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2. Artin, M. Algebra. Prentice-Hall, Englewood Cliffs, NJ, 1994.

3. Berberian, SK. A First Course in Real Analysis. Springer Verlag, New York, 1974.

4. Brown, AL, Page, A. Elements of Functional Analysis. Van Nostrand Rein-hold, London, 1970.

5. Burkill, JC, Burkill, H. A Second Course in Mathematical Analysis. Cam-bridge University Press, Cambridge, 1970.

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9. Drager, LD, Foote, RL. The contraction mapping lemma and the inverse func-tion theorem in advanced calculus. American Mathematical Monthly, 93(1):53–54, 1986.

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Index

# ∃, 1 ∀, 1 ⇒, 1 ⇔, 1 ∅, 1 ∈, 1 ∉, 1

, 2 , 2 , 2

R, 2 , 2

∪, 2 ∩, 2

A

Absolute maximum, 151 Absolute minimum, 151 Absolute value, 5 Additive set function, 229, 236 Adjoint, 21 Akilov, 130 Apostol, 26, 127 Ascending, 258, 259 Associative law, 17, 18

B

Balloon, 239, 243, 244 Banach–Cacciopoli principle, 119 Basic k-form, 254 Berberian, 152 Bessel's equation, 107 Bijection, 3 Bijective, 3, 131 Bolzano–Weierstrass theorem, 7, 45 Boundary, 209 Boundary of a k-chain, 277

Boundary of a set, 44 Bounded, 5 Bounded above, 5 Bounded below, 5 Bounded sequence, 7 Brown, 127 Burkill, 138

C C 1 function (or C 1 map), 99 Cartesian product, 3, 24 Cauchy complete, 30 Cauchy convergence criterion, 8 Cauchy sequence, 7, 30 Cauchy–Schwarz inequality, 26 Centre of a ball in Rn, 40 Centred at a, 40, 124 Chain rule, 11, 85, 86, 135, 235 Change of variables, 14, 217 Change of variables formula, 14 Chaudhary, 130 Cheng, 174 Class, 1 Class C 1, 99 Closed ball, 42, 125 Closed cuboid, 177 Closed cuboid formed by, 180 Closed differential form, 270 Closed interval, 6 Closed r-ball in Rn, 42 Closed subset of Rn, 41 Closure of a set, 43, 247 Closure point, 43 Collection, 1 Column matrix, 16 Commutative law, 17 Compact, 230 Compact set, 10, 48, 57, 161, 185, 209,

223, 235 Compactness, 9, 45, 127

Index 390

Complement, 2, 41 Component, 24 Composite function, 4, 85 Composition, 4, 168, 233 Composition of linear maps, 34, 35 Conformable for addition, 16 Conformable for multiplication, 17 Connected set, 51, 145 Connectedness, 145 Constant map, 35, 79 Constrained extrema, 152 Constrained local maximum, 152 Constrained local minimum, 153 Constrained local strict maximum, 169 Constrained local strict minimum, 169 Constrained optimisation, 151 Constraint equations, 157 Constraint function, 157 Constraint set, 157 Constraints, 151 Content of a set, 203 Continuity, 8 Continuous at a point, 9, 52 Continuous everywhere, 52 Continuous on a set, 9 Continuously differentiable, 99, 124,

137, 144 Contraction (mapping or map), 117 Contraction mapping theorem, 119 Contraction principle, 119, 126 Converge (double sequence), 70 Converge (sequence), 7, 30 Converge absolutely, 71 Convergent double series, 71 Convergent improper integral, 15 Convergent sequence, 7, 30 Convex set, 49, 90, 112, 224 Coordinate, 24 Coordinate transformation, 247 Cover (or covering), 47 Cube, 177 Cuboid, 94, 177 Cuboid formed by, 180 Curl, 296

D

De Morgan’s laws, 3 Decreasing function, 12 Decreasing sequence, 7 Derivative at a point, 11, 78 Derivative function, 11 Determinant, 15, 19, 98, 137, 232 diam E , 217, 219 Diameter, 217, 218, 219, 233 Differentiable, 11 Differential form, 256 Differentiation, 8 Directional derivative, 77, 82 Dirichlet function, 197 Divergence, 296 Divergence theorem, 284 Divergent improper integral, 15 Domain, 3 Dot product, 25 Double limit, 70 Double sequence, 69 Double series, 69, 71 Drager, 123

E

Edge of a cuboid, 177 Elementary matrix, 20, 37, 233 Elementary real analysis, 1 Elementary row and column

operations, 20 Empty set, 1 Equivalent parametrisation, 252 Euclidean norm, 25 Euclidean n-space, 24 Euler’s relation, 103 Euler's theorem, 103 Exact differential form, 270 Exponential function, 11 Exterior derivative, 266 Exterior product, 261 Extremum, 151

Index 391

F

Family, 1, 4, 47, 183, 224 Field axioms, 4 Finite subcover (or subcovering), 10,

48, 241 Finney, 153 First partial derivative, 109 Fixed point, 117 Foote, 123 Fréchet differentiable at a point, 79 Fubini’s theorem, 200, 245 Functions, 1 Fundamental ttheorem of integral

calculus, 14

G

Gateaux derivative, 79 Gauss’ theorem, 250 General Stokes theorem, 274 Gilsdorf, 246 Gopalkrishnan, 21 Gradient, 77, 295 Graves, 138 Greatest lower bound, 6 Green’s theorem, 250

H

Have content, 203 Heine–Borel theorem, 10, 48 Hessian form, 165, 169 Hessian matrix, 165 Hoffman, 21 Homogeneous of degree p, 102, 103

I

Identity map, 34 Identity matrix, 18, 36 Image, 3 Implicit function theorem, 134, 136,

144

Implicit function theorem in two dimensions, 146

Improper integral, 15 Increasing function, 11 Increasing sequence, 7 Induced mapping, 270 Infimum, 6 Infinite-dimensional space, 127 Injective, 3, 124, 131 Inner product in Rn, 25 Integrable, 13, 203 Integral test, 15, 74 Integration, 8 Integration by parts, 14, 278 Interior of a set, 42, 178 Interior point, 42 Intermediate value theorem, 117, 147 Intermediate value theorem (Bolzano),

9 Intersection, 2 Intersection of open sets, 40 Inverse, 3 Inverse function theorem, 121, 124 Inverse image, 3 Inverse of a differentiable map, 124 Inverse of a linear map, 34, 226 Inverse of a matrix, 21, 36 Inversion, 18 Invertible, 3 Invertible linear map, 34, 124, 232 Invertible matrix, 21, 36 Isolated point, 53 Iterated (or repeated) integral, 200, 213

J

Jacobian, 98, 232 Jacobian matrix, 98, 137

K Kantorovich, 130 k-chain, 275 k-index, 257

Index 392

k-surface, 251 Kumaresan, 138 Kunze, 21

L

L(Rn,Rm), 61, 64 Lagrange equations, 157 Lagrange multipliers, 138, 151, 157 Lagrangian, 157, 169 Lang, 127 Leading diagonal, 15 Least upper bound, 5 Least upper bound axiom, 4 Left distributive aw, 18 Leibnitz’s formula, 107 Limit of a function, 8, 54 Limit of a sequence, 7, 30 Limit point, 8, 53, 55, 56 Linear (or Fréchet) derivative, 79 Linear map (operator, transformation),

32 Linear ordering axioms, 4 Lipschitz condition, 115 Local inverse, 127 Local maximum, 11, 151 Local minimum, 11, 151 Local solution, 23, 137 Local strict maximum, 151, 161 Local strict minimum, 151, 161 Locally invertible, 127 Logarithm function (natural), 11 Loomis, 130 Lower bound, 5 Lower face, 179 Lower integral, 13 Lower Riemann integral, 188, 203, 204 Lower sum, 13, 187

M Main diagonal, 15 Map, 3 Mapping, 3

Matrices and determinants, 1 Matrix, 15 Matrix of a linear map, 34 Mean value theorem, 12, 110 Möbius band, 61, 108 Monotone function, 12 Monotone sequence, 7 Morrey, 238 Multi-index, 257

N Nanda, 130 Negative definite, 165 Negative semidefinite, 165 Newton’s method, 12, 129 Nonoverlapping, 180, 185, 231 Nonsingular matrix, 21 Norm of a linear map, 62 Null matrix, 17 n-vector, 24

O

Objective function, 151, 157 One-to-one, 3 One-to-one correspondence, 3 Onto, 3 Open ball, 42, 124, 126 Open cover (or covering), 10, 47, 48 Open cuboid, 177 Open cuboid formed by, 180 Open interval, 6 Open mapping, 131 Open set, 10, 40, 41, 67 Optimisation problem, 151 Ordered field, 5

P

Page, 127 Pairwise disjoint, 2 Partial derivative, 96 Partial sums of a double series, 71 Partition, 12

Index 393

Paving of a cuboid, 180 Polar coordinates, 124, 222, 239, 253 Positive definite, 165 Positive semidefinite, 165 Product of a scalar and a linear map, 35 Product of linear maps, 35 Product of matrices, 17 Product of scalar and vector, 24 Proper subset, 2 Protter, 238 Pugh, 14

Q Quadratic form, 165, 167, 168

R Radius of a ball in Rn, 40, 125 Range, 3 Range space, 3 Rational components, 203 Rational polar coordinates, 246 r-ball in Rn, 40 Rectangular coordinates, 124, 242 Refinement, 188, 190, 191, 195, 220 Repeated (or iterated) limit, 70 Restriction, 3 Riemann integrable, 13, 189 Riemann integral, 189, 203, 204 Right distributive law, 18 Row matrix, 16 Rudin, 26, 127

S Scalar, 24 Scalar product, 25 Schwarz, 109 Schwarz’s theorem, 110, 269 Second partial derivative, 109, 169 Self map, 117 Sequence, 6 Set, 1 Set algebra, 1

Set function, 229 Shirali, 53, 74 Shrinking lemma, 119 Shrinking map, 117 Simple differential form, 254 Singh, 21 Sohrab, 26 Spivak, 26 Square matrix, 15 Standard basis, 25 Standard representation, 258, 260 Stereographic projection, 39 Sternberg, 130 Stokes theorem, 250 Strictly decreasing function, 12 Strictly increasing function, 12 Subcover (or subcovering), 10, 47 Submatrix, 21 Subsequence, 7, 45, 46 Subset, 1 Substitution, 217 Substitution form of the FTC, 250 Substitution rule, 14, 249 Sum in Rn, 24 Sum of a double series, 71 Sum of linear maps, 35 Sum of two matrices, 16 Supremum, 5 Surjective, 3, 131 System of (or simultaneous) equations,

23

T Taylor’s theorem, 12, 109, 112 Term of a double series, 71 Terms of a double sequence, 69 Thomae function, 197 Thomas, 153 Transformation, 32 Transformation formula, 14, 217, 228,

231, 235, 250 Transformation of content, 222 Transpose, 16

Index 394

Transposition, 16 Triangle inequality in R, 5 Triangle inequality in Rn, 26, 28, 41 Trigonometric functions, 11, 246 Twice continuously differentiable, 112

U Unconstrained extremum, 167 Unconstrained optimisation, 151 Uniformly continuous on a set, 10, 57 Union, 2 Union of open sets, 40 Unique local solution, 137 Upper bound, 5 Upper face, 179 Upper integral, 13 Upper Riemann integral, 188, 203, 204 Upper sum, 13, 187

V Vasudeva, 53, 74 Vector, 24 Volume of a cuboid, 177

W Wedge product, 261 Wildberger, 246

Y Young, 109 Young’s theorem, 111

Z Zero vector, 25