo tion in volves sliding at the pont of con tact Y A sphere of ra dius ‘‘a’’ where C.G. is at a dis tance c from

its cen tre C is place on a rough plane so that C.G. is hor i zon tal ; show that it will be gin to roll or slide ac cord ing as µ < or ac

k a2 2+ where k is the ra dius of

gy ra tion about a hor i zon tal axis through G. If µ equal to this value what hap pens ? Y Mo tion of one sphere over an other sphere which if fixed Y Mo tion of

solid cyl in der in sider a hol low cyl in der Y Mo tion of one body on an other, when the lower body is free to turn about its axis Y Mo tion of one body on

an other when both bod ies are free to move Y Mo tion in Two Di men sions (Un der Im pul sive Forces) Y To ob tain the equa tions of mo tion of a rigid

body un der im pul sive forces Y A rod of length 2a is held in a po si tion in clined at an an gle α to the verticle and is the let fall on the smooth in elas tic

hor i zon tal plane will have it im me di ately af ter the im pact if the height through which the rod falls is greater than ( / ) sec cos ( sin )1 8 1 32 2 2a ecα α α+ Y An

im per fectly elas tic sphere im ping ing on a fixed plane Y Work done by an Im pulse Y Con ser va tion of Mo men tum and En ergy Y Prin ci ple of

con ser va tion of lin ear mo men tum (Fi nite Forces) Y Prin ci ple of An gu lar Mo men tum Y Con serva tion of Lin ear Mo men tum (Im pul sive forces)

Con ser va tion of An gu lar Mo men tum (Im pul sive fores) Y Prin ci ple of con ser va tion of en ergy Y Vis-Viva Y Prin ci ple of Vis-Viva Y Con ser va tive forces Y

The o rem : When a body moves un der the ac tion of a sys tem of con ser va tive force, the sum of its ki netic and po ten tial en er gies is con stant through out the

mo tion Y The ki netic en ergy of a Rigid body, mov ing in any man ner is at any in stant equal to the ki netic en ergy of the whole mass, sup posed to be

col lected at its cen tre of in er tia and mov ing with it, to gether with the ki netic en ergy of the whole mass rel a tive to its cen tre of in er tia Y Ini tial Mo tion Y

Def i ni tion Y Lagrange's Equa tions of Mo tion, Small Os cil la tions, Nor mal Co-or di nates Y Gen er al ised Co or di nates Y De grees of free dom Y

Trans for ma tion of equa tions Y Clas si fi ca tion of Me chan i cal sys tems Y Ki netic en ergy and gen er al ised ve loc i ties Y Gen er al ised forces Y Lagrangian

equa tions Y Lagrangian func tion Y Gen er al ised mo men tum Y Ki netic en ergy as a qua dratic func tion of ve loc i ties Y To re duce the prin ci ple of en ergy from

the Lagrange’s equa tions (Con ser va tive field) Y Small os cil la tions Y Lagrange’s equa tions with im pul sive forces Y Eu ler Dynamical Equa tions Y

Mov ing axes and the fixed axes Y Eu ler dy nam i cal equa tions Y Ki netic en ergy of a Rigid Body about a fixed point Y Eu ler’s equa tions (im pul sive forces) Y

Eulerian an gles and geo met ri cal re la tions Y In stan ta neous axis to ro ta tion Y In vari able line Y Lo cus of the in vari able line Y De duc tion of Eu ler’s equa tions

from Lagrange’s equa tions Y Hamiltonian For mu la tion and Variational Prin ci ples Y Ham il ton’s form of the equa tions of mo tion Y Phys i cal

35

–P.P. Gupta & G.S. Malik242-26 Rigid Dynamics-I (Dynamics of Rigid Bodies)

sig nif i cance of the Hamiltonian Y Pas sage from the Hamiltonian to the Lagrangian Y Variational meth ods Y Tech niques of Cal cu lus of vari a tions Y

Brachistochrone Prob lem Y Ex ten sion of the variational meth ods Y Ham il ton’s variational prin ci ples Y Der i va tion of Ham il ton’s Equa tions from the

variational prin ci ple Y Prin ci ple of least ac tion Y Dis tinc tion in be tween Ham il ton’s prin ci ple and principle of least action Y Deduction of Hamilton’s

principle using D’Alembert’s principle Y Extension of Hamilton’s principle to non-conservative and non-holonomic systems Y Motion of Top Y

Definition Y Equation or motion of a top (Derived from Euler’s equations) Y Equation of motion of a top (deduced from the principle of energy and

momentum) Y Equation of motion of top (deduce from Lagrange’s equations) Y Steady motion Y Stable motion (axis vertical) Y Stable motion (axis is not

vertical) Y Limits of θ.

Y Mechanics of a particle Y Velocity of a particle Y Acceleration of a particle Y Linear momentum of the particle Y Moment of force (or torque) and

angular momentum of the particle Y Work done by the force acting on a particle and kinetic energy Y Power Y Impulse Y Conservative force and force

field Y Conservative theorem for a particle Y The equation of motion of a particle D’ Alembert’s Principle Y Motion of a particle under resisting force Y

Motion in a resisting medium Y The simple Harmonic oscillator Y Damped Harmonic oscillator Y Two and three dimensional harmonic oscillator Y To

discuss the motion of a particle executing harmonic vibrations and to find its orbit and frequency Y Forced Harmonic oscillator Y Central Force Field

Motion Y Central forces Y Central orbit Y h pv= Y To obtain the law of force, velocity and period time when the orbit is an ellipse Y General features of the

centre force field motion Y Equations of motion for a particle in a central force field Y Conservation of energy for central force field Y Orbit under a central

force (contd.) Y Determination of the central force Y Kepler’s Laws of planetary motion -bounded motion under in an inverse square filed Y First integrals

of the two body motion under a central force Y Reduction of two body problem to one body problem Y Orbit under inverse square law Y Stability of a

nearly circular orbit Y Unbounded motion-scattering in a central forces field Y Rutherford’s scattering Y Centre of mass and Laboratory co-ordinates Y

Transformation of scattering data from C-system to L-system Y Motion of a System of Particles Y D’Alembert’s Principle Y Linear momentum of a

system of particles Y Torque on a system of particles Y Angular momentum of a system of particles Y Kinetic energy of a system of particles Y Potential

energy of a system of particles Y Conservation of energy for the system of particles Y Collision Problems Y Lagrangian-Dynamics Y Constraints and

Generalised co-ordinates Y Degrees of freedom Y Transformation Equations Y Classification of Mechanical system Y Kinetic energy and generalised

velocities Y Generalised forces Y Lagrange’s equations Y Lagrangian function Y Generalized momentum Y Kinetic energy as quadratic function of

velocities Y Equilibrium configuration for conservative holonomic dynamical system Y To deduce the principle of energy form the Lagrange’s equations

(conservative field) Y Theory of small oscillations of conservative Holonomic Dynamical systems Y Lagrange’s equations with Impulsive Forces Y

Lagrange’s equations for non-holonomic systems with moving constraints Y First integral of motion Y Velocities depending potentials Y Lagrangian for a

charged particle in an electromagnetic field Y Lagrange’s equations for electrical circuits Y Rigid Body Motion (Including Motion in Three

Dimensions) Y Degrees of freedom Y Orthogonal Transformations Y Eulerian angles Y Moving Frames of Reference Y Kinematics of a rigid body Y

Kinetic energy of a rigid with a fixed point Y Kinetic energy of body in general Y Angular momentum of particle and of a system of particles Y Angular

momentum of a rigid body Y Motion of a system Y Moving frames of Reference (continued) Y Motion of a rigid body (continued ) Y General motion of a

rigid body Y General equations of impulsive motion Y Theory of Small Oscillations Y Equations of Motion for small oscillations Y Normal

co-ordinates and normal modes of vibration Y Systems with a few and may degrees of freedom Y Hamiltonian Formulation, Transformations and

Hamilton-Jacobi Theory Y Hamiltonian Formulation Y Phase-space Y If the hamiltonian H is independent of t explicitly, prove that H is constant and

equal to the total energy of the system Y Passage from the Hamiltonian to the Lagrangian Y Ignoration of co-ordinates and Routh’s Procedure Y

Variational Methods (Hamilton’s principle etc.) Y Derivation of Hamilton’s equation from the variational principle Y Extension of Hamilton’s principle to

non conservative and non-holonomic system Y Principle of least action Y Principle of least action in terms of arc length of the particle trajectory Y Jacobi’s

form of the Principle of least action Y Fermat’s Principle Y Distinction between Hamilton’s Principle and principle of least action Y Derivation of

Lagrange’s equations from Hamilton’s Principle Y Conservation theorems and Symmetry Properties Y Homogeneity and Isotropy of space and time

conservation laws Y Virial Theorem Y Liouville’s Theorem Y Transformations and Brackets Y Point and canonical transformations Y Bilinear Invariant as

the condition for canonical transformations Y Generating Functions Y Poincare’s Integral Invariants Y Lagrangian and Poisson Brakcets Y Relation

between Lagrange’s and Poisson Brackets Y Equation in P.B. notation Y Infintiesimal contact transformation (I.C.T.) and Generators Y Generator of

translatory motion Y Contact transformation possesses the group property Y Point transformation Y To obtain an analytic expression for a contact

transformation Y Sub-groups of Mathieu transformations and extended point transformation Y Hamilton-Jacobi Theory Y Hamilton-Jacobi equation Y

Hamilton-Jacobi equation for Hamilton’s characteristic function Y The Hamiltonian being given by H m p p r rr=

+ −θ1

2

2 2 2{ ( / )} ( / )λ and to use

Hamilton-Jacobi theory to solve Kepler’s Problem for a particle in an inverse square central force field Y Harmonic oscillator problem as an example of the

Hamilton-Jacobi method Y Separation of variables Y Action angle variables Y Motion of Spinning Tops and Gyroscopes Y Simple motion of a top

(or Steady precession of a top) Y General motion of a top Y Steady motion Y Stability Investigation Y Gyroscopic compass Y Mechanics of Continuous

Media Y Equation of motion for the vibrating string Y Propagation of waves along a string Y String as a limiting case of a system of particles Y Lagrange’s

equations for vibrating string.

36

–P.P. Gupta & Sanjay Gupta243-14 Rigid Dynamics-II (Analytical Dynamics)

Y Basic Concepts of Set Y Sets Y No ta tion Y Fam ily of sets Y Equal ity of sets Y Fi nite and In fi nite sets Y Null set Y Power set Y Com pa ra bil ity of sets Y

Uni ver sal set Y Sin gle ton set Y In dexed set Y Ba sic Set Op er a tions Y Set op er a tions Y Un ion Y In ter sec tion Y Dis joint sets Y Dif fer ence of Sets.

Venn-Eu ler di a grams Y Sym met ric dif fer ence of a set Y Op er a tions on Com pa ra ble sets Y Dis trib u tive law Y As so cia tive law Y De-Mor gan’s the o rem

Y Or dered pairs Y Prod uct sets Y Re la tions Y In verse re la tion Y Types of Re la tions Y Equiv a lence re la tion Y Par tial or der Y Equiv a lence class Y

Quo tient sets Y Par ti tion of a sets Y An im por tant the o rem on equiv a lence re la tion Y Do main and Range Y Func tions Y Map Y Do main Y Range Y

Co-do main Y Im age Y Preimage Y Onto and Into maps Y One-one map Y Manyone map Y Types of map ping Y In verse of an el e ment Y In verse of a set.

In verse func tion Y Equal func tions Y Iden tity maps Y Con stant maps Y Prod uct of func tions Y As so cia tive op er a tion on prod uct of func tions Y Ex ter nal,

com po si tion Y Nat u ral map ping Y Bi nary op er a tions Y Com mu ta tive and As so cia tive laws Y Al ge braic struc ture narray op er a tion Y Graph of a func tion Y

Re stric tion and ex ten sion of a map Y Choice func tion Y Real val ued map Y Char ac ter is tic map Y Zero map Y Pro jec tion map Y The o rem of in verse

func tion Y Ab stract Char ac ter iza tion of In te gers and Rationals Y Def i ni tions Y Group Y Semi-group Y Com mu ta tive group Y Ring Field In te gral

do main Y Lin early or dered Archimedian field Y Or der Y Com plete Y Ab stract char ac ter iza tion of in te gers and rationals Y The o rems and solved ex am ples

on to tally or dered field and in te gral do mains Y Count abil ity of Sets and Car di nal Num bers Y Equiv a lent sets Y Car di nal num bers Y Sum and

prod uct of car di nal num ber Y Fi nite and in fi nite sets Y Denumerable sets Y Un count able Y Power of Con tin uum Y Com par i son of car di nal num bers Y

The o rems on car di nal num bers and equiv a lent sets Y Schroeder-Bernstein the o rem Y The set of all real num bers in [0, 1] is un count able Y Q and Z are

enumerable sets Y Can tor’s the o rem c.c.=c Y Can tor’s tennary set Y Con tin uum Hy poth e sis Y Or dered Sets Y Par tially or dered sets Y Uncomparable Y

Com pa ra ble Y To tally or dered set Y Bounds of an or dered set Y Max i mal and min i mal el e ments Y Least and great est el e ments Y Nest Y Or der com plete

Y Ini tial seg ment Y First and last el e ments Y Sim i lar ity map Y Im me di ate suc ces sor Y Pre de ces sor Y Or der type Y Sum and prod uct of or der types Y Well

or dered sets Y Prin ci ple of transfinite in duc tion Y Or di nal num ber Y Sum and prod uct of or di nal num bers Y Lex i co graph i cal or der ing Y The or der type

of the or dered set of all real num bers in (a, b) is λ Y Zermelo’s the o rem Y Prin ci ple of transfinite in duc tion Y Burali-Forti Par a dox Y The o rems and solved

ex am ples on or dered sets Y Ax iom of Choice, Zorn's Lemma and Kuratowoski's Lemma Y State ments of Ax iom of choice Y Choice func tion

Zermelo’s pos tu late Y Hausdorff max i mal Prin ci ple Y Zorn’s lemma Y Proof of Ax iom of choice as sum ing the truth of Zermelo’s postuate Y Proof of

Hausdorff max i mal prin ci ple as sum ing the truth of Tukey’s lemma Y Zorn’s lemma ⇒ Ax iom of choice Y Zorn’s lemma ⇒ Kuratowoski’s lemma Y

Kuratowoski’s lemma ⇒ Zorn’s lemma Y Limit of Se quences Y Cauchy’s first the o rem on lim its Y Cauchy’s sec ond the o rem on limit Y Cesaro’s

the o rem Y Quo tient the o rem Y Stolz’s the o rem Y Real Se quences (Se quences con tin ued) Y Kinds of se quences Y Fun da men tal the o rems on

se quences Y The Num ber Sys tem (Real Num bers) Y Peano ax i oms Y Ra tio nal num bers Y Tran scen den tal num bers Y Al ge braic num bers Y

Ar chi me dean the o rems Y Or der com plete ness the o rem Y Q and R dense sets Y Dedekind's The ory of Real Num bers Y Fun da men tal prop er ties of the

set of ra tio nal num bers Y Dedekind’s method of in tro duc ing an ir ra tio nal num ber Y Set of real num bers Y Real-ra tio nal num bers Y Real-ir ra tio nal

num bers Y Some the o rems Y Pos i tive, neg a tive and zero real num bers Y Or der re la tions in the set of real num bers Y Four ar ith met i cal op er a tions in the

set of real num bers Y The set of all real num bers is dense Y Or der com plete ness of the set of real num bers Y Can tor's The ory of Real Num bers and

Ab stract Char ac ter iza tion of Ra tio nal and Real Num bers Y Se quence of real num bers and ra tio nal num bers Y Con ver gent se quences

Y Ar ith met i cal the ory of lim its Y Null se quences Y Pos i tive se quences Y Cauchy se quences Y Cauchy’s gen eral prin ci ple of con ver gence Y The o rem on

Cauchy se quences and ab stract char ac ter iza tion of ra tio nal and real num bers Y Can tor’s def i ni tion of real num bers Y Pos i tive, neg a tive and zero real

num bers Y The four ar ith met i cal op er a tions in the set of real num bers Y The set of real num bers is dense Y Dedekind’s sec tion cor re spond ing to Cauchy

se quence of rationals Y Equiv a lence of the def i ni tions of Dedekind and Can tors Y Ar ith met i cal the ory of lim its based on Can tor’s the ory of real num bers Y

Bounded, De rived, Com pact, Open and Closed Sets Y Interval Y Neighbourhood of a point Y Lim it ing point Y Derived set Y Closed set Y Species

and or der of a set Y Interior point Y Open set Y Closure of a set Y Boundary of a set Y Dense set Y Lvery where dense set Y Non-dense set Y Perfect set Y

Isolated sets Y Bounds of a lin ear set Y Bounded and un bounded sets Y Cover Y Open cover, Sub cover Y Com pact set Y Nested closed In ter val property

Y Bolzano-Weierstrass theorem Y Heine Borel theorem Y Perfect set Theorems open and closed sets Y Theorems on derived sets.

Y Spherical Trigonometry Y Sphere Y Section of a Sphere by a Plane Y Great and small circles Y Shortest distance Between two Points on a Sphere Y

Axis and Poles of a Circle Y Secondaries Y Two Great Circles Bisect each other Y Arc Joining Poles Y Measurement of the Spherical Angle Y Lengrh of

arc of a Small Circle Y Spherical Triangles Y Polar Triangles Y Some Properties of Spherical Triangles Y Formulae Relating to an Oblique Spherical

Triangle Y Napier’s Analogies Y Delambre’s Analogies Y Right Angled Triangles Y Lune Y Application of Spherical Trignometry Y Cagnoli’s Theorem Y

L’Huilier’s Theorem Y Expressions for cos (E/2) and tan (E/2) Y Location of a Point on Earth’s Surface Y The Celestial Sphere Y Definitions Y Diurnal

Motion of Heavenly Bodies Y Annual motion of the Sun Y System of Co-ordinates Y Hour Angle Y The Altitude of the Pole Y Conversion of co-ordinates

from one System to Another Y Equinoxes and Solstices Y Rectangular Co-ordinates Y The Geo-centric Celestial Sphere Y Sidereal Time Y

37

–K.P. Gupta244-10 Set Theory and Related Topics

–S.K. Sharma, R.K. Gupta & D. Kumar245-18 Spherical Astronomy

Trigonometrical Ratios of small Angles Y Rising and Setting of Stars Y Relation between Circular measure and Radians Y Rate of Change of Zenith

Distance (z) and Azimuth (A) Y Motion of the Sun Y Twilight Y Refraction Y Laws of Refraction Y Apparent and True Positions Y Refraction of a Star

near the Zenith Y Representation of true and Apparent Positions on the Celestial Sphere Y Cassini’s hypothesis or homogeneous shell Y Differential

Equation for Refraction Y Simpson’s Hypothesis Y Bradley’s Formula Y Effect of Refraction on Sunrise and Sunset Y Effect of Refraction in

Right-Ascension and Declination Y Refraction in any direction Y Effect of refraction on the distance between two neighbouring stars Y Effect of refraction

on the shape of the disc of the Sun Y Aberration Y Definition Y Aberration Varies as Sine of the Earth’s Way Y Position of the Apex Y Representation of

Apparent and true Position on Account of Aberration on the Celestial Sphere Y Effect of Aberration on Longitude and Latitude Y The Aberration Ellipse Y

Effect of Aberration on Right Ascension and Declination Y Independent Day Numbers Y Aberration in any Direction Y Effect of Aberration on the

Distance between Two Stars Y Diurnal Aberration Y Effect of Diurnal Aberration in Declination, right Ascension and Hour Angle Y Planetary Aberration Y

Precession and Nutation Y Precession Y Nutation Y Physical Cause of Precession and Nutation Y Luni Solar Precession Y Planetary Precession Y

Effect of Precession on right Ascension and Declination Y Effect of Nutation on right Ascension and Declination Y Combined Effect of Precession and

Nutation in right Ascension and Declination Y Independent Day numbers Y Double Stars Y Position Angle of a Double Star Y Parallax Y Definitions

Y Geo-centric Parallax Y Geo-centric Parallax in Right Ascension and Declination Y Gec-centric Parallax in Azimuth and Zenith Distance Y The Moon’s

Size Y Steller or Annual Parallax Y Representation on the Celestial Sphere Y Annual Parallax in Longitude and Latitude Y The Parallactic Ellipse Y Steller

Parallax in right Ascension and Declination Y Time Y Definitions Y The Sun’s Apparent Orbit Y The Mean Sun Y The Equation of Time Y To Prove that

the Equation of time Vanishes four Times in a Year Y Seasons Y Length of the Seasons Y Kepler's Laws of Planetary Motion Y Deduction of Kepler’s

laws from Newton’s Law of Gravitation Y The Definitions Y To Express true Anomaly in terms of Eccentric Anomaly Y Kepler’s Equation Y Kepler’s

Problem Y To Express true Anomaly in terms of Mean Anomaly Y Velocity and Position of a Body in an Elliptic Orbit Y Lambert’s theorem, Time of

Describing an Elliptic Orbit Y Velocity and Position of a Body in a Parabolic Orbit Y Euler’s Theorem, Time of Describing a Parabolic Orbit Y Velocity and

Position of a Body in Hyperbolic Orbit Y Planetary Phenomena Y Conjunctions Y Sidereal Period Y Relation between the Sidereal and Synodic Period

Y Direct and Retrograde Motion Y The Geo-centric Motion of a Planet Y Elongation Y Elongation of a Planet when Stationary Y Phases Y Phase of the

Moon Y Brightness Y The Meridian Circle Y The Three Errors Y Correction in the Observed time of Transit due to the Errors YThe Total Correction to

the Observed time of Transit Y Bessel’s Formula Y Eclipses Y Eclipses of the Moon Y Section of the Shadow Y Notations Y The Angular Radius of the

Earth’s Shadow at the Moon’s Distances Y Duration of an Eclipse Y Length of the Earth’s Shadow Y The Ecliptic Limits Y Calculation of the Lunar Eclipse

Y Points on the Moon where the Eclipse Commences Y Solar eclipse Y The Angle Subtended at the Earth’s centre by the centres of the Sun and the Moon

at the beginning or end of a Solar Eclipse Y Solar Ecliptic Limits Y Frequency of eclipses Y The Saros Y The Metonic Cycle Y Determination of

Position Y The Dip of the Horizon Y Dip of the Horizon Taking Refraction Under Consideration Y The Position Circle Y Artificial Satellites and

Atmospheric Drag Y Theory of an Orbit in Space Y Clairaut’s Formula for the Shape of the Earth Y The Effect of Atmospheric Drag on an Artificial

Satellite Y Rocket Dynamics Y Motion of a Rocket in gravity free Space Y Motion of a Rocket in a Gravitation field Y Motion of a Rocket or Step Rocket

Y Two Stage Rocket Y Transfer Between Orbit Y Changes in the Orbital Elements due to a small and Large Impulses.

Y Forces in Three Di men sions Part I—Cen tral axis Y Forces in three di men sions Y Mo ment of a force about a point Y Def i ni tions Y Gen eral

con di tions of equi lib rium of a rigid body Y Forces in Three Di men sion Part 2—Con strained Bod ies Y Def. Con strained Bod ies Y Con di tion of

equi lib rium of a rigid body with one points fixed Y If a rigid body is con strained to turn about two fixed points under the in flu ence of ex ter nal forces, then

to de ter mine the func tions of equi lib rium Y Forces in Three Di men sion Part 3– Screws and Wrenches Y Forces in Three Di men sions Part 4–

Null lines and Null Planes Y Stable and Unstable Equilibrium (Two and Three Dimension) Y Definitions Y Theorems Y Strings in Two

Dimensions Y Suspension bridge Y Equilibrium of a light inextensible string resting on a smooth curve Y Equilibrium of a heavy inextensible string on a

smooth curve in a vertical plane Y Equilibrium of a light inextensible string resting in equilibrium on a rough plane under the action of no external force Y

Equilibrium of a heavy inextensible string resting in limiting equilibrium on rough plane under the action of no external fores Y Central forces Y Extensible

strings Y Theorem Y Elastic string, Theorem Y Strings in Three Dimensions Y Equilibrium of a string under any forces Y Equilibrium of string on any

surface Y String on the surface of revolution Y Heavy string on a sphere Y Heavy string on a cylindrical surface Y String on a right cone Y String on a rough

surface Y Elastic string on a surface Y Virtual Work Part I–Virtual work in two dimensions Y Work Y Theorem Y Virtual Work and Virtual

Displacement Y Principle of Virtual Work ( a system of coplanar forces acting on a rigid body) Y Forces which may be omitted Y Tension of a string or

Thrust in a rod Y Method of Solving the problems Y Attraction and Potential Y The law of attraction Y Attraction Y Attraction of a rod Y Attraction of a

thin uniform spherical shell Y Attraction of a solid sphere Y Potential (Definition) Y Relation between the attraction and potential Y Potential of a finite rod

Y Potential of an infinite rod Y Potential of a Circular disc Y Potential of a spherical shell Y Potential of a solid sphere.

38

–J.K. Goyal K.P. Gupta246-15 Statics (With Attraction & Potential)

Y Tensor Algebra Y Space of N-dimensions Y Curve Y Transformation of co-ordinates Y Summation convention Y Indicial (or Range) convention Y

Dummy suffix Y Contravariant and covariant vectors (Tensor of first order) Y Tensors of second order (or of rank two) Y The kroneckor delta Y Tensor of

higher rank (or higher orders) Y Invariant or scalar Y Algebraic operations with tensors Y Addition and subtraction of tensors Y Contraction Y Product of

tensors Y Inner Product Y Symmetric Tensor Y Skew-symmetric Tensor Y Quotient law Y Generalized Quotient law Y Conjugate (or Reciprocal)

symmetric tensor Y Relative tensor Y Some theorems on groups Y Tensor field Y Metric Tensor and Riemannian Space Y The metric tensor,

Riemannian metric, Riemannian space Y Fundamental contravariant tensor Y Length of a curve and Null curve Y Associated tensors, Raising and

Lowering of indices Y Magnitude of a vector, Unit vector, Null vector, Scalar product Y Angle between two vectors Y Co-ordinates curves Y Hypersurface

Y Angle between two hypersurfaces Y Angle between two co-ordinate hypersurfaces Y N-ply Orthogonal system of hypersurfaces in a VN Y Congruence

of curves Y Orthogonal ennuple Y Principal directions for a symmetric covariant tensor of second order Y Homogeneous space Y Euclidean space of

m-dimensions Class of VN Y Gradient Y Christoffel's Three-Index Symbols (or Brackets), Covariant Differentiation Y The Christoffel three

Index symbols Y Transformation of Christoffel symbols Y Covariant differentiation of vectors Y Covariant differentiation of tensors Y Intrinsic derivative of

tensor Y Laws of covariant differentiation of tensors Y Covariant derivative of a scalar Y Derived vector Y Tendency of a vector Y Cross product of two

vectors in tensor notation Y Covariant constants, Ricci’s theorem Y Derived vector Divergence of a vector Y Curl of vector Y Laplacian operator Y Some

Important Identities Y Curvature of a Curve, Geodesics Y Curvature of a curve Y First curvature Y Principal normal Y Geodesics Y Euler’s condition Y

Differential equations of Geodesics in a VN Y Geodesic co-ordinates Y Riemannian co-ordinates Y Geodesic form of the line (or linear) element Y

Geodesics in Euclidean space SN Y Parallelism, Generalised Covariant Differentiation Y Parallelism of vector of constant magnitude (Levi Civita’s

Concept) Y Some theorems Y Parallelism for vector of variable magnitude along a curve Y Sub-spaces of a Riemannian Manifold Y Some theorems on

Subspaces Y Parallelism in subspace Y Properties of Vm Y The Fundamental Theorem of Local Riemannian Geometry Y Generalised covariant

Differentiation or Tensor Differentiation Y Laws of Tensor Differentiation Y Riemann Symbols and Curvature Tensor Y Riemann Christoffel Tensor

or curvature tensor Y Riemannian’s symbols of the second kind Y Ricci Tensor Y Covariant curvature tensor and Riemannian symbol of first kind Y

Bianchi ldentity Y Riemannian curvature of VN at a point Y Formula for Riemannian curvature Y Flat space Y Schur’s Theorem Y Mean Curvature

Geometrical interpretation of the Ricci tensor Y Ricci’s Principal directions Y Einstein space Y Weyl Tensor Y Ricci's Coefficients of Rotation,

Congruences Y Ricci’s coefficients of rotation Y Geometrical interpretation of link Reason for the name, "Ricci Coefficient of Rotation" Y To find

curvature of a congruence Y Geodesic congruence Y Normal Congruence Y Curl of congruence lrrotational Congruence Y Congruence canonical with

regard to a given congruence Y Linear sum and difference of two congruences Y Hyper Surfaces Y Gauss’s Formulae Y Second Fundamental form Y

Curvature of a curve in a hypersurface and normal curvature of a hypersurface Y Meunier’s Theorem Y Dupin’s Theorem Y Some definitions Y Euler’s

Formula Y Conjugate directions in a hypersurface Y Asymptotic direction in a hypersurface Y Umbilical points Y Totally geodesic hypersurface Y Tensor

derivative of the unit normal Y The Gauss and Codazzi equations Y Gauss Formulae Y Curvature of a curve in a subspace Y Line of curvature for a given

normal Y Central quadric hypersurfaces in Euclidean space Y Polar hyperplane Y Evolute of hypersurface vn in a Euclidean space SN + Y Hypersphere Y

Y The e-Systems and Generalized Kronecker Delta Y Completely symmetric Y Completely skew-symmetric Y e-System Y Generalized Kronecker

Delta Y Two theorems Y Contraction of δlmn

pqrY Appendix, Some Preliminaries Y Tensor Analysis Y Determinants Y Differentiation of a

determinant Y Jacobian or Functional determinant Y Matrices Y Linear Equation.

Y Classical Theory of Relativity: Speed of Light Y Inertial Frame (Galilean Frame) Y Galilean Transformations Y Fictitious Force Y

Electrodynamics Y Fizeau's Experiment Y Michelson and Morley Experiment Y Explanation of Negative Results (Null Results) Y Lorentz

Transformations Y The New Concept of Space and Time Y Postulates of Special Theory of Relativity Y Lorentz Transformation Equations Y

Consequences of Lorentz Transformations Y Time Dilation or Apparent Retardation of Rest Y An Interesting Example of Time Dilation Y Experimental

Verification on Time Dilation Y Simultaneity Y Relativistic Formulae for Composition of Velocities Y Relativistic Formulae for Composition of

Accelerations Y Relativity of Time: Proper Time Y Lorentz Transformation Form a Group Y Aberration (Relativistic Treatment) Y Doppler's Effect Y

Confirmation of Doppler Effect Y Relativistic Mechanics Y Mass and Momentum Y Newton's Laws of Motion Y Measurement of Different Units Y

Experimental Verification of the Relation Y Transformation Formula for Mass Y Transformation Formula for Momentum and Energy Y Particle with Rest

Mass Zero Y Binding Energy Y Transformation Formula for Force Y Relativistic Transformation Formula for Density Y Minkowski Space (Four

Dimensional Continuum) Y Geometrical Interpretation of Lorentz Transformation Y Space and Time Like Intervals Y World Points and World Lines Y

Light Cone Y Proper Time Y Energy Momentum Four Vector Y Four Vector (World Vectors) Y Relativistic Equations of Motion Y Minkowski's Equation of

Motion Y Special Relativity in Classical Mechanics Y Lorentz Transformation Y Relativistic Lagrangian and Hamiltonian Y Relativistic Hamiltonian

39

–D.C. Agarwal247-27 Tensor Calculus and Riemannian Geometry

–J.K. Goyal & K.P. Gupta248-27 Theory of Relativity

Y Tensor Calculus Y Tensor and Line Element Y Summation Convention Y Dummy Suffix Y Real Suffix Y Kronecker Delta Y Determinant Y Four

Vectors (World vectors) Y Transformation of Co-ordinates Y Tensor Y Symmetric Tensor Y Anti-symmetric Tensor Y Addition of Tensor Y Inner Product

of Two Vectors Y Multiplication of Tensors Y Contraction Y Reciprocal Symmetric Tensor Y Relative Tensor Y Riemannian Metric Y Associate Tensors Y

Magnitude of a Vector Y Angle between Two Vectors Y Geodesic Curves, Covariant Differentiation Y Christoffel Symbols Y Geodesic Y Covariant

Differentiation of Tensor Y Gradient of a Scalar Y Derived Vector Projection Y Tendency of Vector Y Curl of a Vector Y Divergence of a Vector Y Parallel

Displacement of Vectors Y Principal Normal Y Geodesic Co-ordinates Y Natural Co-ordinates Y Curvature Tensor Y Riemannian Christoffel's Tensor Y

Covariant Curvature Tensor Y Flat Space Time Y Space of Constant Curvature Y The General Theory of Relativity Y Principle of Covariance Y

Principle of Equivalence Y Relativistic Field Equations Y Energy Momentum Tensor Y Schwarzschild Solution Y Isotropic Co-ordinates Y Crucial

Tests in General Relativity Y Cosmology Y Cosmological Models Y Electrodynamics Y Introduction (Maxwell's Equations) Y Gauge Transformation Y

Covariant Form of Lorentz Condition and Equation of Continuity Y Electromagnetic Energy Momentum Tensor Y Energy and Momentum of the

Electromagnetic Field Y Electromagnetic Stress Y Some Applications of Special Theory of Relativity Y Compton Effect Y Experiment on Compton

Scattering Y To Discuss De-Broglie Hypothesis of Matter Waves Y Non-Static Cos mo log i cal Model Y Der i va tion of the Rob ert son Walker Line

El e ment.

Y Elements of Set Theory Y Sets and Subsets Y Quantifiers ∀ ∃, Y Set Y Sets of Numbers Y Venn-Euler Diagrams Y Basic Operations on Sets Y

Cartesian Product of Two Sets Y Relations Y Equivalence Classes Y Partitions Y Quotient Set Y Functions Y Sequences Y Real-valued Functions Y

Characteristic Function Y Intervals Y Inverse Mapping Y Product or Composition of Mappings Y Partial Order Relations Y First and Last; Maximal and

Minimal Elements Y Some Properties of Real Numbers Y Zorn's Lemma, Well Ordering and Countability Y The Axiom of Choice and Its Equivalents

Y Cartesian Product of Arbitrary Collection of Sets Y Lemma Y *Tukey’s Lemma Y *Hausdorff Maximal Principle Y *Zorn’s Lemma Y Well Ordering

Theorem Y Well-ordered Set Y Complete Order Y Cardinality and Denumerability Y Schroeder-Bernstein Theorem Y Denumerable Sets Y Decimal,

Ternary and Binary Representations Y Cardinal Arithmetic Y Exponentiation Y Cantor’s Ternary Set Y Order Types and Ordinal Numbers Y Initial

Segments Y Metric Spaces Y The Real Line R Y Sequences in R Y Metric Y Euclidean Spaces Y Some Important Inequalities Y Bounded and

Unbounded Metric Spaces Y Some Important Metric Spaces Y Sequence Spaces Y Sphere (or Balls) Y Open Sets Y Closed Sets Y Neighbourhood Y

Accumulation Points Y Closure, Interior, Exterior and Boundary of a Set Y Dense and Non-dense Sets Y Sequences and Subsequences in a Metric Space

Y Cauchy Sequences Y Complete Metric Spaces Y Baire Category Theorem Y Completeness and Contracting Mappings Y Banach’s Fixed Point Theorem

Y Some Complete Metric Spaces Y Completion of Metric Space Y Topological Spaces Y Topologies Y Intersection and Union of Topologies Y Metric

Topologies Y Metrizable Spaces Y Equivalent Metrics Y Closed Sets Y Neighbourhood Y Base for the Neighbourhood System of a Point, Base for a

Topology Y Topological Space Generated by Collection of Sets Y Limit Points, Adherent Points and Derived Sets Y Hausdorff Spaces Y Closure Y

Interior, Exterior and Frontier of a Set Y Separable Spaces Y Relations between Closure, Interior and Frontier Y Sub-spaces Y Finite Product of

Topological Space Y Continuity and Homeomorphism Y Continuity Y Continuity of the Composite Function Y Sequential Continuity Y The Pasting

Lemma Y Homeomorphism Y Topological Property Y Uniform Continuity Y Connectedness Y Separated Sets Y Connected and Disconnected Sets Y

Continuity and Connectedness Y Components Y Totally Disconnected Space Y Locally Connected Spaces Y Arcwise Connectivity Y Compactness Y

Open Cover (or covering) Y Compact Y Reducible to Finite Subcover Y Basic and Sub-basic open covers Y Non-compact Y Compact Sub-space Y Finite

Intersection Property (FIP) Y Bolzano Weierstrass Property (BWP) Y Compactness in R Y Compactness in Rn Y Countable, Sequential and Local

Compactness Y Compactness in Metric Space Y Continuity and Compactness Y Uniform Continuity and Compactness Y Continuity and Local

Compactness Y Countability and Separation Axioms Y First Countable Spaces Y Second Countable Spaces Y Separable Spaces Y T0 -Spaces or

Kolmogorov Spaces Y T1 - Spaces or Frechet’s Separation Axiom Y T2 - Spaces or Hausdorff Spaces Y Regular Spaces: T3 - spaces Y Normal Space:

T4 - spaces Y Completely Normal Spaces Y Completely Regular Spaces, Tychonoff Spaces Y One-point Compactification Y Product and Quotient

Spaces Y Weak Topologies Y Lattice Y Product Space of Two Spaces Y Projection Mappings Y Product Invariant Properties for Finite Products Y

General Product Spaces (Tychonoff Topology) Y Product Topology (or Tychonoff Topology) Y Product Invariant Properties Y Tychonoff Theorem Y

Embedding Theorems and Metrizability Y Quotient Spaces Y Upper Semi-continuous Decomposition Y The Stone-Cech Compactification Y

Convergence (Net and Filters) Y Sequence in Topological Spaces Y Nets Y Subnets and Cluster Points Y Filters Y Filters Generated by Collection of

Sets Y Filter Base Y Ultrafilters Y Convergence of Filters Y Cluster Points of a Filter Y Metrization Theorems and Paracompactness Y Local Finiteness

Y The Nagata-Smirnov Metrization Theorem Y Paracompactness Y The Smirnov Metrization Theorem Y The Fundamental Group and Covering

Spaces Y Homotopy of Paths Y Homotopy Equivalence Y Retraction and Deformation Y The Fundamental Group Y α-hat Y Covering Spaces Y

Fundamental Group of The Circle Y Lifting Lemma Y Covering Homotopy Lemma Y The Fundamental Group of a Product Space Y The Fundamental

Theorem of Algebra.

40

–J.N. Sharma & J.P. Chauhan249-45 Topology (General & Algebraic)

Y Mathematical Logic Y Statements Y Negation of a Statement Y Conjunction Y Disjunction Y Truth Tables Y Conditional and Bi-conditional

Statements Y Propositional Functions and Propositional Variables Y Tautologies and Contradictions Y Equivalent Statements or Functions Y Law of

Duality Y Functionally Complete Set of Operations Y Quantifiers Y Arguments Y Sets, Re la tions and Func tions Y Methods of Describing Sets Y

Equality of Sets Y Operations on Sets Y Laws of Algebra of Sets Y Cartesian Product of Sets Y Functions Y Binary Operations or Binary Compositions Y

Composition Table Y Relations Y Difference between Relations and Functions Y Properties of Relations on a Set Y Equivalence Relations Y Equivalence

Classes Y Partitions Y Partially Ordered Sets Y Countable and Uncountable Sets Y Cardinality of Sets Y Lattices Y Lattices as Partially Ordered Sets Y

Some Properties of Lattices Y Lattices as Algebric Systems (second definition of Lattices) Y Sublattices Y Direct Product of Lattices Y Isomorphic Lattices Y

Bounded Lattices Y Complements Complemented Lattices Y Cover of an Element, Atoms and Irreducible Elements Y Modular Lattices Y Distributive

Lattices Y Discrete Numeric Functions and Generating Functions Y Discrete Numeric Functions Y Asymptotic behaviour of Numeric Functions Y

The big O-Notations Y Generating Functions Y Solution of Combinatorial Problems using Generating Functions Y Solution of Recurrence Relations by

Generating Functions Y Boolean Al ge bra Y Subalgebra Y Boolean Algebra as Lattices Y Representation Theorem for Finite Boolean Algebra Y

Boolean Functions Y Conjugate Normal form Y Minimization of Boolean Functions/Karnaugh Maps Y Switch ing Cir cuits and Logic Cir cuits Y

Switching Circuits Y Simplification of Circuits Y Non-series Parallel Circuits Y Relay Circuits Y Logic Circuits Y Design of Circuits from given Properties Y

Ba sic Con cepts in Graph The ory Y Graphs Y Isomorphic Graphs Y Subgraphs and Complements Y Walk, paths and circuits Y Connected Graphs

and Components Y Operations on Graphs Y Special Graphs Y Eulerian Graphs Y Hamiltonian Graphs Y Weighted Graphs Y Trees and Cut-sets Y

Trees Y Distance and centres in a Tree Y Spanning Trees Y Fundamental circuits Y Minimal (shortest) Spanning Trees Y Cut-sets Y Edge Connectivity,

Vertex Connectivity and Separability Y Planer Graphs and Colourings Y Planer Graphs Y Euler's Formula Y Detection of Planarity Y Dual Graphs Y

Thickness and Crossings Y Colouring of Graphs Y Five Colour Problem Y Chromatic Partitioning and Independent Sets Y Chromatic Polynomial Y

Ma tri ces as so ci ated with Graphs Y Incidence Matrix Y Adjacency Matrix Y Path Matrix Y Circuit Matrix Y Cut-set Matrix Y Di rected Graphs Y Basic

Definitions and Concepts Y Euler Digraphs Y Rooted Trees and Binary Trees Y A cyclic Digraphs Y Matrices in Digraphs Y Gen eral Count ing Meth ods

Y Sum and Product Rules Y Permutations Y Combinations Y The Pigeonhole principle Y The Inclusion-exclusion Principle Y Formal Languages,

Grammars and Finite State Machines Y Languages and Grammars Y Types of Grammars and Languages Y Regular Sets and Regular Languages Y

Finite state Machines Y Machine Minimization Y Finite State Machines as Language Recognizers Y Semigroups and Monoids Y Algebraic Structure Y

Semigroups Y Homomorphism of Semigroups Y Monoids Y Homomorphism of Monoids Y Congruence Relation and Quotient Semigroups.

Y Differential Equations Y Elementary Integration Y Definitions Y Constant of Integration Y Some properties of integral Y Fundamental integration

formulae Y Extended forms of fundamental formula Y Methods of integration Y Integration by substitution Y Integral of the product of two functions

Y Integration by parts as applied to the functions of the type e f x f xx [ ( ) '( )]+ Y Integrals of e bxax sin and e bxax cos Y Some Special Integrals YThree

special integrals ( ) ( )dx

a x

dx

x ax

dx

a xx a

2 2 2 2 2 2+ −>

−<

∫∫∫ ; , ; ,a Y Evaluation of integrals of various types by using standard results Y Three more special

integrals dx

a x

dx

x a

dx

x a2 2 2 2 2 2− − +

∫∫∫ ; ; Y Evaluation of integrals of various types ( ) ( )1 2 2/ ; /ax bx c px q ax bx c+ + + + +

by using standard

results Y Three more special integrals a x dx x a dx x a dx2 2 2 2 2 2− −

+ ∫ ∫ ∫; ; Y Evaluation of integrals of varius types

ax bx c2 + +∫ ; ( )px q ax bx c+ + +∫ 2 ; ( )/px qx r ax bx c2 2+ + + +

∫ by using standard results Y Integration of some special irrational algebraic

fractions ( )

1 12ax b cx d ax bx c Ax B+ + + + +

;( )

;

( ) ( ) ( )1 1

2 2 2px q ax bx c Ax B Cx D+ + + + +

; Y Integrals of the type

dx

a b x

dx

a b x

dx

a x b x

dx

a x b x c+ + + + +∫cos,

sin,

sin cos,

sin cos∫∫∫ Y Integration of P x Q x R

a x b x c

cos sin

cos sin

+ ++ +

Y Integration of sin cosm nx x Y Integration

Using Partial Fractions Y Rational fractions Y Partial fractions Y Integration using partial fractions Y Definite Integrals Y Definition Y Evaluation of

definite integrals Y Results regarding trignometric functions Y Subsitutions in the case of definite integrals Y Fundamental properties of definite integralas Y

41

–A.R. Vasishtha & Others251-04 (B) Advanced Mathematics for Pharmacists

–M.K. Gupta250-15 Discrete Mathematics

The definite integral as the limit of a sum Y Differential Equations of First Order and First Degree Y Definitons Y Differential equations of first

order and first degree Y Variables separable Y Homogeneous equations Y Equations reducible to homogeneous form Y Linear differential equations Y

Equations reducible to the linear form Y Exact differential equations Y Integrating factors Y Change of variables Y Linear Differential Equations with

Constant Coefficients Y Defintions Y Determination of complementary function (C.F.) Y The Particular Integral (P.I.) Y Particular integral in some

special cases Y To find P.I. when Q e Va x= , where V is any function of x Y To find P.I. when Q eax= and F a C( ) = Y To find P.I. when Q ax= sin or

cos ax and F a( )− =2 0 Y To find P.I. when Q xV= , where V is any function of x Y The operator 1

D − αα, being a constant Y Ordinary Simultaneous

Differential Equations Y Methods of solving simultaneous linear differential equations with constant coefficients Y Number of arbitrary constants Y

Simultaneous equations of the form P dx Q dy R dz P dx Q dy R dz1 1 1 2 2 20 0+ + = + + =, , where P P Q Q R1 2 1 2 1, , , , and R2 are functions of x,y,z Y

Geometrical interpretation of the differrential equations dx

P

dy

Q

dz

R= = Y Linear Dependence and Independence of Solutions of Equations Y

Linear dependence and independence of solutions of an equation Y Fundamental set of solutions Y Wronskian Y Theorem Y Biometrics Y Data

Collection Y Primary and Secondary data Y Collection of primary data Y Collection of secondary data Y Limitations of secondary data Y Census and

Sampling Y Population and Sample Y Census and sample enquiry Y Census versus sample enquiry Y Fundamental principles of sampling theory Y

Method of sampling Y Organisation of Data Y Classification of data Y Object of classification Y Basis of classification Y Classification according to

attributes Y Classification by variables Y Frequency distribution Y Sturge's rule for number of classes and size of class interval Y Cummulative frequency

distribution Y Diagrammatic Representation of Data Y Importance and utility of diagrams Y Limitations of diagrams Y Rules for construction of

diagrams Y Types of diagrams Y Graphic Representation of Data Y The histogram Y The frequency polygon Y The frequency curve Y Cummulative

frequency curve or ogive Y Graphs of time series or line graphs Y Measures of Central Tendency Y Objectives of average Y Characteristics of a good

average Y Various measures of central tendency Y Some special problems relating to arithmetic mean Y Properties of arithmetic mean Y Correcting

incorrect values Y Merits and demerits of arithmetic mean Y Median Y Calculation of median Y Properties of median Y Advantages of the median Y

Partition values Y Graphical determination of median quartiles etc. Y Quartiles Y Mode Y Calculation of mode Y Determination of mode from mean and

median Y Mea sures of Despersion Y Objects and importance of dispersion Y Characteristics for a satisfactory measure of dispersion Y Absolute and

relative measure of variation Y Measures of dispersion combined standard deviation Y Correcting incorrect values of mean and standard deviation Y

Coefficient of variation Y Mathematical properties of standard deviation Y Choice of suitable measure of dispersion Y Mea sures of Skew ness and

Kurtosis Y Skewness Y Measure of skewness Y Moments Y Conversion of moments about an arbitrary origin into moments about mean Y Utility of

moments Y Kurtosis Y Measure of Kurtosis Y Cor re la tion and Re gres sion Y Types of correlation Y Methods of determining correlation Y Regression Y

Linear and non-linear regression Y Regression lines Y Another form of regression lines Y Method of fitting regression lines Y Prob a bil ity Y A priori or

classical definition of probability Y A posteriori or empirical probability Y Algebra of events Y Probability defined on events Y Permutations and

combinations Y Probability of a simple event Y Addition rule of probability Y Addition rule, when events are not mutually exclusive Y Independence and

the multiplication rule Y Conditional probability Y Probability of at least one event Y Odds in favour and odds against Y Probability based on Bernouli's

trials Y Inverse probability Y Baye’s theorem Y Ran dom Vari able and Prob a bil ity Dis tri bu tion Y Random variable Y Probability distribution Y Mean

and variance of random variable Y Binomial and poisson distribution Y Coefficients of the binomials Y Characteristics of binomial distribution Y

Recurrence formula for the probabities of binomial distribution Y Poisson distribution Y Characteristics of poisson distribution Y Recurrence formula for

the probabilities of poisson distibution Y Normal distribution Y Definiton Y Some properties of normal distribution Y Standard form of the normal

distribution Y Area under the normal curve Y Method of consulting table Y Fitting of normal distribution Y Sta tis ti cal In fer ence Y Population and

sample Y Parameter and statistic Y Sampling distribution of the statistic Y Standard error of the statistic Y Utility of standard error Y Statistical inference Y

Errors in hypothesis testing Y Procedure of test of significance Y Various tests of significance Y Test of significance based on t-distribution Y Test of

significance based on F- distribution Y χ2 - distribution Y Tests based on χ2 distribution Y Analysis of variance Y Assumptions Y The basic principle of

Anova Y Analysis of variance of one way classified data Y Short-cut method Y Coding method Y Statistical Tables Y Area Under The Standard Normal

Curve Y Ordinates of The Standard Normal Curve Y Critical Values of t-distribution Y Critical Values of χ2- distribution Y Percentage Points of The F

Distribution (Upper 1% Points) Y Percentage Points of The F Distribution (Upper 5% Points) Y Value of e m− Y Logarithms Y Antilogarithms Y Ap pen dix.

Y Multiple Products of Vectors Y Vectorss Y Addition of Vectors Y Subtraction of Two Vectors Y Multiplication of A Vector By A Scalar or Scalar Multiple of A

Vector Y Rectangular Components of A Vector in Three Dimensions Y The Scalar or Dot Product of Two Vectors Y The Vector Product or Cross Product of

Two Vectors Y Properties of Vector Product Y Triple Products Y Scalar Triple Product Y Distributive Law For Vector Product Y Properties of Scalar Triple

Product Y To express the value of the scalar triple product [abc] in Terms of Rectangular Components of the vectors Y To Express the Scalar Triple Product [a,b,c]

42

–A.R. Vasishtha & A.K. Vasishtha252-03 (B) Basic Mathematics for Chemists

In Terms of Any Three Non- coplanar Vectors l, m, n Y Vector Triple Product Y Vector Triple Product is not Associative Y Scalar Product of Four Vectors Y

Vector Product of Four Vectors Y Reciprocal System of Vectors Y Differentiation and Integration of Vectors Y Vector Function Y Scalar Fields and

Vector Fields Y Limit and Continuity of A Vector Function Y Derivative of A Vector Function With Respect to A Scalar Y Differentiation Formulae Y

Derivative of A function of A Function Y Derivative of A Constant Vector Y Derivative of A Vector Function In Terms of Its Components Y Some Important

Results Y Integration of vector functions Y Some Standard Results Y Gradient, Divergence and Curl Y Partial Derivatives of Vectors Y The Vector

Differential Operator Del ( )∇ Y Gradient of A Scalar Field Y Formulas Involving Gradient Y Equipotential Surfaces or Level Surfaces Y Directional Derivative of

A Scalar Point Function Y Tangent plane And Normal To A level Surface Y Divergence of A vector Point Function Y Curl of A Vector Point Function Y The

Laplacian Operator ∇2 Y Physical Interpretation of Divergence And Curl Y Some Important Vector Identities Y Green's, Gauss's and Stoke's Theorems Y

Some Preliminary Concepts Y Line Integrals Y Surface Integrals Y Volume Integrals Y Green's Theorem In the Plane Y Green's Theorem In The Plane In Vector

Notation Y Applications Of Green's Theorem Y The Divergence Theorem of Gauss Y Some Deductions From Divergence Theorem Y Applications of Gauss’s

Divergence Theorem Y Stoke's Theorem Y Applications of Stoke's Theorem Y Determinants Y Determinants of Order 2 Y Determinants of Order 3 Y

Determinants of Order 4 Y Minors And Cofactors Y Properties of Two Determinants of The Same Order Y Working Rule For Finding The Value of A

Determinant Y Symbols And Notations To Be Employed For Finding The Values of A Determinant Y Application of Determinants In Solving A System of

Linear Equations Y System of Linear Non-homogenous equations in Two Unknowns (Cramer’s Rule) Y System of Linear Non- homogeneous Equations In

Three Unknowns (Cramer's Rule) Y Algebra of Matrices Y Matrix Y Special Types of Matrices Y Submatrices of A Matrix Y Equality of Two Matrices Y

Addition of Matrices Y Properties of Matrix Addition Y Properties of Multiplication of A Matrix By A scalar Y Multiplication of Matrices Y Properties of Matrix

Multiplication Y Positive Integral of A Square Matrix Y Transpose of A Matrix Y Symmetric and Skew-symmetric Matrices Y Conjugate of a Matrix Y Transposed

Conjugate of a Matrix Y Hermitian And Skew -Hermitian Matrices Y Orthogonal and Unitary Matrices Y Singular and Non-singular Matrices Y Adjoint and

Inverse of a Matrix Y Adjoint of a Square Matrix Y Inverse or Reciprocal of a Matrix Y Linear Equations Y Solving Systems of Linear Equations Using

Inverse of a Matrix Y Submatrix of a Matrix Y Rank of a Matrix Y Echelon Form of a Matrix Y Elementary Operations or Elementary Transformations of a

Matrix Y Symbols to be Employed For The Elementary Transformations Y Elementary Matrices Y Vectors Y Linear Dependence And Linear Independence of

Vectors Y Homogeneous Linear Equations Y Some Important Conclusions About The Nature of Solutions of The Equations AX 0= Y Working Rule For

Finding The Solutions of the Equation AX 0= Y Systems of Linear Non- homogeneous Equations Y Condition For Consistency Y Condition For A

Systems of n Equations in n Unknowns To Have A Unique Solution Y Working Rule For Finding The Solution of the Equations AX B= Y Eigen values

and Eigen vectors Y Matric Polynomials Y Characteristic Values and Characteristic Vectors of a Matrix Y Certain Relations Between Characteristic Roots

and Characteristic Vectors Y Nature of the Characteristic Roots of Special Types of Matrices Y The Process of Finding the Eigen values and Eigen vectors of a

Matrix Y Cayley-Hamiliton Theorem Y Diagonalisation of a Matrix Y Introduction to Vector Space Y Some Basic Concepts Y Vector space Y General

Properties of Vector Spaces Y Vector Subspaces Y Algebra of Subspaces Y Linear Combination of Vectors Y Linear Sum of Two Subspaces Y Linear

Dependence and Linear Independence of Vectors Y Basis of a Vector Space Y Introduction to Tensors Y Superscript and Subscript Y Space of

n-dimensions of Subspace Y Curve in n- Dimensional Space Y Einstein 's Summation Convention Y Transformation of Coordinates Y Kronecker

Delta Y Some Properties of Kronecker Delta Y Scalars or Invariants Y Contravariant and Covariant (Tensors of Order One) Y Tensors of Order

Two Y Tensors of Higher order (or Higher Rank) Y Some Properties of Tensors Y Symmetric and Skew - symmetric or Anti-symmetric Tensor Y

Addition and Subtraction of Tensors Y Functions, Limits and Continuity Y Functions Y Examples of Some Real Functions Y Some Definition

and Basic Concepts Y Limit of a Function at a Point Y Algevra of Limits Y Some Important Expansions Y Some Important Properties of Limits Y

Factorisation Method Y Evaluation of a Limit When the Direct Subsitution Gives The Indeterminate Form ∞ − ∞ Y Some Standard Limits Y One

Sided Limits i.e., Right Hand and Left Hand Limits Y Limits at Infinity and Infinite Limits Y Continuity Y Discontinuity Y Jump of a Function at a

Point Y Working Rule For Checking the Continuity of a Function f x( ) At A Point a of its Domain Y Cauchy's Definition of Continuity Y

Differentiability Y Relation Between Continuity and Differentiability Y Differentiation Y Increments Y The Differential Coefficient Y Some

Standard Results Y List of Standard Results to be Committed to Memory Y Differential Coefficient of the sum of two Functions Y Differential

Coefficient of the Product of Two Functions Y Differential coefficient of the Quotient of Two Functions Y Differential Coefficient of a Function Y

Hyperbolic Functions Y Inverse Hyperbolic Functions and their Derivatives Y Inverse Functions Y Differential Coefficients of Inverse

Trignometric Functions Y Trigonometric Transformation Y Logarithmtic Differentiation Y Differential Coefficient of the Product of Any Number

of Functions Y Implict Functions Y Parametric Equations Y Differentiation of a Function With Respect to a Function Y Differentiation of Infinite

Recurring Expressions Y Partial Differentiation Y Partial Differential Coefficient Y Partial Differential Coefficients of Higher orders Y

Homogeneous Function Y Euler's Theorem on Homogeneous Functions Y Total Differential Coefficient Y First Differential Coefficient of an

Implicit Function Y Applications to Thermodynamics Y Applications of Exact and Inexact Differentials to Thermodynamics Y Maxima and

Minima Y Working Rule For Maxima and Minima of f x( ) Y Applications of Maxima and Minima to Geometrical and Other Problems Y Most Probable Speed From

Maxwell's Distribution Y Bohr's Radius Y Curve Sketching Y Concavity and Convexity Y Point of Inflexion Y Test for Point of Inflexion Y Multiple

Points Y Singular Points Y Classification of Double Points Y Species of Cusps Y Tangents At Origin Y Change of Origin Y Tangents at the Point

( )h k, To A Curve Y Curve Tracing. Cartesian Equations Y Curve Tracing . Polar Equations Y Parametric Equations Y Integration Y Definitions Y

Constant of Integration Y Some Properties of Integral Y Standard Results Y Extended Forms of Fundamental Formulae Y Methods of Integration

43

Y Integration by Subsitution Y Integral of the Product of two Functions Y Integration by Parts as Applied to the Functions of the Type [ ]e f x f xx ( ) '( )+ Y

Integrals of e bxa x sin and e bxax cos Y Three Special Integrals Y Evaluation of Integrals of Various Types by Using Standard Results Y Three More

Special Integrals Y Evaluation of Integral of Various Types 1 2 2/ ( );( ) /ax bx c px q ax bx c+ + + + +

by Using Standard results Y Three More Special

Integrals a x dx x a dx x a dx2 2 2 2 2 2− − +∫∫∫ ; ; Y Evaluation of Integrals of Various types

( ) ( )( ); ; /ax bx c px q ax bx c px qx r ax bx c2 2 2 2+ + + + + + + + +

by Using Standard Results Y Integration of Some Special Irrational Algebraic

Fractions

( ) ( )1 1

2ax b cx d ax bx c Ax B+ + + + +; ;

1

2( ) ( )

;

px q ax bx c+ + +

( )1

2 2Ax B Cx D+ + Y Integrals of the Type

dx

a b x

dx

a b x

dx

a x b x

dx

a x b x c+ + + + +∫cos,

sin,

sin cos,

sin cos∫∫∫ Y Integration of P x Q x R

a x b x c

cos sin

cos sin

+ ++ +

Y Integration of sin cosm nx x Y Partial

Fractions Y Definite Integral Y Evaluation of Definite Integrals Y Results Regarding Trigonometric Functions Y Reduction Formulae Y

Reduction Formulae sin ,n xdx∫ and cos ,n x dx n∫ being a ive+ integer Y Walli's Formula Y To Find Reduction Formula for tann x dx∫ and

cotn x dx∫ Y To Obtain the Reduction Formulae For secn x dx∫ and cos ec x dxn∫ Y To Find A Reduction Formula For sin cosm nx x dx∫ Y Gamma

Functions to Show That sin cosm nx x dx

m n

m n=

+

+

+ +

Γ Γ

Γ

1

2

1

2

22

20

π / 2

∫ , where m and n are Positive Integers Y Integration of x mxn sin Y Reduction

Formulae for x x dxnsin∫ and x x dxncos∫ Y Reduction Formulae for e bx dxax nsin∫ and e bx dxax ncos∫ Y Reduction Formulae for

x e bx dxn ax sin∫ and x e bx dxn ax cos∫ And x x dxncos∫ Y Reduction Formula for cos sinm x nx dx∫ Y Reduction Formula for cos cosm x nx dx∫ Y

Applications of Integral Calculus Y Quadrative Y Areas of Curves Given by Cartesian Equations Y Areas of Curves Given by Polar

Equations Y Rectification Y Lengths of Curves Y Differential Equations of First Order and First Degree Y Definitons Y Formation of a

Differential Equation Y Differential Equations of First Order and First Degree Y Variables Separable Y Homogeneous Equations Y Linear

Differential Equations Y Equations Reducible to the Linear Form Y Exact Differential Equations Y Integrating Factors Y Applications of

Differential Equations to Chemical Reactions and Solutions Y Second Order Differential Equations with Constant Coefficients Y Second

Order Differential Equations Y Determination of Complementary Function (C.F) Y Power Series Solutions of Differential Equations Y Power

Series Y Power Series Method For Solving Linear Differential Equations With Variable Coefficients Y Legendre's and Bassel's Differential

Equations Y Solution Near A Regular Singular Point Y Some Cases of Failure of the Method of Frobenius Y Spherical Harmonics Y Fourier

Series Y Periodic Functions Y Dirichlet's Conditions Y Fourier Series Y Some Important Results Y Determination of Fourier Coefficients Y

Fourier Series Expansion of an Even or Odd Function in ( )− π π, Y Permutations and Combinations Y The Factorial Function Y Fundamental

Principles of Counting Y Permutations Y Restricted Permutations Y Permutations of Objects Not All Distinct Y Permutations When Objects Can

Repeat Y Circular Permutations Y Combination Y Difference Between A Permutation and a Combination Y Combinations of n Different Objects

Taken r at a time Y Division into Groups Y Practical Problems On Combinations Y Probability Y Some Basic Concepts and Definitions Y

Probability of an Event Y Odds in Favour and Odds Against an Event Y Addition Theorems of Probability (Or Theorems of Total Probability) Y

Conditional Probability Y Multiplication Theorem of Probability Y Inverse Probability : Baye's Theorem Y Errors Y Curve Fitting Y Method of

Least Squares Y Some Particular Cases Y Change of Origin and Scale.

Y Preliminaries Y Well ordering principle Y Mathematical induction Y Binomial coefficient Y Pascal’s triangle Y Basis representation theorem Y

Divisibility Theory Y Division algorithm Y Greatest common divisor Y Euclidean algorithm Y Least common multiple Y Fibonacci sequence Y Lame’s

theorem Y Kroneeker’s theorem Y The linear diophantine equation Y Primes and Their Distribution Y Prime number Y Fundamental theorem of

arithmetic Y Sieve of eratosthenes Y The Goldback conjecture Y Theory of Congruence Y Basic properties of congruence Y Residue system Y Tests of

divisibility Y Linear congruence Y Solvability condition of a system of linear congruences Y Fermat’s Theorem Y Fermat’s factorization method Y

44

–Hari Kishan254-10 Number Theory

Fermat’s little theorem Y Wilson’s theorem Y Euler’s factorization method Y Number Theoretic Functions Y The function τ and σ Y The Mobius

function (or the Mobius inversion formula) Y The greatest integer function Y Euler’s function Y Euler’s theorem Y Some properties of Euler’s function Y

Function T(n) Y Function S(n) Y Function ζ( )s Y Function φ( )n Y Square free integer Y Application to cryptography Y Primitive Roots and Indices Y

The order of an integer modulo n Y Primitive roots Y Primitive roots for primes Y Composite numbers having primitive roots Y The theory of indices Y

Quadratic Congruence and Quadratic Reciprocity Law Y Quadratic congruence Y Quadratic residue Y Euler’s residue Y Legendre symbol and its

properties Y Quadratic reciprocity law Y Quadratic congruences with composite module Y Jacobi symbol Y Perfect Numbers Y Mersenne primes Y

Fermat number Y Pythagorean triples Y Other diophantine equation Y Fermat’s last theorem Y Sum of Squares of Integers Y Sum of two squares Y

Sum of more than two squares Y Waring’s problem Y Additional Topics Y Types of Number theory Y Theta function Y ψ-Function Y π-Function Y

Elementary properties of π(x) Y Bertrand’s conjecture Y Gaussian integer Y Properties of Gaussian integer Y Partition Y Graphical representation of

partition Y Conjugate partition Y Generating function Y Some Important Tables Y List of prime numbers less than 10,000 Y Squares and Cubes of

integers, n, where 1 200≤ ≤n Y Least primitive root r of each prime p, where 2 1 000≤ ≤p , Y τ( )n , σ( )n , φ( )n and µ( )n where 1 100≤ ≤n .

Y Mathematical Aspects of Population Biology Y Some Fun da men tal Con cepts Y Mod els Y Math e mat i cal Mod el ling Y For mu la tion of a

Math e mat i cal Model Y So lu tion of a Math e mat i cal Model Y In ter pre ta tion of the So lu tion Y Types of Mod els Y Lim i ta tion of Mod els Y Ar eas of Mod el ling

Y Some Sim ple Math e mat i cal Mod els Y Math e mat i cal Mod el ling in Bi ol ogy or Bio-mathematics Y Sin gle Spe cies Mod els Y Sta bil ity and Clas si fi ca tion of

Equi lib rium Points Y Re la tion ship be tween Eigen val ues and Crit i cal Points Y Single-species Models (Non-age structured) Y Exponential Growth

Model Y Formulation of the Model Y Solution and Interpretation Y Limitations of the Model Y Effects of Immigration and Emigration on Population Y

Logistic Growth Model Y Solution and Interpretation Y Limitation of Logistic Model Y Extension of the Logistic Model Y Single-species Models (Age

Structured) Y Continuous-time Continuous-age Scale Population Models Y Discrete-time Discrete-age Scale Population Models Y Density Dependent

Model Y Two-sex Models Y Continuous-time Discrete-age Population Model Y Mc Kendrick Approach to age Structure Y Two Species Populations

Models Y Predator Prey Model Y Secular Equation for Determining Stability Y Some Other Prey-predator Models Y Two Dimensional Models and

Competition Models Y Two Dimensional Model Without Carrying Capacity Y Two Dimensional Model with Carrying Capacity Y Competition Models

Y General Continuous Model for Competition Y Competition Model with Time Delays Y Simple Competition Model Y Mathematical Models in

Epidemiology Y Basic Concepts Y SI Model Y SIS Model with Constant Coefficient Y SIS Model with coefficient is a function of time t Y SIS Model with

Constant Number of Carriers Y SIS Model when the Carriers is a Function of Time t Y General Deterministic Model with Removal (SIR Model) Y Epidemic

Model with Vaccination Y Biological Fluid Mechanics Y Some Basic Concepts of Fluid Dynamics Y Poiseuille’s Flow Y Model for Blood Flow Y

Properties of Blood Y Bifurcation in an Artery Y Pulsatile Flow of Blood Y Tans-capillary Exchange Y Sedimentation.

Y Introduction to Security Attacks Y Services and Mechanisms Y Introduction to cryptology Y Conventional Encryption: Conventional Encryption

model Y Classical encryption techniques - Substitution ciphers and transposition ciphers cryptanalysis Y Steganography Y Stream and block ciphers Y

Midern Block Ciphers: Block Ciphers principles Y Shannon's Theory of Confusion and diffusion Y Fiestal Structure Y Data Encryption Standards

(DES) Y Strength of DES Y Differential and Linear Cryptanalysis of DES Y Block Cipher Modes of Operation Y Triple DES Y IDEA encryption and

decryption Y Strength of IDEA Y Confidentiality using Conventional Encryption Y Traffic confidentiality Y Key distribution Y Random number generation

Y Introduction to group Y Ring and field Y Prime and Relative Prime numbers Y Modular arithmetic Y Fermat's and Euler's Theorem Y Primality Testing

Euclid's Algorithm Y Chinese Remainder Theorem Y Discrete Logarithms Y Principles of public key cryptosystems Y RSA algorithm, Security of RSA

Y Key management Y Diffle-Hellman key Exchange algorithm Y Idea of Elliptic Curve cryptography Y Elgemal Encryption Y Message Authentication

and Hash Function: Authentication requirements Y Authentication functions Y Message Authentication codes Y Hash functions Y Birthday attack Y

Security of Hash function and MACSM, MD5 message digest algorithm Y Secure Hash Algorithm (SHA) Y Digital Signatures: Digital Signatures Y

Authentication Protocol Y Digital Signature Standard (DSS) Y Proof of digital signature algorithm Y Authentication Applications: Kerberos and

X.509 Y Directory authentication service Y Electronic Mail security-Pretty Good Privacy (PGP), S/MIME Y IP Security: Architecture Y Authentication

Header Y Encapsulating security payloads Y Combining security associations Y Key management Y Web Security: Secure Socket Layer and Transport

Layer Security Y Secure Electronic Transaction (SET) Y System Security: Intruders Y Viruses and related threats Y Firewall design principles Y Trusted

systems.

45

–Bhupendra Singh & Neenu Agarwal255-02 (C) Bio-Mathematics

–Dr. Manoj Kumar336-05 Cryptography and Network Security

Y Laplace and Inverse Laplace Transforms (Elementary Idea) Y Laplace transform (definitions) Y Linearity property of Laplace transformation Y

Laplace transforms of some elementary functions Y Laplace transforms of some elementary functions table Y Laplace transform theorem Y Two important

theorems Y Some special functions an their Laplace transforms Y Inverse Laplace Transform Y Inverse Laplace transform (definition) Y Linearity

property of inverse Laplace Transform Y Inverse Laplace transform of some elementary functions Y Inverse Laplace transform theorems Y Convolution Y

Convolution theorem (or convolution property) Y Heaviside’s expansion theorem or formula Y The complex inversion formula Y Laplace transforms of

partial derivatives Y Fourier Transforms Y Dirichlet’s conditions Y Fourier series Y Fourier’s integral formula Y Fourier transform or complex fourier

transform Y Inversion theorem for complex fourier transforms Y Fourier sine transform Y Fourier cosine transform Y Linearity property of fourier

transform Y Change of scale property Y Shifting property Y Multiple four transforms Y Convolution Y The convolution or Falting theorem for fourier

transforms Y Relationship between fourier and Laplace transforms Y Fourier transform of the derivative of a function Y Finite fourier transform Y Finite

fourier sine transform Y Inversion formula for finite fourier sine transform Y Finite fourier cosine transform Y Inversion formula for fourier cosine transform

Y Finite fourier sine and cosine transforms of the derivatives of a function f x( ) Y Convolution Y Partial Differential Equations of the First Order Y

Definitions Y Derivation of partial differential equations Y Some definitions Y Lagrange’s linear partial differential equation Y Lagrange’s solution of the

Lagrange’s linear equation (Lagrange’s method of solving the linear partial differential equation of order one namely P Q Rp q+ = ) Y Working method Y

The linear partial differential equation with n Independent variables Y Integral surfaces passing through a given curve Y Compatible system of first-order

equations Y Non-linear Partial Differential Equations of First Order (Charpit's and Jacobi's Methods) Y Solution of partial differential

equations of first order and any degree in some standard forms Y Standard form I, equations involving only p and q and no x y, and z Y Standard form II :

equations involving only p q, and z Y Standard from III;. equations of the form f x p f y q1 2( , ) ( , )= Y Standard form IV equations of the form

z px qy f p q= + + ( , ) Y Charpit’s Method : general method of solution of non-linear partial differential equation of order one with two independent

variables Y Jacobi’s methods Y Envelopes and Characteristics Y Integral strip and characteristic strip Y Cauchy’s method of characteristics for solving

a non linear partial differential equation Y An important theorem Y Envelope Y Two important theorems on envelops Y Partial Differential

Equations of the Second Order with Variable Coefficients Y The origin of second order partial differential equation Y Special types of second

order partial differential equations Y Solutions of equations under given condition Y Classification of Linear Partial Differential Equations Y

Classification of linear partial differential equations of second order in n-independent variations Y Classification of linear partial differential equation of

second order in two independent variables Y Laplace Trans for ma tion, Ca non i cal Forms, Lin ear Hy per bolic Equa tions Y Laplace

transformation (canonical forms) Y Linear hyperbolic equations (existence theorem) Y Riemann method of solution of general linear Hyperbolic equation

of the second order Y Wave, Heat, Laplace and Dif fu sion Equa tions Y One-dimensional wave equation Y Two dimensional wave equations Y One

dimensional heat equation Y Heat equation Y Laplace’s equation Y Two dimensional Laplace (or Harmonic) equation in terms of plane polar

co-ordinates ( , )r θ Y Laplace’s equation in terms of spherical co-ordinates Y Laplace’s equation in terms of cylindrical co-ordinates Y Diffusion equation Y

Ap pli ca tions of Laplace Trans form in the So lu tions of Par tial Dif fer en tial Equa tions (Ini tial and Bound ary Value Problems) Y A

boundary value problem Y Laplace transforms of some partial derivatives Y Application of Laplace transform to mechanics Y Ap pli ca tions of Fou rier

Trans form in the So lu tions of Par tial Dif fer en tial Equa tions (Ini tial and Bound ary Value Prob lems) Y Application of infinite fourier

transforms Y Choice of infinite fourier sine or cosine transform Y Application of finite four transforms Y The choice of finite fourier sine or cosine transform

Y The Wave Equation Y Wave equation in different forms Y Solution of linear partial differential equation by separation of variable method Y Some

important and useful series Y Solution of one dimensional wave equation by using the method of separation of variables Y Solution of one dimensional

wave equation under the given conditions Y The Riemann- Volterra solution of the one-dimensional wave equation Y Some important and useful

differential equations and their solutions Y Solution of two dimensional wave equation Y Vibration of a circular membrane (Solution of two dimensional

wave equation in Polar co-ordinates) Y Solution fo three dimensional wave equation by method of separation of variables Y Solution of wave equation is

cylindrical co-ordinates by the method of separation of variables Y Solution of wave equation in spherical polar co-ordinates by the method of separation

of variables Y The Heat (or Dif fu sion) Equa tion Y Heat (or Diffusion) equation in different forms Y Solution of one dimensional heat equation by

separation of variables Y Solution of one dimensional heat equation under given boundary conditions Y Solution of two dimensional heat equation in

cartesian coordinates Y Solution of heat equation in plane polar co-ordinates by separation of variables Y Solution of three dimensional heat equation by

the method of separation of variables Y Solution of heat (diffusion) equation in spherical polar coordinates by the method of separation of variables Y

Laplace Equation Y Laplace’s (or potential) equation in different forms Y Solution of two dimensional Laplace’s (Harmonic) equation by using the

method of separation of variables Y Solution of two dimensional Laplace’s equation under the given conditions Y Solution of Laplace equation in plane

polar coordinates by separation of variables Y Solution of Laplace’s equation in rectangular cartesian co-ordinates ( , , )x y z by the method of separation of

variables Y Solution of Laplace’s equation in cylindrical co-ordinates by the method of separation of variables Y Solution of Laplace’s equation in

46

–Dr. R.K. Gupta526-05 Partial Differential Equations

spherical co-ordinates by the method of separation of variables Y Green’s Func tions and Prop er ties of Har monic Func tions Y Some definitions Y

Green’s function Y Green’s function for Laplace equation Y Symmetric property of the Green’s function Y Helmholtz’s first theorem Y Green’s function for

the wave equation Y Determine the Green’s function for the Helmholtz equation for the half-space z ≥ 0 Y Green’s function for the heat equation

(diffusion equation) Y Harmonic function Y Properties of Harmonic function Y The spherical mean Y Mean value theorem for Harmonic Functions Y

Cal cu lus of Variations Y Euler’s equation Y Another form of Eulers equation Y Field of extremals Y Jacobi condition and Jacobi equation Y Legendre

condition Y Hamiltonian equations Y The Hamilton-Jacobi equation Y Transport Equation Y Generalised or weak solution Y Transport equation for

a linear hyperbolic system.

Y Concepts of Set Theory Y Func tions or Mappings Y Types of Mappings Y Sets of Num bers Y Al ge braic Struc ture Y Re la tion Y Equiv a lence Re la tion Y

Pro logue to Groups Y Groupoid Y Semigroups Y Monoids (Semigroups with iden tity) Y Sub Semigroups Y Com mu ta tive Monoid Y Morphisms of

Monoids Y Con gru ence Re la tion and Quo tient Semigroups Y Groups Y Abelian or Com mu ta tive Group Y Fi nite and In fi nite Groups Y In te gral Pow ers

of an El e ment Y Or der of an El e ment of a Group Y Modulo Sys tems Y Di vi sion Al go rithm Y Res i due Classes Modulo n Y Trans for ma tion, Per mu ta tion

and Per mu ta tion Groups Y Cy clic Per mu ta tion Y Even and Odd Per mu ta tions Y Sub groups of a Group Y Un ion and In ter sec tion of Sub groups Y Cosets Y

Cy clic Groups Y Nor mal Sub group Y Con ju gate El e ment Y Normalizer of an El e ment Y Quo tient Group Y Homomorphism of a Group Y Ad vanced

The ory of Groups Y Op er a tions of a Group on a Set Y Rep re sen ta tions of G as a group of Per mu ta tions Y Iso tropy Group Y Iso tropy Sub groups Y

Ap pli ca tions of G-sets to Count ing Y Max i mal Sub groups and Com po si tion Se ries Y Se ries of Groups : Nor mal and Sub nor mal Se ries Y Sub nor mal Se ries

Y Com po si tion Se ries Y As cend ing and De scend ing Sub nor mal Chain Y Sylow’s p-Sub group Y Sylow’s Gen eral The o rems Y Nilpotent and Solv able

Groups Y Rings and Fields Y Ring Y El e men tary Prop er ties of a Ring Y Ring with and With out Zero Di vi sors Y Can cel la tion Laws in a Ring Y Field,

In te gral Do main and Skew Field Y Subrings : Rings within Rings YProp er ties of Subrings Y Subfield : Field within Field Y Char ac ter is tic of a Ring Y

Char ac ter is tic of a Field Y Or dered In te gral Do main Y Or dered Re la tions in an in te gral Do main Y Poly no mial Rings Y Set of All Poly no mi als Over a Ring Y

Ide als Y Homo morph ism of Rings Y The o rems on Homomorphisms Y Ker nel of a Ring Homo morph ism Y Isomorphisms and Quo tient Rings Y Prin ci pal

Ideal Y Prin ci pal Ideal Ring Y Divisibility in an In te gral Do main Y Units and As so ci ates Y Prime Ide als Y Max i mal Ide als Y Em bed ding of Rings Y

Eu clid ean and Factorization Do mains Y Con cepts of Divisibility in a Ring Y Prime and Ir re duc ible El e ments Y Method of Find ing the g.c.d. of Any

Two Mem bers of F(x) Y Eu clid ean Rings (or Eu clid ean Do main) Y Unique Factorization Do main Y Poly no mial Rings over Unique Factorization Do main

Y Field of Quo tients of a Unique Factorization Do main Y Eisenstein’s Cri te rion of Irreducibility Y Ad vanced The ory of Rings Y Pri mary De com po si tion

of Ide als Y Gröbner Bases for Ide als Y Rings of Frac tion Y Rings with ORE Con di tion Y Equiv a lence Re la tion and Equiv a lence Class with ORE Con di tion

Y Wedder burn’s The o rem on Fi nite Di vi sion Ring Y Noetherian Rings (Rings With Chain Con di tions) Y Noetherian Rings Y Ba sic Prop er ties of

Noetherian Rings Y De com po si tion of Ide als in Noetheriai Rings Y Artinian Rings Y Ba sic Prop er ties of Artinian Rings Y Vec tor Spaces Y El e men tary

Prop er ties of Vec tor Spaces Y Vec tor Subspaces Vec tor Spaces within Vec tor Spaces Y El e men tary Prop er ties of Vec tor Subspaces Y Al ge bra of

Subspaces Y Lin ear Sum of Two Subspaces Y Di rect Sum of Vec tor Subspaces Y Lin ear Com bi na tion of Vec tors Y Lin ear De pend ence and

In de pend ence of Vec tors Y Ba sis of a Vec tor Space Y Fi nite Di men sional Vec tor space Y Di men sion of a Subspace of Vec tor Space Y Cosets Y Ad di tion

and Mul ti pli ca tion of Two Cosets Y Quo tient Space Y lsomorphism Y Lin ear Trans for ma tions and their Ma trix Rep re sen ta tions Y Lin ear

Trans for ma tion Y Al ge bra of Lin ear Trans for ma tions Y Lin ear Op er a tor Y Al ge bra of Lin ear Op er a tors Y Range and Null Space of a Lin ear

Trans for ma tion Y In vert ible Lin ear Trans for ma tion Y Non-sin gu lar Lin ear Trans for ma tion Y Co-or di nate Vec tor Y Ma trix Rep re sen ta tion of a Lin ear

Trans for ma tion Y Change of Ba sis Y Lin ear Functionals Y Dual Spaces Y Dual Ba sis Y Sec ond Dual Space: Bidual Space Y Nat u ral Map ping Y

An ni hi la tor Y An ni hi la tor of an An ni hi la tor Y Eigen values and Eigen vector or Lin ear Trans for ma tion Y Min i mal Poly no mial Y Invariance of Lin ear

Op er a tor Y Diagonalization Y In ner Prod uct Spaces Y Orthogonality and Orthonormality Y The Adjoint of a Lin ear Trans for ma tion Y Prop er ties of the

Adjoint Y Self-adjoint Trans for ma tion Y Struc ture of Bilinear Forms Y Bilinear Forms Y Bilinear Forms and Ma tri ces Y Qua dratic Forms Y Real

Sym met ric Bilinear and Qua dratic Forms: Law of In er tia Y Or thogo nal Diagonalization of the Qua dratic Form Y Hermitian Forms Y Ma trix

Rep re sen ta tion of a Hermitian Form Y Ca non i cal Form Y Sim i lar ity of Ma tri ces Y Sim i lar ity of Lin ear Trans for ma tion Y In vari ant Subspace Y In vari ant

Di rect-Sum De com po si tions Y Nor mal Form Y Tri an gu lar Form Y Nilpotent Trans for ma tion Y Jor dan Ca non i cal Form Y Ra tio nal Ca non i cal Form Y

Mod ules Y Coset-R Mod ule Y Gen eral prop er ties of mod ules Y Submodules Y Lin ear Sum of two mod ules Y Homo morph ism of Mod ules (lin ear

trans for ma tions) Y Quo tient Mod ules Y Cy clic mod ule Y Ad vanced The ory of Mod ules Y Sim ple and Semi-sim ple mod ules Y Free Mod ules Y

Noetherian and Artinian Mod ules Y Fil tered and Graded Mod ules Y Pro jec tive and Injective Mod ules Y Smith Nor mal Form Over a PID and Rank Y

47

–S.K. Pundir529-06 Advanced Abstract Algebra

Fi nitely Gen er ated Mod ules over a PID Y Ex ten sion Fields (Al ge braic, Nor mal and Sep a ra ble Ex ten sions) Y Field Ex ten sions Y Field Adjunctions

Y Sim ple Ex ten sion of a Field Y Al ge braic Ex ten sion of a Field Y Roots of a Poly no mial Y Split ting or De com po si tion Field Y Mul ti ple Roots Y Nor mal and

Sep a ra ble Ex ten sions of a Field Y Al ge bra ically Closed Fields and Al ge braic Clo sure Y Galois The ory and Its Ap pli ca tions Y Automorphisrn and

Group of Automorphism of Fields Y Nor mal Ex ten sions and El e men tary Sym met ric Y Func tions of the El e ments of a Field Y Galois Group Y Galois

Group of a Sep a ra ble Poly no mial Y Galois Group of a Poly no mial Rep re sented as a Group of Per mu ta tion of its Roots Y Fi nite Fields Y Galois Field Y

Con struc tion of Galois Field and its Subfields Y The Galois Group of Cyclotomic Ex ten sions Y Solvability by Rad i cals Y Cy clic Ex ten sions Y Con struc tion

with Ruler and Com pass Y Insolvability of the Gen eral Equa tion of Degree 5 (Quintic) by Radicals.

Y Spher i cal Trig o nom e try Y Sphere Y Axis and Poles of a Cir cle Y Two great cir cles bi sect each other Y Mea sure ment of the Spher i cal An gle Y Length

of the arc of a small cir cle Y Spher i cal Tri an gles Y Po lar Tri an gles Y Re la tion be tween Sides and An gles of a Spher i cal Tri an gle Y Right An gled Tri an gles Y

Lune Y Spher i cal Ex cess Y So lar Sys tem Y Black Hole Y Ce les tial Sphere Y System of Co-ordinates Y Advantage of this Second System of

Co-ordinates Y Hour Angle Y Equinoxes and Solstices Y The Geocentric Celestial Sphere Y Location of a point on Earth’s surface Y Set ting of Stars

and Twilight Y Rate of change of Zenith distance (z) and Azimuth (A) Y Motion of the Sun Y Twilight Y Refraction Y Refraction of a Star near the

Zenith Y Representation of True and Apparent Positions on the Celestial Sphere Y Differential Equation for Refraction Y Aberration Y Precession and

Nutation Y Precession (Precession of the equinoxes) Y Nutation Y Effect of precession on right ascension and declination Y Combined effect of

precession and nutation in right ascension and declination Y Double stars Y Position angle of a double star Y Par al lax Y Relation between v and φ Y

Geocentric Parallax Y Geocentric Parallax in Right Ascension and Declination (Earth Taken as Spheriod) Y Geocentric Parallax in Azimuth and Zenith

Distance Y The Moon’s Size Y Steller or Annual Parallax Y Annual Parallax in Longitude and Latitude Y The Parallactic Ellipse Y Time Y The Equation of

Time Y Seasons Y The Me rid ian Cir cle Y The Three Errors Y The total correction to the to the observed time of transit or Mayer’s formula Y Brassel’s

formula Y Eclipses of the Moon Y The Angular Radius of the Earth’s Shadow at the Moon’s Distance Y Solar Eclipse Y De ter mi na tion of Position Y

The Dip of the Horizon Y The Position Circle Y Space Dy nam ics Y Central Or bits Y Central Force Y Central Orbit Y Elliptic Orbit (Centre of force

being the focus) Y Hyperbolic Orbit (Centre of force being the focus) Y Parabolic Orbit (Central of force being the focus) Y Velocity and Position of a body

in an Elliptic Orbit Y Velocity and Position of a Body in Hyperbolic Orbit or Find an expression for velocity of body moving in hyperbolic orbit Y Kepler's

Laws of Plan e tary Motion Y Kepler’s Law Y Newton’s Law of Gravitation Y Ar ti fi cial Satellites Y History of Artificial Satellites Y Orbital Classification Y

Centric Classifications Y Orbital Plane Co-ordinate System Y Orbit Space Or The orbit in Rectangular and Spherical Co-ordinates; Heliocentric and

Geocentric System Y The Effect of Atmospheric Drag on an Artificial Satellite Y Mo tion of Rocket and Trans fer Orbits Y Motion of Rocket in Vacuum

(Gravity free space) Y Motion of Rocket in a Gravitational Field Y Motion of a Rocket in Atmosphere Y Transfer between Circular, Coplanar Orbits Y

Parabolic and Hyperbolic Transfer Orbit Y Inter-Planetary and Lunar Trajectories Y Heliocentric and Geocentric Latitudes and Longitudes Y

Conjunctions Y Synodic and Orbital Periods Y Interplanetary Trajectories Y Direct and Retrograde Motion Y The Geo-centric Motion of a Planet Y

Phases.

Y Cen tral Or bits Y Cen tral Force Y Cen tral Or bit Y El lip tic Or bit (Cen tre of force be ing the fo cus) Y Hy per bolic Or bit (Cen tre of force be ing the fo cus) Y

Par a bolic Or bit (Cen tral of force be ing the fo cus) Y Apse, Apsidal Dis tance and Apsidal An gle Y Ve loc ity from In fin ity Y Ve loc ity of fall to the point

pro jec tion Y Ve loc ity of Cir cle Y Ve loc ity and Po si tion of a body in an El lip tic Or bit Y Ve loc ity and Po si tion of a Body in Hy per bolic Or bit or Find an

ex pres sion for ve loc ity of body mov ing in hy per bolic or bit Y Velocity of a body Y Kep ler's Laws of Plan e tary Mo tion Y Kep ler’s Law Y New ton’s Law

of Grav i ta tion Y Ar ti fi cial Sat el lites Y His tory of Ar ti fi cial Sat el lites Y Or bital Clas si fi ca tion Y Cen tric Clas si fi ca tions Y Or bital Plane Co-or di nate Sys tem

Y Or bit in Space or the or bit in Rect an gu lar and Spher i cal Co-or di nates; He lio cen tric and Geo cen tric Sys tem Y The Ef fect of At mo spheric Drag on an

Ar ti fi cial Sat el lite Y Mo tion of Rocket and Trans fer Or bits Y Mo tion of Rocket in Vac uum (Grav ity free space) Y Mo tion of Rocket in a Grav i ta tional

Field Y Mo tion of a Rocket in At mo sphere Y Trans fer be tween Cir cu lar, Coplanar Or bits Y Par a bolic and Hy per bolic Trans fer Or bit Y Inter-Plan e tary

and Lu nar Tra jec to ries Y He lio cen tric and Geo cen tric Lat i tudes and Lon gi tudes Y Con ju nctions Y Syn odic and Or bital Pe ri ods Y In ter plan e tary

Tra jec to ries Y Direct and Retrograde Motion Y The Geo-centric Motion of a Planet Y Phases.

48

–J.P. Chauhan538-02 Spherical Astronomy & Space Dynamics

–J.P. Chauhan539-01 (B) Space Dynamics

Y Fourier Series Y In ner Prod ucts of Func tions Y Or thogo nal Func tions Y Or thogo nal Set of Func tions Y Orthonormal Set of Func tions Y Gram-Schmidt

Pro cess of Orthonormalization Y Even and Odd Func tion Y Fou rier Se ries Y Eu ler For mula for Fou rier Se ries Y Piecewise Smooth Func tions Y Dirichlet

Con di tions Y Jump Dis con ti nu ities Y Con ver gence of Fou rier Se ries Y Uni form Con ver gence of Fou rier Se ries Y The Half-Range Se ries Y Com plex Form of

a Fou rier Se ries Y The Change of In ter val Y Ap pli ca tion of Fou rier Se ries in Ini tial and Bound ary Value Prob lem Y Bound ary Value Prob lem Y

Dif fer en ti at ing Fou rier Se ries Y In te gra tion of a Fou rier Se ries Y Parseval’s Iden tity Y So lu tion of Vi brat ing String us ing Fou rier Se ries Y So lu tion of One

Di men sional Heat (Dif fu sion) Equa tion Y Vi bra tions of an Elas tic String Y String with Ini tial Dis place ment Y String with Ini tial Ve loc ity Y Prob lem

Vari a tions Y Steady-State Prob lems Y The Sturm-Liouville Prob lem Y In te gral Equa tions; Ba sic Con cepts and Fredholms Equa tions Y In te gral

Equa tion Y Dif fer en ti a tion of a Func tion Un der an In te gral Sign Y Re la tion Be tween Dif fer en tial and In te gral Equa tions Y Fredholm In te gral Equa tions Y

Fredholm First The o rem Y Prove that the So lu tionφ λ ξ λ ξ ξ( ) ( ) ( , ; ) ( ) ( )x f x R x f da

b

= + ∫ Y Ev ery Zero of Fredholm Func tion D( )λ is a Pole of the Resolvent

Ker nel Y If a Real Ker nel K x( , )ξ has a Com plex Eigen Value λ µ +0 = iv, then it Also Con tains the Con ju gate Eigen Value to λ µ0 = – iv Y Fredholm

Sec ond The o rem Y Fredholm's As so ci ated Equa tion Y Char ac ter is tic So lu tions Y Fredholm's Third The o rem Y In te gral Equa tions with De gen er ate

Ker nels Y It er ated Kernels and Volterra's In te gral Equa tions Y So lu tion of Non ho mo ge neous Volterra's In te gral Equa tion of Sec ond kind by the

Method of Suc ces sive Sub sti tu tion Y So lu tion of Non-ho mo ge neous Volterra's In te gral Equa tion of Sec ond Kind by the Method of Suc ces sive

Ap prox i ma tion Y De ter mi na tion of Some Resolvent Ker nels Y Volterra In te gral Equa tion of the First Kind Y So lu tion of the Fredholm In te gral Equa tion

by the Method of Suc ces sive Sub sti tu tions Y It er ated Ker nels Y So lu tion of the Fredholm In te gral Equa tion by the Method of Suc ces sive Ap prox i ma tion Y

Re cip ro cal Func tions Y Volterra's So lu tion of Fredholm's Equa tion Y Hilbert Schmidt The ory Y All It er ated Ker nels of a Sym met ric Ker nel are also

Sym met ric Y Orthogonality Y Orthogonality of Fun da men tal Func tions Y Eigen Val ues of Sym met ric Ker nel are Real Y Real Char ac ter is tic Con stants Y

Ex pan sion of a Sym met ric Ker nel in Eigen Func tions Y Sym met ric Ker nels with a Fi nite Num ber of Eigen Val ues Y Sym met ric Ker nels with a Fi nite Eigen

Val ues λ λ+ +m 1 m 2... Y Fou rier Se ries of Power of the Eigen Val ues of the It er ated Ker nel Y Hilbert-Schmidt The o rem Y The in equal i ties of Schwarz and

Min kow ski Y Hilbert's The o rem Y So lu tion of the Fredholm In te gral Equa tion of First Kind Y Ap pli ca tion of In te gral Equa tions Y Ini tial Value Prob lem Y

Bound ary Value Prob lems Y Green's Func tion Y Laplace Trans form Y Prop er ties of the Laplace Trans form Y Ap pli ca tion to Volterra In te gral Equa tion Y

Fou rier Trans form Y Ap pli ca tion of Fou rier Trans form Y Mellin Transform.

Y Fuzzy Set Theory Y Fuzzy Set The ory Ver sus Crisp Set The ory Y Dis trib u tive Prop erty of Un ion and In ter sec tion Y Law of Ex cluded Mid dle and Law

of Con tra dic tion Y Fuzzy Sets Y Rep re sen ta tion of Mem ber ship Func tion Y Types of Fuzzy Sets Y Prop er ties of Fuzzy Sets Y De gree of Subsethood Y

Ham ming Dis tance Y Con vex Fuzzy Sets Y Law of Con tra dic tion and Law of Ex cluded Mid dle Y Sig nif i cance of Fuzzy Vari ables Y Crisp Set Ver sus

Fuzzy Sets Y De com po si tion of Fuzzy Sets or Spe cial Fuzzy Set Y Rep re sen ta tion of Fuzzy Sets Y Sig nif i cance of The o rem Y Op er a tion on Fuzzy Sets

Y Fuzzy Com ple ments Y Fuzzy In ter sec tions: t-Norms Y Ar chi me dean t-norm Y Yager Class of t-norms Y Fuzzy Un ions: t-Conorms Y Du al ity With Re spect

to Fuzzy Com ple ment Y Ag gre ga tion Op er a tions Y Norm Op er a tions Y Fuzzy Num bers and Fuzzy Arith me tic Y Fuzzy Num bers Y Con vex

Nor mal ized Fuzzy Set Y Fuzzy Car di nal ity for Fuzzy Sets Y Fuzzy Arith me tic Op er a tions on In ter vals Y Bi nary Op er a tion on Fuzzy Num bers Y Spe cial

Ex tended Op er a tions Y L-R Rep re sen ta tion of Fuzzy Sets Y Arith me tic Op er a tions on Fuzzy Num bers Y Lat tice of Fuzzy Num bers Y Fuzzy Re la tions

and Fuzzy Re la tions Equa tions Y Fuzzy Re la tion Y Pro jec tion and Cy lin dric Ex ten sion Y Cy lin dric Ex ten sion Y Bi nary Fuzzy Re la tions Y Re la tional

Join Y Fuzzy Com pat i bil ity Re la tions Y Fuzzy Or der ing Re la tion Y Fuzzy Morphisms Y Strong Homo morph ism Y Sup-i Com po si tion of Fuzzy Re la tions Y

Inf-w i Com po si tion of Fuzzy Re la tions Y Fuzzy Re la tion Equa tions Y Fuzzy Re la tion Equa tions Based on Sup-i Com po si tion Y Fuzzy Re la tion Equa tions

Based on Inf-wi Com po si tions Y Pos si bil ity The ory Y Fuzzy Mea sures Y Semi-con tin u ous Fuzzy Mea sures Y Ev i dence The ory Y Ba sic Prob a bil ity

As sign ment (BPA) Y To tal Ignorence Y Body of Ev i dence Y Dempster's Rule Y Com mon al ity Func tion Y Pos si bil ity The ory Y Pos si bil ity Dis tri bu tion Y

De gree of Con fir ma tion and De gree of Disconfirmation Y Fuzzy Sets and Pos si bil ity Dis tri bu tions Y Pos si bil ity The ory vs. Prob a bil ity The ory Y Fuzzy

Logic Y State ment Y Truth Value of a State ment Y Pred i cates and Quan ti fi ers Y Fuzzy Quan ti fi ers Y Lin guis tic Hedges Y Mod i fi ers Y In fer ence from

Con di tional Fuzzy Prop o si tions Y Gen er al ized Mo dus Ponens Y Gen er al ized Mo dus Tollens Y Gen er al ized Hy po thet i cal Syl lo gism Y In fer ence from

Con di tional and Qual i fied Prop o si tions Y In fer ence from Quan ti fied Prop o si tions Y Un cer tainty Based In for ma tion Y Amount of In for ma tion Needed

to Char ac ter ized the El e ment of Set A Y Sim ple, Joint and Con di tional Un cer tain ties Y Non-in ter ac tive Sets Y In for ma tion Trans mis sion Y Non Spec i fic ity

49

–Shiv Raj Singh & Chaman Singh679-03 Fuzzy Set Theory

–Shiv Raj Singh592-03 Advanced Ma thematical Methods

of Fuzzy Sets Y De gree of Va lid ity or Cred i bil ity Y Spec i fic ity for Pos si bil ity Dis tri bu tion Y Sim ple, Joint and Con di tional Un cer tain ties of Fuzzy Sets Y

Fuzz i ness Y Un cer tainty in Ev i dence The ory Y Mea sure of Dis so nance and Con fu sion Y Mea sure of Dis cord Y Strife Y To tal Un cer tainty Y Fuzz i ness in

Ev i dence The ory Y Prin ci ples of Un cer tainty Y Fuzzy De ci sion Mak ing Y Clas si fi ca tion of De ci sion Mak ing Y Fuzzy De ci sion Mak ing with Weighted

Coefficient Y Multiperson De ci sion Mak ing Y Con struc tion of an Or der ing of Given Al ter na tives Y Multicriteria De ci sion Mak ing Y Mul ti stage De ci sion

Mak ing Y Fuzzy Rank ing Meth ods Y Fuzzy Lin ear Pro gram ming Y Types of Fuzzy Lin ear Pro gram ming Prob lems Y So lu tion of Fuzzy Lin ear

Pro gram ming Problems.

Y The Calculus of Finite Differences Y Fi nite Dif fer ences Y Dif fer ences Y Dif fer ence For mu lae Y Fun da men tal The o rem of Dif fer ence Cal cu lus Y The

Dif fer ence Ta ble Y The Op er a tor E Y Prop er ties of the Op er a tors E and ∆ Y Re la tion be tween Op er a tor E of Fi nite Dif fer ences and Dif fer en tial Co ef fi cient

D of Dif fer en tial Cal cu lus Y One or More Miss ing Terms Y Fac to rial No ta tion Y Meth ods of Rep re sent ing any Given Poly no mial in Fac to rial No ta tion Y

Dif fer ences of Zero Y Leibnitz’s Rule Y Ef fect of an Er ror in a Tab u lar Value Y Stirling Num bers Y In ter po la tion with Equal In ter vals Y The Fol low ing

In ter po la tion Meth ods are Used Y Sub-di vi sion of In ter vals Y In ter po la tion with Un equal In ter vals Y Di vided Dif fer ences Y Prop er ties of Di vided

Dif fer ences Y New ton’s For mula for Un equal In ter vals Y Re la tion be tween Di vided Dif fer ences and Or di nary Dif fer ences Y Sheppard’s Rule Y Lagrange’s

In ter po la tion For mula for Un equal In ter vals Y It er a tive Method Y Hermite In ter po la tion For mula Y Spline In ter po la tion Y Cen tral Dif fer ence

In ter po la tion For mu lae Y Gauss’s In ter po la tion For mu lae Y Stirling’s For mula Y Bessel’s For mula Y Laplace-Everett For mula Y Use of Var i ous

In ter po la tion For mu lae Y Nu mer i cal Dif fer en ti a tion Y Di rect Meth ods (us ing for mula) Y Max ima and Min ima of a Tab u lated Func tion Y Nu mer i cal

In te gra tion Y A Gen eral Quad ra ture For mula for Equi dis tant Or di nates Y The Trap e zoidal Rule Y Simpson’s One-Third Rule Y Simpson’s

Three-Eighth’s Rule Y Boole’s Rule Y Weddle’s Rule Y Er ror in Quad ra ture For mu lae Y Cote’s Method Y The Eu ler-Maclaurin’s Sum ma tion For mula Y

Stirling’s For mula for Ap prox i ma tion to Fac to ri als Y Method of Un de ter mined Co ef fi cients Y In te gra tion For mula Y Quad ra ture For mu lae Based on

Cen tral Dif fer ences Y Loz enge Di a grams for Quad ra ture For mu lae Y Romberg In te gra tion Y Hardy’s For mula Y Nu mer i cal Dou ble In te gra tion Y

Gaussi an In te gra tion Y Ap prox i ma tions and Er rors in Com pu ta tion Y Float ing Point Rep re sen ta tion of Num bers Y Arith me tic op er a tions with

Nor mal ized Float ing Point num bers Y Num bers and their Ac cu racy Y Er rors and their Anal y sis Y Gen eral Method of Find ing Re main der Term Y Sources

of Er ror Y Chop ping Y Ab so lute, Rel a tive and Per cent age Er rors Y Er ror in the Ap prox i ma tion of a Func tion Y Er ror Com mit ted in a Se ries Ap prox i ma tion

Y Or der of Ap prox i ma tion Y Re main der Term of Var i ous In ter po la tion For mu lae Y Er rors in Dif fer ent Quad ra ture For mu lae Y Nu mer i cal So lu tions of

Or di nary Dif fer en tial Equa tions of First and Sec ond Or der Y Picard’s Method of Suc ces sive Ap prox i ma tions Y Eu ler’s Method Y Im proved Eu ler’s

Method Y Mod i fied Eu ler’s Method Y Tay lor’s Se ries Method Y Runge’s Method Y Runge Kutta Method Y Pre dic tor and Corrector Method Y Milne’s

Method Y Ad ams-Bash Forth Method Y Gen eral Ap proach to Pre dic tors and Correctors Y Si mul ta neous Dif fer en tial Equa tion (first or der)Y Dif fer en tial

Equa tion of Sec ond Or der Y Numerov’s Method Y Bound ary Value Prob lems Y Er ror Anal y sis Y Con ver gence of a Method Y Sta bil ity Anal y sis Y

So lu tion of Al ge braic and Tran scen den tal Equa tions Y Some Prop er ties of Equa tions Y Nearly Equal Roots Y Rate of Con ver gence of New ton’s

Method When there Ex ist Dou ble Roots Y So lu tion of Nu mer i cal Equa tions (Contd.) Y Con trac tion of Hor ner’s Method Y So lu tion of Si mul ta neous

Lin ear Al ge braic Equa tions Y Dif fer ent Meth ods of Ob tain ing the So lu tions Y New ton-Raphson Method for Solv ing Non-lin ear Si mul ta neous

Equa tions Y Matrix In ver sion Y Gauss Elim i na tion Method Y Gauss-Jor dan Method Y Triangularization Method Y Crout’s Triangularisation Method Y

Doolittle Method Y Choleski’s Method Y It er a tive Method Y Es ca la tor Method for Ma trix In ver sion Y Com plex Ma tri ces and Inversion.

Y The Riemann-Stieltjes Integral Y A Gen er al iza tion of the Riemann In te gral Y Riemann-Stieltjes Sums Y Riemann Stieltjes Integrals Y Nec es sary and

Suf fi cient Con di tion for RS-Integrability Y The RS-In te gral as a Limit of Sums Y Some Classes of Riemann-Stieltjes Integrable Func tion Y Al ge bra of

RS-Integrable Func tions Y RS-Integrability of Com pos ite Func tions Y Mean Value The o rem Y Rectifiable Curves Y In te gra tion of Vec tor Val ued Func tions

Y A Re la tion Be tween R-Integrable and RS-Integrable Y In te gra tion and Dif fer en ti a tion Y Fun da men tal The o rem of Cal cu lus Y In te gra tion by Parts Y

Change of Vari able Y Func tion of Bounded Vari a tion Y Uni form Con ver gence of Se quences and Se ries of Func tions Y Pointwise Con ver gence

[in Met ric Space] Y Uni form Con ver gence Y Cauchy’s Gen eral Prin ci ple of Uni form Con ver gence Y Dini’s Cri te rion for Uni form Con ver gence of a

Se quence of Con tin u ous Func tions Y Tests for Uni form Con ver gence Y Uni form Con ver gence and Con ti nu ity Y Uni form Con ver gence and In te gra tion Y

Uni form Con ver gence and Dif fer en ti a tion Y Uni form Con ver gence of an In fi nite Prod uct Y Weierstrass Ap prox i ma tion The o rem Y Arzela's

The o rem an Equicontinuous Fam i lies Y Power Se ries Y Def i ni tion and Some El e men tary The o rem Y Ra dius of Con ver gence Y Uni form Con ver gence

of Power Se ries Y Prop er ties of Power Se ries Y Unique ness for Power Se ries Y Abel's Summability Y Func tions of Sev eral Vari ables Y In tro duc tion of

50

–J.P. Chauhan852-02 Analysis-I (Real Analysis)

– Prof. P.P. Gupta, G.S. Malik & J.P. Chauhan851-01 (B) Advanced Numerical Analysis

Two Vari ables Y The Neigh bour hood (nbd) of a Point Y The Limit of a Func tion Y Con ti nu ity of Func tion of Two Vari ables Y Par tial De riv a tives Y

In ter change of the Or der of Dif fer en ti a tion Y Differentiability Y Com pos ite Func tion Y Lin ear Trans for ma tion Y Dif fer en ti a tion Y Dif fer en ti a tion in Rn Y

Unique ness of the De riv a tive Y The Chain Rule Y Par tial Dif fer en ti a tion Y The In verse Func tion The o rem Y Im plicit Func tion Y Some Con se quences of

The o rem 16 Y Con tin u ously Dif fer en tia ble Map ping Y Re peated Par tial De riv a tives Y In ter change of Or der of Dif fer en ti a tion Y De riv a tives of Higher

Or der Y Equal ity of Mixed Par tial De riv a tives for Vec tor Val ued Func tions Y Tay lor's The o rem Y Ex treme Prob lems with Con straints Y

Dif fer en ti a tion of Integrals Y Ex treme Prob lems with Con straints Y Sta tion ary, Ex treme and Sad dle Points Y Sad dle Point Y Work ing Rule for Max ima and

Min ima Y Lagrange's Mul ti plier Method Y Work ing Method for Lagrange Mul ti pli ers Y Lagrange's Meth ods of Un de ter mined Multipliers Y Jacobians.

Y Real Number System (Bounded and unbounded sets of real numbers. Neighbourhoods and limit points) Y Field axioms Y Some

properties of real numbers Y Absolute value or modulus of a real number Y Bounded and unbounder subsets of real numbers Y Least upper bound or

supermum Y Greatest lower bound or infimum Y Some properties of supremum and infimum Y Completeness axiom, Existence of supremium and

infimium of bounded sets Y The set of real numbers as a complete ordered field Y Archimedean property of real numbers Y The denseness property of

the real number system Y Neighbourhood of a point Y Limit poitns of a set Y Bolzano Weierstrass theorem Y Sequences Y Subsequences Y Bounded

sequences Y Convergent sequences Y Divergent sequences Y Algebra of convergent sequences Y Monotonic sequences Y Limit points of a sequence Y

Cauchy sequences Y Cauchy’s general principle of convergence Y Limit superior and limit inferior of a sequence Y Nested interval theorem or Cantor’s

intersection theorem Y Infinite Series Y Convergence and divergence of Series. Y Cauchy’s general principle of convergence for series Y The auxiliary

Series ∑( / )1 n p Y Comparison test Y Cauchy’s Root test Y D’ Alembert’s ratio test Y Cauchy’s condensation test Y Raabe’s test Y Logarithmic test Y De

Morgan’s and Bertrand’s test Y Gauss’s test Y Cauchy Maclaurin’s integral test Y Alternating series Y Alternating series test (Leibnitz’s test) Y Absolute

convergence and conditional convergence Y Limits and Continuity Y Definition of limit Y Algebra of limits Y Right hand and left hand limits Y Infinite

limits Y Cauchy’s definition of continuity Y Types of discontinuity Y Algebra of continuous functions Y Properties of continuous functions Y Uniform

continuity Y Differentiability Y Derivative at a point Y Derivative of a function Y A necessary condition for the existence of finite derivatives Y Algebra of

derivatives Y Rolle’s theorem Y Lagrange’s mean value theorem Y Cauchy’s mean value theorem Y Taylor’s theorem with Lagrange’s form of remainder

Y Taylor’s theorem with Cauchy’s form of remainder Y Taylor’s series Y Maclaurin’s series Y Maclaurin’s expansion of some basic functions e x etcx .sin .

Y The Riemann Integral Y Partitions and Riemann sums Y Upper and lower Riemann integrals Y Riemann integrability Y Riemann’s necessary and

sufficient conditions for R-integrability Y Some classes of integrable functions Y Fundamental theorem of Integral Calculus Y The Riemann-Stieltjes

Integral Y A generalisation of the Riemann Integral Y Partitions Y Lower and upper Riemann-Stieltjes sums Y The lower and upper Riemann-Stieltjes

Integrals Y The Riemann-Stieltjes integral Y The RS-integrals as a limit of sums Y Some classes of RS-integrable functions Y A relation between R-integral

and RS-integral Y Uniform convergence of sequences and series of functions Y Uniform Convergence Y Cauchy’s general principle of uniform

convergence Y Tests for uniform convergence Y Uniform convergence and continuity Y Uniform convergence and integration Y Uniform convergence

and differentiation Y Convergence of Improper Integrals Y Tests for convergence of improper integrals of the first kind Y Absolute convergence

Y Tests for convergence of improper integrals of the second kind.

Y Variational Problems with Fixed Boundaries Y Calculus of Variation Y Functionals Y Extremal Y Euler's Equation Y Other Form of Euler's

Equation Y Solutions of Euler's Equation Y Particular Cases of Euler's Equation Y Geodesics Y Functional Dependent on Higher Derivatives Y Functional

for Several Dependent Variable Y Functionals Dependent on Several Independent Variables Y Isoperimetric Problems Y Invariance of Euler's Equation

under Co-ordinate Transformation Y Variational Problems with Moving Boundaries Y Transversality Conditions Y Orthogonality Conditions Y

Variational Problem with a Moving Boundary for a Functional Dependent on Two Functions Y One Sided Variations Y Sufficient Conditions for an

Extremum Y Jacobi Condition Y Sufficient Condition for Extremum (Legendre Condition) Y Weak and Strong Extremum Y Application of the

Calculus of Variation Y Hamilton's Principle Y Lagrangian of a System Y Lagrange's Equation Y Hamiltonian Y Hamilton's Canonical Equation of Motion

Y Principal of Least Action Y Variational Method for Boundary Value Problems Y Rayleigh-Ritz Method (For Ordinary Differential Equation) Y

Galerkin's Method Y Partial Differential Equation (By Rayleigh-Ritz Method) Y Kantorovich Method.

51

–Mukesh Kumar Singh864-01 Calculus of Variations

–A.R. Vasishtha & Vipin Vasishtha260-10 (B) Real Analysis

Krishna's

Fully Solved Series on

MATHEMATICSfor

(All Indian Universities and Competitive Examinations)

Y Inverse Circular Functions Y General and Principal Values of Inverse Circular Functions Y Relations between Inverse Functions Y Some Important

Results about Inverse Functions Y Complex Numbers Y Addition of Complex Numbers Y Multiplication of Complex Numbers Y Conjugate of a

Complex Number Y Modulus of a Complex Number Y Modulus Argument form or Polar Standard form or Trigonometric form of a Complex NumberY

The Points on the Argand Plane Representing the Sum, Difference, Product and Division of two Complex Numbers Y More Properties of Moduli and

Arguments Y Integral and Rational Powers of a Complex Number Y De-Moivre's Theorem Y Deductions from De-Moivre's Theorem

(Expansions of sine and cosine functions in power series) Y Expansions of cos nθ and sin nθ in Powers of cos θ and sin θ (n being a positive

integer) Y Expansion of tan nθ in Powers of tan θ Y Expansion of tan ( ....... )θ θ θ θ1 2 3 4+ + + + Y Expansions of cos α and sin α in Series of Powers of α Y

Expansions of cos α° and sin α° Y Expansion of tan α in Powers of α Y Evaluation of Limiting Values of Indeterminate Forms Y Exponential,

Trigonometric and Hyperbolic Functions of a Complex Variable Y (Separation into real and imaginary parts) The Exponential Function of a

Complex Variable Y Index Law for the Exponential Functions Y Trigonometrical Functions or Circular Functions of a Complex Variable Y Euler’s

Theorem Y Periodicity of Functions Y De-Moivre's Theorem for Complex Argument Y Some Standard Trigonometrical Results for Complex Arguments Y

Hyperbolic Functions Y Relations between Hyperbolic and Circular Functions Y Properties of Hyperbolic Functions Y Expansions in Series for sin h x

and cos h x Y Periods of Hyperbolic Functions Y Separation into Real and Imaginary Parts Y Logarithms of Complex Quantities Y Logarithms in the

Set of Real Numbers Y Logarithms of Complex Numbers Y Principal and General Values of Logarithm of a Non-zero Complex Number Y Properties of

the Logarithmic Function Y Working Rule to Evaluate Log ( )x iy+ Y Logarithm of a Positive Real Number in the Set of Complex Numbers Y Logarithm of

a Negative Real Number Y The General Exponential Function az Y To Separate ( )α β+ +i p iq into Real and Imaginary Parts Y Inverse Circular and

Hyperbolic Functions of Complex Quantities Y Inverse Circular Functions of Complex Quantities Y Inverse Hyperbolic Functions Y Relations

between Inverse Hyperbolic Functions and Inverse Circular Functions Y Expansion of Some Trigonometrical Functions Y Expansion of cosn θ in

Terms of Cosines of Multiples of θ, n being a Positive Integer Y Expansion of sinn θ in a Series of Cosines or Sines of Multiples of θ, According as n (a

Positive Integer) is Even or Odd Y Expansion of sin cosm nθ θ Y Gregory's Series and Trigonometrical Expansions Y Gregory's Series Y General

Theorem on Gregory's Series Y Value of π Y Summation of Trigonometrical Series Y C+iS Method for Summing up Trigonometric Series Y Series

Based on Geometric Progression or Arithmetico-Geometric Series Y Series Based on Binomial Expansions Y Series Based on Exponential Series Y Series

Based on Logarithmic Series and its Sub-case Gregory's Series Y The Difference Method Y Angles in Arithmetical Progression.

Y Algebra of Matrices Y Basic concepts Y Matrix Y Special types of matrices Y Submatrices of a matrix Y Equality of two matrices Y Addition of matrices

Y Properties of matrix addition Y Multiplication of a matrix by a scalar Y Multiplication of two matrices Y Properties of matrix multiplication Y A useful way

of representing matrix products Y Associative law for the product of four matrices Y Positive integral powers of matrices Y Triangular, Diagonal and Scalar

matrices Y Transpose of a matrix Y Conjugate of a matrix Y Transposed conjugate of a matrix Y Symmetric and skew-symmetric matrices Y Hermitian and

Skew – Hermitian matrices Y Determinants Y Determinants of order 2 Y Determinants of order 3 Y Minors and cofactors Y Working rule for finding the

value of a determinant Y Determinants of order n Y Determinant of a square matrix Y Properties of Determinants Y Determinants of order 4 Y Product of

two determinants of the same order Y System of non-homogeneous linear equations (Cramer's Rule) Y Inverse of a Matrix Y Adjoint of a square matrix

Y Invertible matrices; Inverse or Reciprocal of a Matrix Y Singular and non-singular matrices Y Reversal law for the inverse of a product Y Use of the

inverse of a matrix to find the solution of a system of linear equations Y Orthogonal and Unitary matrices Y Partitioning of matrices Y Rank of a Matrix Y

Sub-matrix of a matrix Y Rank of a matrix Y Elementry transformations of a matrix Y Symbols to be employed for the elementry transformations Y

Elementry matrices Y Invariance of rank under elementary transformation Y Reduction to normal form Y Equivalence of Matrices Y Row and column

equivalence of matrices Y Employment of only row transformations Y Employment of only column transformations Y The rank of a product Y Use of

elementry transformations to find the inverse of a non–singular matrix Y Working rule for finding the inverse of a non-singular matrix by E-row

transformations Y Linear Equations Y Vectors Y Linear dependance and linear independence of vectors Y A vector as a linear combination of vectors Y

Row rank and column rank of a matrix Y Homogeneous linear equations Y Some important conclusions about the mature of selutions of the equation

AX = 0 Y Fundamental set of solutions of the equation AX = 0 Y Working rule for finding the solutions of the equation AX = 0 Y System of linear non

52

–A.R. Vasishtha & A.K. Vasishtha443-21 Series: Matrices

–A.R. Vasishtha, S.K. Sharma & A.K. Vasishtha442-17 Series: Trigonometry

homogeneous equations Y Condition for consistency Y Condition for a system of n equations in n unknowns to a unique solution Y Eigenvalues and

Eigenvectors or Characteristic roots and Characteristic vectors Y Matrix polynomials definition Y Characteristic polynomial and characteristic

equation of a matrix Y Cayley–Hamilton theorem.

Y General Properties of Equations Y Synthetic Division Y Some Important Results Y Relations between the Roots and Coefficients of an

Equation Y Relations between the Roots and the Coefficients Y Particular Cases Y The Cube Roots of Unity Y Symmetric Functions of the Roots Y

Order and Weight of a Symmetric Function Y The Number of Terms in any Symmetric Function Y Symmetric Functions of the Roots Expressed in Terms

of the Coefficients Y Newton's Theorem on the Sums of the Powers of the Roots of an Equation Y Newton’s Fundamental Theorem on Symmetric

Functions Y Transformation of Equations Y To Transform an Equation into Another whose Roots are the Roots of the Given Equation with their Signs

Changed Y To Transform an Equation into Another whose Roots are the Roots of the Given Equation Multiplied by a Constant m Y To Transform an

Equation into Another whose Roots are the Reciprocals of the Roots of the Given Equation Y To Transform an Equation into Another whose Roots are

any Powers of the Roots of the Given Equation Y To Transform an Equation into Another whose Roots are the Roots of the Given Equation Diminished

by a Constant h Y Removal of Terms Y Reciprocal Equation Y To Reduce the Cubic with Binomial Coefficients Y To Form an Equation whose Roots are

the Symmetric Functions of the Roots of a Given Equation Y Solution of Cubic Equations Y Cardan’s Method of Solving the Cubic Equation Y

Application of Cardan’s Method to Numerical Equations Y Inequalities Y Some Elementary Properties of Inequalities Y Factorisation Y Arithmetic and

Geometric Means of two Positive Numbers and the Inequality of these Means Y Sum and Product of two Positive Numbers Y Arithmetic and Geometric

Means of n Positive Numbers and the Inequality of these Means Y Arithmetic Mean of the mth Powers Y Arithmetic Mean of mth Powers Y

Cauchy-Schwarz Inequality Y Tchebychef’s Inequalities Y Tchebychef’s Inequalities for Sets of n Real Numbers Y Weirstrass’s Inequalities Y Application

of Inequalities to Problems of Maxima and Minima Y Let a b c, , ... be Any Positive Real Numbers whose Sum a b c+ + + ... Y Continued Fractions Y

To Convert a Given Ordinary Fraction into a Simple Continued Fraction Y To Convert a Quadratic Surd into a Simple Continued Fraction Y Recurring

Continued Fraction Y Law of Formation of Successive Convergents Y Relation between Successive Convergents Y Properties of convergents of a simple

continued fraction Y Limits to the Error: To Find the Limits to the Error Made in Taking any Convergent for the Continued Fraction Y To Find Two

Positive Integers P and Q Such that QA PB− = ± 1 Y Convergence of Infinite Series Y Series Y Convergence and Divergence of Series Y To Discuss

the Convergence of a Geometric Series Y General Theorems Y Cauchy’s Root Test Y The Series Σ 1 / np Y Comparison Test Y Working Rule for Applying

Comparison Test Y D’ Alembert’s Ratio Test Y Comparison of Ratios Y Raabe’s Test Y Cauchy’s Condensation Test Y The Auxiliary Series Σ 1

n n p(log ) Y

Kummer’s Test Y Gauss’s Test Y Logarithmic Test Y De Morgan’s or Bertrand’s Test Y An Important Test Y Summary of the Tests for a Series of Positive

Terms Y Alternating Series Y Leibnitz Test (Alternating Series Test) Y Absolute Convergence Y Cauchy’s General Principle of Convergence of a Series Y

Important Theorems about Absolute Convergence Y Re-arrangement of the Terms of an Absolutely Convergent Series Y Insersion and Removal of

Brackets Y To Discuss the Convergence and Absolute Convergence of the Binomial Series.

Y Differentiation Y Derivative of a Function Y Some Standard Results Y Differential Coefficient of a Function of a Function Y Hyperbolic Functions

Y Inverse Hyperbolic Functions Y Some More Methods of Differentiation (Logarithmic Differentiation, Implicit Functions, Parametric Equations,

Trigonometrical Transformations, Differentiation of a Function w.r.t. a Function) Y Successive Differentiation Y Standard Results Y Leibnitz’s

Theorem Y nth Derivative for x = 0 Y Expansions of Functions Y Taylor’s Series Y Maclaurin’s Series Y Expansions of Functions (Maclaurin’s

Theorem, Taylor’s Theorem) Y Some Important Expansions (Exponential Series, Sine Series, Cosine Series, Binomial Series, Logarithmic Expansion) Y

Partial Differentiation Y Functions of Two or More Variables Y Homogeneous Functions Y Euler’s Theorem on Homogeneous Functions Y Total

Derivatives Y Indeterminate Forms Y The Form 0 0/ Y Algebraic Methods Y Form ∞ ∞/ Y Form 0 × ∞ Y Form ∞ − ∞ Y The Forms 0 10, ,∞ ∞o Y

Tangents and Normals Y Tangent Y Tangents Parallel and Perpendicular to the x-axis Y Normal Y Angle of Intersection of Two Curves Y Length of

Cartesian Tangent, Normal, Subtangent and Subnormal Y Polar Co-ordinates Y Angle between Radius Vector and Tangent Y Angle of Intersection of two

Polar Curves Y Polar Sub-tangent and Polar Sub-normal Y Length of Perpendicular from Pole to Tangent Y Pedal Equation Y Differential Coefficient of

arc Length (Cartesian Co-ordinates) Y Differential Coefficient of arc Length (Polar Co-ordinates) Y Curvature Y Radius of Curvature of Intrinsic Curves Y

Radius of Curvature of Cartesian Curves Y Radius of Curvature of Parametric Curves Y Radius of Curvature of Pedal Curves Y Radius of Curvature of

Polar Curves Y Tangential Polar Formula for Radius of Curvature Y Miscellaneous Formulae for Radius of Curvature when x and y are Functions of arc

Length Y Radius of Curvature at the Origin (Newton’s Method, Expansion Method, Radius of Curvature at the Pole) Y Co-ordinates of Centre of

53

–A.R. Vasishtha, R.K. Gupta & A.K. Vasishtha444-12 Series: Algebra

–A.R. Vasishtha, S.K. Sharma & A.K. Vasishtha446-25 Series: Differential Calculus

Curvature Y Chord of Curvature through the Origin Y Chords of Curvature Parallel to Co-ordinate axes Y Asymptotes Y Determination of Asymptotes Y

Asymptotes of General Algebraic Curves Y Non-existence of Asymptotes Y Case of Parallel Asymptotes Y Asymptotes Parallel to the Co-ordinates axes Y

Total Number of Asymptotes of a Curve Y Complete Working Rule of Finding the Asymptotes of Rational Algebraic Curves Y Asymptotes by ExpansionY

Alternative Methods of Finding Asymptotes of Algebraic Curves Y Asymptotes by Inspection Y Intersection of a Curve and its Asymptotes Y Asymptotes of

Polar Curves Y Circular Asymptotes Y Change of the Independent Variable Y To Change the Independent Variable into the Dependent Variable Y To

Change Independent Variable x into Another Variable z, when x f z= ( ) Y Change of Both the Depedent and Independent Variables Y Transformation in

Case of two Independent Variables Y Transformation from Cartesian to Polar Co-ordinates and Vice Versa Y Maxima and Minima (of Functions of a

Single Independent Variable) Y Properties of Maxima and Minima Y Condition for Maximum and Minimum Values Y Working Rule of Maxima and

Minima of f x( ) Y Application of Maxima and Minima to Geometrical and other Problems Y Maxima and Minima (of Functions of Two Independent

Variables) Y Necessary Conditions for the Existence of a Maximum or a Minimum of f x y( , ) at x a y b= =, Y Stationary and Extreme Points Y Sufficient

Conditions for Maxima or Minima Y Working Rule for Maxima and Minima Y Maxima and Minima (of Functions of Several Variables) Y Maxima

and Minima of Functions of Several Variables and Lagrange’s Method of Undetermined Multipliers Y Envelopes and Evolutes Y One Parameter

Family of Curves Y Envelope of a One Parameter Family of Curves Y Method for Finding Envelope Y Envelope in case the Equation of the Family of

Curves is a quadratic in the Parameter Y Geometrical Significance of the Envelope Y Evolute of a Curve Y Length of Arc of an Evolute Y Jacobians Y

Case of Functions of Functions Y Jacobian of Implicit Functions Y Condition of a Jacobian to Vanish Y Singular Points Y Concavity and Convexity Y

Point of Inflexion Y Test for Point of Inflexion Y Multiple Points Y Classification of Double Points (Node, Cusp, Conjugate Point) Y Species of Cusps Y

Tangents at Origin Y Change of Origin Y Tangent at Any Point to a Curve Y Position and Character of Double Points Y Nature of a Cusp at the Origin Y

Nature of a Cusp at any Point Y Curve Tracing Y Curve Tracing (Cartesian Equations) Y Polar Equations Y Parametric Equations Y Functions of a

Real Variable, Limits, Continuity and Differentiability Y Limits Y Continuity Y Differentiability Y Rolle's Theorem, Mean Value Theorems,

Taylor's and Maclaurin's Theorems Y Rolle’s Theorem Y Lagrange’s Mean Value Theorem Y Some Important Deductions from the Mean Value

Theorem Y Cauchy’s Mean Value Theorem Y Taylor’s Theorem with Lagrange’s form of Remainder After n Terms Y Taylor’s Theorem with Cauchy’s

form of Remainder.

Y Elementary Integration (Standard Forms, Integration by parts) Y Constant of Integration Y Two Simple Theorems Y Hyperbolic Functions Y

Fundamental Formulae Y Extended Forms of Fundamental Formulae Y Methods of Integration Y Three Important Forms of Integrals Y Some more

standard Integrals Y Integral of the Product of two Functions Y Successive Integration by Parts Y Integrals of eax cos bx and eax sin bx Y Integration by

Partial Fractions Y Reduction Formulae (Application of Successive Integration by Parts) Y Definite Integrals (Simple) Y Integration of Rational

Fractions Y Rational Fractions Y Integration of ( ) / ( )px q ax bx c+ + +2 Y Integration of 1 2/ ( )x k n+ Y To Integrate ( ) / ( )px q ax bx c n+ + +2

Y Integration of Irrational Algebraic Fractions Y Integration by Rationalisation Y Integration of 1

( ) ( )ax b cx d+ √ + Y Integration of

1 2/ {( ) ( )}ax bx c Ax B+ + √ + Y Integration of 1 2/ ( )√ + +ax bx c Y Integration of √ + +( )ax bx c2 Y Integration of ( ) / ( )px q ax bx c+ √ + +2 Y

Integration of ( ) ( )px q ax bx c+ √ + +2 Y To Evaluate ∫ + √ + +dx

px q ax bx c( ) ( )2 Y Integration of 1 2 2/ {( ) ( )}Ax B Cx D+ √ + Y Integration of

Trigonometric Functions Y Integration of sinm x and cosm x Y Integration of sin cosm nx x Y Integration of 1 / ( cos )a b x+ Y Integration of

1 / ( sin )a b x+ Y To Evaluate ∫ +dx

a x b xsin cos Y To Evaluate ∫ + +

dx

a b x c xcos sin Y Integration of

P x Q x R

a x b x c

cos sin

cos sin

+ ++ +

Y Integration of

1 / ( tan )a b x+ Y Reduction Formulae (For Trigonometric Functions) Y Reduction Formulae Y Reduction Formulae for ∫ sinn x dx and

∫ cos ,n x dx n being a +ive Integer Y Walli’s Formula Y To Find Reduction Formula for ∫ tann x dx and ∫ cotn x dx Y To Obtain the Reduction Formulae

for ∫ secn x dx and ∫ cosecn x dx Y To Find a Reduction Formula for ∫ sin cosm nx x dx Y Gamma Function Y To Show that

0

21

2

1

2

22

2

π /

∫ =

+

Γ +

Γ + +

sin cosm nx x dx

m n

m n

Γ

Y Integration of x mxn sin and x mxn cos Y Reduction Formulae for ∫ x x dxnsin and ∫ x x dxncos Y

Reduction Formulae for ∫ e bx dxax nsin and ∫ e bx dxax ncos Y Reduction Formulae for ∫ x e bx dxn ax sin and ∫ x e bx dxn ax cos Y Reduction

Formula for ∫ cos sinm x nx dx Y Reduction Formula for ∫ cos cosm x nx dx Y Reduction Formulae Continued [For Irrational, Algebraic and

Transcendental Functions] Y Reduction Formulae for ∫ +x a bx dxm n p( ) Y Reduction Formula for ∫ +dx

x a n( ),

2 2 where n is Positive Y Reduction

54

–A.R. Vasishtha, S.K. Sharma & A.K. Vasishtha447-21 Series: Integral Calculus

Formula for ∫ √ −x ax x dx mm ( ) ;2 2 being a Positive Integer Y Reduction Formulae for ∫ e x dxmx n and ∫ >e

xdx n

mx

n( )0 Y Reduction Formulae for

∫ a x dxx n and ∫ ( / )a x dxx n Y Reduction Formula for ∫ x log x dxm n( ) Y Definite Integrals (Properties of Definite Integrals, Definite Integral

as the limit of Sum, Summation of Series with the help of Definite Integrals) Y Fundamental Properties of Definite Integrals Y The Definite

Integral as the Limit of a Sum Y Summation of Series with the Help of Definite Integrals Y Areas of Curves (Quadrature) Y Areas of Curves Given by

Cartesian Equations Y Areas of Curves Given by Polar Equations Y Lengths of Curves (Rectification) (Lengths of Curves and Intrinsic

Equations) Y Lengths of Curves Y Intrinsic Equations Y Volumes and Surfaces of Solids of Revolution (Volumes of Solids of Revolution,

Surfaces of Solids of Revolution, Theorem of Pappus and Guldin) Y Volumes of Solids of Revolution Y Surfaces of Solids of Revolution Y

Theorems of Pappus or Guldin Y Multiple Integrals (Double Integrals, Triple Integrals, Change of Order of Integration, Change of

Variables in a Double Integral) Y Double Integrals Y Properties of a Double Integral Y Evaluation of Double Integrals Y To Express a Double Integral

in Terms of Polar Coordinates Y Triple Integrals Y Evaluation of Triple Integrals Y Change of Order of Integration Y Change of Variables in a Double

Integral Y Euler’s Integrals (Beta and Gamma Functions) Y Some Simple Properties of Beta Function Y Another Form of Beta Function Y Gamma

Function Definition Y Fundamental Property of Gamma Function Y Some Transformations of Gamma Function Y Relation between Beta and Gamma

Functions Y The Value of Γ

1

2 Y To Prove that for all Values of m and n Such that m n> − > −1 1, Y Some Important Transformations of Beta Function

Y Duplication Formula Y To Find the Value of Γ

Γ

Γ

1 2 3

n n n ... Γ −

n

n

1 Y To Find the Values of the Integrals

0

1∞

− −∫ e bx x dxax mcos .

0

1∞

− −∫ e bx x dxax msin . Y Dirichlet’s Theorem for Three Variables Y Dirichlet’s Theorem for n Variables Y Liouville’s Extension of Dirichlet’s Theorem

Y Convergence of Improper Integrals Y Convergence of Improper Integrals Y Tests for Convergence of Improper Integrals of the First Kind Y

Comparison Test Y The µ-Test Y Abel’s Test for the Convergence of Integral of a Product Y Dirichlet’s Test for the Convergence of Integral of a Product Y

Absolute Convergence Y Test for Convergence of Improper Integrals of the Second Kind Y Comparison Test Y The µ-Test Y Abel’s Test Y Dirichlet’s Test.

Y Introduction Y Differential Equation Definition Y Order and Degree of a Differential Equation Y Ordinary and Partial Differential Equations Y Linear

and Non-linear Differential Equations Y Solutions of Differential Equations Y Differential Equations of First Order and First Degree Y Integrating

Factors Y Geometrical Problems Y Linear Differential Equations with Constant Coefficients Y Determination of Complementary Function (C.F.)

Y The Particular Integral (P.I.) Y Particular Integral in Some Special Cases Y To Find P.I. when Q e Vax= , where V is any Function of x YTo Find P.I.

when Q eax= and F a( ) = 0 Y To Find P.I. when Q ax= sin or cos ax and F a( )− =2 0 Y To Find P.I. when Q xV= , where V is any Function of x Y The

Operator 1

D − αα, being a Constant Y Orthogonal Trajectories Y Trajectory Y Trajectories-Cartesian Co-ordinates Y Orthogonal Trajectories-polar

Coordinates Y Homogeneous Linear Differential Equations Y Method of Solution Y Equations Reducible to Homogeneous Form Y Differential

Equations of the First Order but not of the First Degree Y Equations Solvable for p Y Equations Solvable for y Y Equations solvable for x Y

Clairaut’s Equation Y Equations Reducible to Clairaut’s Form Y Geometrical Meaning of a Differential Equation of the First Order Y Singular Solutions Y

Determination of Singular Solutions with the Help of c-discriminant and p-discriminant relations Y Working Rule for Finding the Singular Solution Y The

Singular Solution of Clairaut’s equation Y Linear Equations of Second Order with Variable Coefficients Y An Equation of the Form

d y

dxP

dy

dxQy R

2

2+ + = Y The Complete Solution in Terms of a Known Integral Y Removal of the First Derivative Y Transformation of the Equation by

Changing the Independent Variable Y Method of Variation of Parameters Y Method of Operational Factors Y Guidelines of the Procedure for the Solution

of Linear Differential Equations of Second Order Y Ordinary Simultaneous Differential Equations Y Methods of Solving Simultaneous Linear

Differential Equation with Constant Coefficients Y Number of Arbitrary Constants Y Simultaneous Equations of the Form Y Geometrical Interpretation of

the Differential Equations dx

P

dy

Q

dz

R= = Y Total Differential Equations Y Necessary and Sufficient Condition for Integrability of Total Differential

Equation P dx Q dy R dz+ + = 0 Y The Conditions for Exactness Y Methods for Solving the Differential Equation Pdx Qdy Rdz+ + = 0 Y Geometrical

Interpretation of the Single Differential Equation P dx Q dy R dz+ + = 0 which is Integrable Y The Locus of P dx Q dy R dz+ + = 0 is Orthogonal to

the Locus of dx

P

dy

Q

dz

R= = Y The Non-integrable Single Differential Equation Y Equations Containing More than Three Variables.

55

–A.R. Vasishtha & S.K. Sharma448-17 Series: Differential Equations

Y Change of Axes Y Transformation of Co-ordinates Y Change of Origin (Translation of Axes) Y Rotation of Axes (Change of Directions of Axes)

Y Origin Shifted and Axes Rotated Y Hyperbola Y Standard Equation of a Hyperbola Y Second Focus and Second Directrix Y A Geometrical Property of

the Hyperbola Y Some Definitions and Results Y Asymptotes Y Rectangular Hyperbola Y Equation of the Rectangular Hyperbola Referred to Asymptotes

as Co-ordinate Axes Y Equation of a General Hyperbola Referred to Asymptotes as Axes of Co-ordinates Y Parametric Representation Y Equations of the

Chord, Tangent and Normal for the Hyperbola Y Equations of the Chord, Tangent and Normal for the Rectangular Hyperbola Y Conjugate Hyperbola Y

Properties of Conjugate Hyperbolas Y Properties of Conjugate Diameters of Conjugate Hyperbolas Y Polar Equation of a Conic (Chord of Contact)

Y Conic Section Y Polar Coordinates Y To Find the Polar Equation Y To Find the Equation to the Directrix of the Conic l r e/ cos= + θ1 Y To Find the

Polar Equation Y Chord Joining any Two Points on the Conic Y Tangent to the Conic at a Given Point on it Y Asymptotes Y Auxiliary Circle Y To Find the

Point of Intersection Y Director Circle Y Pair of Tangents Y Chord of Contact Y Polar Y Perpendicular Lines Y Normal Y Tracing of Conics (General

Equation of Second Degree) Y Conic Sections Y To Prove that the General Equation of the Second Degree Always Represents a Conic Section in

General Y Centre Definition Y Centre of a Conic Y Asymptotes Y Nature of a Conic Y Lengths and Equations of the Axes of a Central Conic Y Eccentricity,

Coordinates of the Foci and the Equation of the Directrices of the Central Conic Y Working Rule to Trace an Ellipse or a Hyperbola Y Tracing of a

Parabola Y General Conics, Contacts and Confocals Y Equation of a Conic Section Y Conic Through Five Points Y Intersection of a Straight Line

and a Conic Y Tangent to the General Conic Y Condition of Tangency Y Polar of a Point or Chord of Contact Y Conjugate Lines Y Chord with a Given

Middle Point Y Diameter Y Conjugate Diameters Y To Find the Condition that the two straight lines Y Pair of Tangents Y Director Circle of a Conic Y Foci

Y Tangents From a Focus to a Conic Y To Find the Foci of a Conic Y Axes of the Conic Y Directrices of the Conic Y Contact of Conics Y The Equation of a

Family of Conics Y Tangents from an External Point to a Conic Found by the Method of Double Contact Y To Find the General Equation Y To Find the

Equation of the Circle Y Equation of a Conic Referred to Tangent and Normal as Coordinate Axes Y Confocal Conics Y Propositions on Confocals Conics

Y Ellipse Y Standard Equation of an Ellipse Y Second Focus and Second DirectrixY Some Definition and Results for the Ellipse whose Equation is

x

a

y

b

2

2

2

21+ = Y General Equation of the Ellipse Y The Tangent and the Normal Y Intersection of the Line y mx c= + and the Ellipse

x

a

y

b

2

2

2

21+ = Y

Auxiliary circle Y The Equation of the Chord Joining the Points ‘φ1’ and ‘φ 2’ Y Sub-tangent and Sub-normal Y Tangents from a Given Point (h k, ) to the

Ellipse x a y b2 2 2 2 1/ /+ = Y Director Circle Y Chord of Contact Y Some Propositions on Pole and Polar Y Pole and Polar Y The Equation of the Pair of

Tangents from a Point Y Diameter Y Propositions on Diameters Y Propositions on Conjugate Diameters Y Equi-conjugate Diameters Y A Property of

Equi-conjugate Diameters Y Supplemental Chords Y Oblique Axes Y Co-normal Points Y Concurrency of Three Normals Y Concyclic Points on an Ellipse

Y Centre of a Circle Passing Through Three Given Points on an Ellipse.

Y Systems of Co-ordinates Y Definitions Origin, Axes and Co-ordinate Planes Y Co-ordinates of a Point in Space Y Properties of Co-ordinates of a

Point P Y Octants Y Change of Origin Y The Distance between two Given Points Y Division of a Line Y Centroid of a Triangle Y Spherical Polar

Co-ordinates Y Direction Cosines and Projections Y Angle between two Non-coplanar (i.e., Non-intersecting Lines) Y Direction Cosines of a Line Y

If the Length of a Line OP Y If l m n, , are Direction of cosines of only line AB Y Direction Ratios Y Projection of a Point on a Given Line Y Projection of a

Segment of a Line on Another Line (in the Same Plane or Another) Y Projection of a Broken Line on a Given Line, Or Given n Points C C1 2, , ..., Cn (say)

in Space, to Find the Projection of C Cn1 on a Given Line Y Direction Cosines of a Line Joining two Points P x y z( , , )1 1 1 and Q x y z( , , )2 2 2 Y If O and P are

two Points ( , , )0 0 0 and ( , , ),x y z1 1 1 then to Prove that the Projection of OP on a Line whose Direction Cosines are l m n, , is l x my nz1 1 1+ + Y To Find the

Projection of the Line Joining two Points P x y z( , , )1 1 1 and Q x y z( , , )2 2 2 on Another Line whose d.c.’s are l m n, , Y Angle between two Lines Y To Find the

Perpendicular Distance of a Point P x y z( , , )′ ′ ′ from a Line Through A a b c( , , ) and whose Direction Cosines are l m n, , Y The Plane Y The Equation of a

Plane (Normal Form) Y To Prove that the General Equation of the first degree Y The Reduce the General Equation of the Plane to the Normal Form Y

Intercepts Form Y Plane Through a Given Point and Perpendicular to a Given Line Y Equation of a plane through Three points Y Equations of the

co-ordinate Planes Y Angle between Two Planes Y The Two Sides of a Plane Y To Find the Length of the Perpendicular from the Point ( , , )x y z1 1 1 to a

Given Plane Y To Find the Distance between two Parallel Planes Y A Plane Through the Intersection of two Given Planes Y To Find the Condition that a

Line Y The Angle between a Line and a Plane Y Equations of the Planes Bisecting the Angles between two Given Planes Y Combined Equation of a Pair of

Planes Y Projection on a Plane Y Area of a Triangle Y The Straight Line Y General Equations of a Straight Line Y Symmetrical Form of the Equations of

a Straight Line Y Line Through two Points Y To Transform the General form of the Equations of a Straight Line to Symmetrical Form Y The Plane and the

Straight Line Y To Find the Equation of the Plane Y To Find the Equation of the Plane Through a Given Line and Parallel to Another Line Y Foot and

Length of Perpendicular from a Point to a Line Y Coplanar Lines Y To Determine the Equations of a Straight Line Intersecting two Given Lines Y To Find

56

–A.R. Vasishtha, D.C. Agarwal & A.K. Vasishtha450-16Series: Analytical Geometry of Three Dimensions

–A.R. Vasishtha, D.C. Agarwal & A.K. Vasishtha449-13Series: Analytical Geometry of Two Dimensions

the Perpendicular Distance of a Point from a Line and the Co-ordinates of the Foot of the Perpendicular Y Intersection of Three Planes Y Shortest

Distance Y Skew Lines Y Length the Equations of the Line of Shortest Distance Y Volume of Tetrahedron Y To Find the Volume of a Tetrahedron,

whose three Coterminous Edges in the Right-handed Orientation are a b c, , where a b c, , are Vectors Y To Find the Volume V of a Tetrahedron, in terms of

the Lengths of Three Concurrent Edges and their Mutual Inclinations Y To Find the Volume V of a tetrahedron Y Skew Lines Y The Equations of Two

Skew Lines Y Change of Axes Y Transformation of Co-ordinates Y Change of Origin (Translation of Axes) Y Change of Directions of Axes (Rotation of

Axes) Y Relations between the Direction Cosines of three Mutually Perpendicular Lines YThe Sphere Y Equation of a Sphere Y Plane Section of a

Sphere Y Intersection of two Spheres Y The System of Spheres Through a Given Circle Y The Intersection of a Straight Line and a Sphere Y The Equation

of the Tangent Plane Y Plane of Contact Y Pole and Polar Plane Y Properties of the pole and the polar plane Y The Polar line Y The Angle of Intersection

of Two Spheres Y Touching Spheres Y The Length of the Tangent Y The Radical Plane Y The Properties of the Radical Plane Y The Radical Line (or

Radical Axis) Y Radical Centre Y Coaxial System of Spheres Y The Cylinder Y Right Circular Cylinder Y Tangent Plane to a Cylinder Y Enveloping

Cylinder Y The Cone Y The Cone with the Vertex at the Origin Y The Line x l y m z n/ / /= = Y To Find the General Equation of a Cone Y The Equation

of the Cone with a Given Vertex and a Given Conic as Base Y To Find the Condition for the General Equation of the Second Degree to Represent a Cone

and to Find the Co-ordinates of its Vertex Y The Tangent Line and the Tangent Plane to a Cone Y The Condition of Tangency Y The Reciprocal Cone Y

The Angle between the Lines in which a Plane Cuts a Cone Y Three Mutually Perpendicular Generators Y Three Mutually Perpendicular Tangent Planes

Y Right Circular cone Y The Enveloping Cone Y Central Conicoids Y The Ellipsoid Y The Hyperboloid of One Sheet Y The Hyperboloid of two Sheets

Y The Tangent Plane Y The Condition of Tangency Y The Director Sphere Y The Polar Plane Y Properties of the Polar Planes and the Polar Lines Y Locus

of Chords Bisected at a Given Point Y Normal to a Conicoid Y Number of Normals Y Cubic Curve Through the Feet of the Normals Y To Find the

Equation of the Cone Through Six Concurrent Normals (The Six Normals Drawn from a Point to an Ellipsoid) Y Diametral Plane Y Conjugate Diameters

and Conjugate Diametral Planes Y The Relationship between the Co-ordinates of the Points P Q R, , where OP OQ, and OR are the Conjugate

Semi-diameters of an Ellipsoid Y Properties of Conjugate Semi-diameters of an Ellipsoid.

Y Mappings, Binary Compositions and Relations Y Functions or Mappings Y 'Into' and 'Onto' Mappings Y 'One-one' and 'Many-one' Mappings

Y Inverse function Y Composite of Mappings Y Binary operation or Binary composition Y Types of binary operations Y Relation Y Equivalence relations

Y Equivalence classes Y Partitions Y Fundamental Theorem on equivalence relations Y Partial order relations Y Groups Y Algebraic structure Y Group

Definition Y Abelian group Y Finite and infinite groups Y Order of a finite group Y General properties of a group Y Definition of a group based upon left

axioms Y Composition tables for finite sets Y Addition modulo m Y Multiplication modulo p Y Residue classes of the set of integers Y An alternative set of

postulates for a group Y Permutations Y Groups of permutations Y Cyclic permutations Y Even and odd permutations Y Integral powers of an element of a

group Y Order of an element of a group Y Homomorphism and Isomorphism of groups Y Complexes and subgroups of a group Y Intersection of

subgroups Y Cosets Y Relation of congruence modulo Y Lagrange's theorem Y Order of the product of two subgroups of finite order Y Cayley's theorem Y

Cyclic groups Y Rings Y Ring with unity Y Elementary properties of a ring Y Rings with or without zero divisors Y Integral domain Y Field Y Division ring

or skew field Y Isomorphism of rings Y Subrings Y Subfields Y Characteristic of a ring Y Ordered integral domains Y Ideals Y Principal Ideal Y Principal

Ideal ring Y Divisibility in an integral domain Y Polynomial rings Y Polynomials over an integral domain Y Vector Spaces Y General Properties of vector

spaces Y Vectors subspaces Y Linear combination of vectors Y Linear span Y Linear sum of two subspaces Y Linear dependance and linear independence

of vectors Y Basis of a vector space Y Finite dimensional vector spaces Y Dimension of a finitely generated vector space Y Dimension of a subspace Y

Homomorphism of vector spaces or Linear transformations Y Isomorpshism of vector spaces Y Direct sum of spaces Y Dimension of a direct sum Y

Complementary subspaces Y Coordinates Y Rings (Continued) Y Divisibility of polynomials over a field Y Division algorithm for polynomials over a

field Y Euclidean algorithm for polynomials over a field Y Unique factorization domain Y Quotient rings or Rings of residue classes Y Homomorphism of

rings Y Maximal ideal Y Prime ideals Y Euclidean rings Y Normal Subgroups Y Conjugate elements Y Normalizer of an element of a group Y Class

equation of a group Y Centre of a group Y Conjugate subgroups Y Invariant subgroups Y Quotient groups Y Homomorphisms of groups Y Kernel of a

homomorphism Y Fundamental theorem on homomorphism of groups Y More results on group homomorphism.

Y Multiple Products Y Scalar triple product Y Vector triple product Y Lagrange's identity for four vectors Y Vector product of four vectors Y Reciprocal

system of vectors Y Differentiation and Integration of Vectors Y Vector function Y Scalar fields and vector field Y Limit and continuity of a vector

function Y Derivative of a vector function with respect to a scalar Y Differentiation formulae Y Curves in space Y Integration of vector functions

57

–A.R. Vasishtha & Kiran Vasishtha451-07 Series: Modern Algebra

–A.R. Vasishtha & A.K. Vasishtha452-11 Series: Vector Calculus

Y Gradient, Divergence and Curl Y Partial derivatives of vectors Y The vector differential operator del. (∇) Y Gradient of a scalar field Y Level surfaces

Y Directional derivative of a scalar point function Y Tangent plane and normal to a level surface Y Divergence of a vector point function Y Curl of a vector

point function Y Important vector identities Y Green's, Gauss's and Stoke's Theorems Y Line integrals Y Surface integrals Y Volume integrals

Y Green's Theorem in the plane Y The Gauss's divergence theorem Y Stoke's Theorem Y Line integrals independent of path.

Y Introduction (Concurrent Forces, Lami's Theorem) Y Action and Reaction Y Resultant Force Y Parallelogram of Forces Y λ µ- Theorem Y

Components of a Force in Two Given Directions Y Resolved Parts of a Force Along two Mutually Perpendicular Directions Y Resultant of a Number of

Coplanar Forces Acting at a Point Y Conditions of Equilibrium of a Number of Forces Acting at a Point Y Triangle Law of Forces Y Converse of the

Triangle of Forces Y Lami’s Theorem Y Polygon of Forces Y Equilibrium of a Rigid Body (Moments, Equilibrium of Coplanar Forces) Y Moment

of a Force About a Point Y General Theorems of Moments Y Couple Y If Three Forces Acting in one Plane Upon a Rigid Body Y Theorem Y Necessary

and Sufficient Conditions for Equilibrium of a Rigid Body Y Equation of the Resultant Y Equilibrium of a Rigid Body Under the Action of Three Forces

Only Y Two Important Trigonometrical Theorems Y Virtual Work Y Displacement Y A Rigid Body Y Kinds of Displacement of a Rigid Body Y Rotation

of a Rigid Body About a Point Y Position Vector of a Point After a General Displacement Y Work Done by a Force Y Work Done by a System of

Concurrent Forces Y Work Done by a Couple During a Small Displacement Y Work Done by a System of Forces During a Small Displacement Y Virtual

Displacement and Virtual Work Y The Principle of Virtual Work Y Forces which are Omitted in Forming the Equation of Virtual Work Y Application of the

Principle of Virtual Work Y Strings in Two Dimensions (Common Catenary) Y The Catenary Y Intrinsic Equation of the Common Catenary Y

Cartesian Equation of the Common Catenary Y Some Important Relations for the Common Catenary Y Sag of Tightly Stretched Wires Y Strings in Two

Dimensions (Catenary of Uniform Strength and Strings Resting on a Smooth and Rough Plane Curve) Y Catenary of Uniform Strength Y

Law of Variation of the Mass of String Y Equilibrium of a Light Inextensible String Resting on a Smooth Plane Curve Y Equilibrium of a Heavy Inextensible

String on a Smooth Curve in a Vertical Plane Y Equilibrium of a Light Inextensible String Resting in Equilibrium on a Rough Plane Under the Action of no

External Forces Y Equilibrium of a Heavy Inextensible String Resting in Limiting Equilibrium on a Rough Plane Curve Under the Action of no External

Forces Y Stable and Unstable Equilibrium Y The Work Function Y Work Function Test for the Nature of Stability of Equilibrium Y Potential Energy

Test for the Nature of Stability of Equilibrium Y z-Test for the Nature of Stability Y Stability of a Body Resting on a Fixed Rough Surface Y Centre of

Gravity Y Determination of the C.G. by Integration Y Centre of Gravity of a Plane Area Y Centre of Gravity of a Solid of Revolution Y Centre of Gravity of

Surface of Revolution Y Centre of Gravity when the Density Varies Y Use of Multiple Integrals to Find the Centre of Gravity of any Volume Y Equilibrium

of Forces in Three Dimensions [(Central Axis) Excluding Wrenches] Y To Find the Resultant of any Given Number of Forces Acting on a Particle

Y Necessary and Sufficient Conditions of Equilibrium of a Particle Under the Action of a System of Forces Y Reduction of a System of Forces to a Single

Force and a Couple Y Necessary and Sufficient Conditions of Equilibrium of a Rigid Body Under the Action of a System of Forces Acting at any Points of it

Y Wrench Y Central Axis Y Characteristics of a Central Axis Y Wrench and Screw Y Invariants Y Conditions for a Single Resultant Force Y Equations of the

Central Axis Y Computation of X Y Z L M N, , : , , YConstrained Bodies Y Conditions of Equilibrium of a Rigid Body with one Point Fixed Y Conditions of

Equilibrium of a Rigid Body with Two Fixed Points Y Forces in Three Dimensions (Screws and Wrenches; Null Lines and Null Planes) Y Null

Lines, Null Plane and Null Point (Definitions) Null Lines Y To Find the Equation to Null Plane of a Given Point ( , , )a b c Referred to Any Axes Ox Oy Oz, , Y

To Find the Null Point of the Plane Y To Find the Condition that the Straight Line Y Conjugate Forces and lines (Def.) Y Screw, Pitch and Wrench

(Definitions) Y To Find the Resultant Wrench of Two Given Wrenches Y Reciprocal Screws (Def.) Y Attraction Y The Law of Attraction (Newtonian Law

of Gravitation) Y Attraction Y Attraction of a Rod Y Attraction of a Curvilinear Rod Y Attraction of a Thin Uniform Spherical Shell Y Attraction of a Solid

Sphere Y Potential Y Relation between the Attraction and Potential Y If V be the Potential of an Attracting Mass M, at any Point P x y z( , , ), then ∂∂

∂∂

∂∂

v

x

v

y

v

z, ,

Y Potential of a Finite Rod Y Potential of an Infinite Rod Y Potential of a Circular Disc Y Potential of a Spherical Shell Y Potential of a Solid Sphere.

Y Kinematics in Two Dimensions Y Kinematics of a Particle (Velocity and Acceleration) Y Angular Velocity and Acceleration Y Rate of Change of a

Unit Vector in a Plane Y Relation between Angular and Linear Velocities Y Components of Velocity and Acceleration Along the Co-ordinate Axes Y

Radial and Transverse Velocities and Accelerations Y Tangential and Normal Velocities and Accelerations Y Rectilinear Motion (S.H.M.) Y Velocity

and Acceleration Y Motion Under Constant Acceleration Y Newton’s Laws of Motion Y Equation of Motion Y Simple Harmonic Motion ( . . .)S H M Y

Geometrical Representation of S H M. . . Y Important Results About S.H.M. Y Hooke’s Law Y Particle Attached to One end of a Horizontal Elastic String Y

Particle Suspended by an Elastic String Y Motion Under Inverse Square Law Y Motion of a Particle Under the Attraction of the Earth Y Motion Under

58

–A.R. Vasishtha, R.K. Gupta & A.K. Vasishtha455-16 Series: Statics

–A.R. Vasishtha, R.K. Gupta & A.K. Vasishtha456-19 Series: Dynamics

Miscellaneous Laws of Forces Y Constrained Motion Y Motion in a Vertical Circle Y Some Important Results of the Motion of a Projectile to be Used in

this Chapter Y Motion on the Outside of a Smooth Vectical Circle Y Cycloid Y Motion on a Cycloid Y Motion on the Outside of a Smooth Cycloid with its

Axis Vertical and Vertex Upwards Y Simple Pendulum Y Oscillations of a Simple Pendulum Y Beat of a Pendulum Y The Second’s Pendulum Y Gain or

Loss of Beats (time) by a Clock Y Central Orbits Y Differential Equation of a Central Orbit Y Rate of Description of the Sectorial Area Y Elliptic Orbit

(Focus as the Centre of Force) Y Hyperbolic and Parabolic Orbits (Centre of Force Being the Focus) Y Velocity from Infinity Y Velocity in a Circle Y Given

the Central Orbit, to Find the Law of Force Y Apse and Apsidal Distance Y Property of the Apse-Line Y Given the Law of Force, to Find the Orbit Y The

Inverse Square Law (Planetary Motion) Y Newton’s Law of Gravitation Y Motion Under the Inverse Square Law Y Kepler’s Laws of Planetary Motion

Y Deductions from Kepler’s Laws Y Some Important Geometrical Properties of an Ellipse Y Time of Description of an Arc of a Central Orbit Y To Find the

Time of Description of a Given Arc of a Parabolic Orbit Starting from the Vertex Y To Find the Time of Description of a Given Arc of an Elliptic Orbit

Starting from the Nearer end of the Major Axis Y To Find the Time of Description of a Given Arc of a Hyperbolic Orbit Starting from the Vertex Y Motion

in a Resisting Medium (In a Straight Line Only) Y Terminal Velocity Y Motion of a Particle Falling Under Gravity Y Motion of a Particle Projected

Vertically Upwards Y Projectiles Y The Motion of a Projectile and its Trajectory Y Latus Rectum, Vertex, Forcus and Directrix of the Trajectory Y Time of

Flight, Horizontal Range and Maximum Height Y Velocity at any Point of the Trajectory Y Locus of the Focus and Vertex of the Trajectory Y Some

Geometrical Properties of a Parabola Y Projections to Hit a Given Point Y Range and Time of Flight on an Inclined Plane Y Range and Time of Flight

Down an Inclined Plane Y Envelope of the Trajectories with the Same Velocity of Projection Y Particles Suffered to Describe Parabolic Paths Y Work,

Energy and Impulse Y The Concept of Work Y Work Done by a Constant Force Y Work done by a Variable Force Y Units of Work Y Power Y Kinetic

Energy Y The Work-energy Principle Y Conservative and Non-conservative Forces Y Potential Energy (P.E.) Y The Principle of Conservation of Energy Y

The Principle of Conservation of Energy for the Motion in Plane Y The Principle of Conservation of Linear Momentum Y Impulse Definition When the

Force is Constant Y D' Alembert's Principle (And Equations of Motion of a Rigid Body) Y Motion of a Particle Y Motion of a Rigid Body Y D’

Alembert’s Principle Y General Equations of Motion of a Body Y Linear Momentum Y Motion of the Centre of Inertia Y Motion Relative to the Centre of

Inertia Y Impulse of a Force Y An Important Rule Y General Equations of Motion Under Impulsive Forces Y Moments of Inertia Y Moments and

Products of Inertia with Respect to Three Mutually Perpendicular Axes Y Some Simple Propositions Y Moment of Inertia of a Uniform Rod of Length 2a Y

Moment of Inertia of a Rectangular Lamina Y Moment of Inertia of a Circular Wire Y Moment of Inertia of a Circular Disc Y Moment of Inertia of an Elliptic

Disc Y Moment of Inertia of a Uniform Triangular Lamina about One Side Y Moment of Inertia of a Rectangular Parallelopiped about an Axis Through its

Centre and Parallel to One of its Edges Y M.I. of a Spherical Shell (i.e., Hollow Sphere) about a Diameter Y M.I. of a Solid Sphere about a Diameter Y M.I.

of an Ellipsoid Y Routh’s Rule Y Theorem of Parallel Axis Y Moment of Inertia of a Plane Lamina about a Line Y Principal Axes Y Motion about a Fixed

Axis Y Moment of the Effective Forces about the Axis of Rotation Y Equation of Motion of the Body about the Axis of Rotation Y Moment of Momentum

about the Axis of Rotation Y Kinetic Energy Y Compound Pendulum Y Time of a Complete Small Oscillation of a Compound Pendulum Y Simple

Equivalent Pendulum Y Minimum Time of Oscillation of a Compound Pendulum Y The Centre of Suspension and the Centre of Oscillation of a

Compound Pendulum are Convertible Y Reactions of the Axis of Rotation Y Centre of Percussion Y Centre of Percussion of a Rod Y Centre of Percussion

(In General Case).

Y Real Number System (Bounded and Unbounded Sets of Real Numbers, Neighbourhoods and Limit Points) Y Field Axioms Y Some

Properties of Real Numbers Y Absolute Value or Modulus of a Real Number Y Bounded and Unbounded Subsets of Real Numbers Y Least Upper Bound

or Supremum Y Greatest Lower Bound or Infimum Y Some Properties of Supremum and Infimum Y Completeness Axiom, Existence of Suprema and

Infima of Bounded Sets Y The Set of Real Numbers as a Complete Ordered Field Y Archimedean Property of Real Numbers Y The Denseness Property of

the Real Number System Y Neighbourhood of a Point Y Limit Points of a Set Y Bolzano Weierstrass Theorem Y Interior of a Set Y Open Sets Y Closed Sets

Y Closure of a Set Y Countability of Sets Y Dedikind’s Property of Real Numbers Y Sequences Y Subsequences Y Bounded Sequences Y Convergent

Sequences Y Divergent Sequences Y Algebra of Convergent Sequences Y Monotonic Sequences Y Limit Points of a Sequence Y Cauchy Sequences Y

Cauchy’s General Principle of Convergence Y Limit Superior and Limit Inferior of a Sequence Y Nested Interval Theorem or Cantor’s Intersection

Theorem Y Infinite Series Y Convergence and Divergence of Series Y Cauchy’s General Principle of Convergence for Series Y The Auxiliary Series

Σ ( / )1 np Y Comparison Test Y Cauchy’s Root Test Y D’ Alembert’s Ratio Test Y Cauchy’s Condensation Test Y Raabe’s Test Y Logarithmic Test Y De

Morgan’s and Bertrand’s Test Y Gauss’s Test Y Cauchy Maclaurin’s Integral Test Y Alternating Series Y Alternating Series Test (Leibnitz’s Test) Y Absolute

Convergence and Conditional Convergence Y Limits and Countinuity Y Definition of Limit Y Algebra of Limits Y Right Hand and Left Hand Limits Y

Infinite Limits Y Cauchy’s Definition of Continuity Y Types of Discontinuity Y Algebra of Continuous Functions Y Properties of Continuous Functions Y

Uniform Continuity Y Differentiability Y Derivative at a Point Y Derivative of a Function Y A Necessary Condition for the Existence of Finite

Derivatives Y Algebra of Derivatives Y Rolle’s Theorem Y Lagrange’s Mean Value Theorem Y Cauchy’s Mean Value Theorem Y Taylor’s Theorem with

Lagrange’s Form of Remainder Y Taylor’s Theorem with Cauchy’s Form of Remainder Y Taylor’s Series Y Maclaurin’s Series Y Maclaurin’s Expansion of

Some Basic Functions e xx , sin etc. Y The Riemann Integral Y Partitions and Riemann Sums Y Upper and Lower Riemann Integrals Y Riemann

59

–A.R. Vasishtha, A.K. Vasishtha & Hemlata Vasishtha457-08 Series: Real Analysis

Integrability Y Riemann’s Necessary and Sufficient Conditions for R-Integrability Y Some Classes of Integrable Functions Y Fundamental Theorem of

Integral Calculus Y The Riemann-Stieltjes Integral Y A Generalisation of the Riemann Integral Y Partitions Y Lower and Upper Riemann-Stieltjes

Sums Y The Lower and Upper Riemann-Stieltjes Integrals Y The Riemann-Stieltjes Integral Y The RS-Integrals as a Limit of Sums Y Some Classes of

RS-Integrable Functions Y Algebra of RS-Integrable Functions Y A Relation between R-Integral and RS-Integral Y Uniform Convergence of

Sequences and Series of Functions Y Some Definitions Y Uniform Convergence Y Cauchy’s General Principle of Uniform Convergence Y Tests for

Uniform Convergence Y Uniform Convergence and Continuity Y Uniform Convergence and Integration Y Uniform Convergence and Differentiation Y

Convergence of Improper Integrals Y Some Definitions Y Convergence of Improper Integrals Y Test for Convergence of Improper Integrals of the

First Kind Y Absolute Convergence Y Tests for Convergence of Improper Integrals of the Second Kind.

Y The Calculus of Finite Differences Y Finite Differences Y Fundamental Theorem Y Differences of a Factorial Polynomial Y Generalized Factorial

Function Y Leibnitz’s Rule Y Differences of Zero Y Separation of Symbols Y Interpolation (With Equal Intervals) Y Methods of Interpolation Y

Newton’s Interpolation Formula Y Newton-Gregory Forward Difference Interpolation Formula Y Newton-Gregory Backward Difference Interpolation

Formula Y Subdivision of Intervals Y Interpolation (With Unequal Intervals) Y Divided Differences Y Divided Difference Table Y Newton’s Divided

Difference Formula Y Confluent Divided Difference Y Lagrange’s Interpolation Formula) Y Solution of Algebraic and Trancendental Equations Y

Descarte’s Rule of Signs Y Graphical Method Y Analytic Method or Bisection Method Y Method of Falsa Position Y Iterative Methods Y Newton-Raphson

Method Y Nearly Equal Roots Y Rate of Convergence Y Graeffe’s Root Squaring Method Y Bairstow’s Method Y Simultaneous Linear Algebraic

Equations Y Gauss’s Elimination Method Y Jordan Method Y Crout’s Method Y Method of Factorization Y Jacobi Iterative Method Y Gauss Sidel

Iterative Method Y Matrix Inversion Y Gauss Elimination Method Y Gauss-Jordan Method Y Crout’s Method Y Doolittle Method Y Choleski’s Method Y

The Escalator Method Y Iterative Methods Y Numerical Solution of Ordinary Differential Equations Y Picard’s Method of Successive

Approximations Y Euler’s Method Y Modified Euler’s Method Y Solution by Taylor’s Series Y Milne’s Method Y Runge-Kutta Methods Y A General

Approach to Predictors and Correctors Y Differential Equations of Second Order Y Central Differences & Gauss Interpolation Formulae Y The

Central Difference Operator δ Y Gauss Forward Formula Y Gauss Backward Formula Y Numerical Integration Y A General Quadrature Formula for

Equidistant Ordinates Y The Trapezoidal Rule Y Simpson’s One-third Rule Y Simpson’s Three-eighths Rule Y Weddle’s Rule Y Cote’s Method Y

Approximations Y Least Squares Approximation Y Uniform Approximation Y Rational Approximation.

Y Preliminary Concepts Y States of Matter Y Distinction Between Solid, Liquid and Gas Y Hydrostatics Y Perfect Fluid and Viscous Fluid Y Fluid

Pressure Y Pressure at a Point Y Equality of Pressure in Different Directions Y Transmissibility of Liquid Pressure (Pascal’s Law) Y Hydraulic or Bramah’s

Press Y Density Y Weight in Terms of Density Y Specific Gravity (Relative Density) Y Weight in Terms of Specific Gravity Y Specific Gravity of Mixtures

Y Theorems on Fluid Pressure under Gravity and Condition of Equilibrium of Fluid Y Fluid at Rest Under Gravity Y Equality of Pressure at all

Points in a Horizontal Plane Y Pressure in Heavy Homogeneous Fluid Y Free Surface of a Liquid Y Surfaces of Equal Density Y Pressure at a Point in the

Lower Layer of two Liquids Y Surface of Separation of two Liquids Y Effective Surface Y Head of Liquid Y Characteristics of Liquids Y Pressure Derivative

in Terms of Force Y Equation of Pressure in Different Systems of Coordinates Y Necessary Condition of Equilibrium Y The Pressure and the Potential

Energy Function Y Surfaces of Equal Pressure Y Lines of Force Y Curves of Equal Pressure and Density Y Whole Pressure on a Plane Surface

Y Thrust on a Surface Y Whole Pressure on a Plane Surface Y Layers of Different Liquids Y Thrust on a Horizontal Area Y Thrusts on Curved Surfaces

Y How to Find Resultant Vertical Thrust Y Resultant Vertical Thrust When the Liquid Presses Upwards Y Resultant Vertical Thrust When the Liquid

Presses Y Partly Upwards and Partly Downwards Y How to Find Resultant Horizontal Thrust Y Resultant Thrust Y Resultant Thrust on a Solid — Principle

of Archimedes Y Force of Buoyancy and Centre of Buoyancy Y Thrust on a Vessel Containing Liquid Y Thrust on a Curved Surface Bounded by a Plane

Curve Y Centre of Pressure Y Position of the C.P. Remains Unaltered by Rotation of its Plane Area Y Geometrical Method for Finding C.P. Y Formulae

for the Depth of the C.P. Y Application to Standard Cases Y Analogy in the Determination of C.P. and C.G. Y Effect of Further Immersion Y The Depth of

the C.P. of a Triangle in Terms of the Depths of its Three Vertices Y Coordinates of the Centre of Pressure Y Depth of the C.P. of an Area Immersed in

Various Liquids of Different Densities Which do not Mix Y Equilibrium of Floating Bodies Y General Considerations for Equilibrium Y Conditions of

Equilibrium of a Body Freely Floating in a Liquid Y Volume Immersed Y Bodies Floating in More than one Liquid Y Tension in the String Supporting a

Body Y Weighing a Body Immersed in a Liquid Y Weighing in Air Y Correction for Weighing in Air Y Bodies Floating Under Constraint.

vvv

60

–A.R. Vasishtha, S.K. Sharma & Hemlata Vasishtha458-13 Series: Numerical Analysis

–A.R. Vasishtha, A.K. Vasishtha459-03 Series: Hydrostatics

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