Index

Absolute differential calculus IAcceleration 6,48, 52, 54, 58, 66, 67,94

centripetal 66convective 82local 82particle 82path 66,91roof components 53

Anglebetween two vectors 8of rotation 10,41

Ant 87,91 ,92, 94Arrow 3Axis 22Axis of rotation 10,41

Balloons 65nBase vectors 25

Cartesian 10, 55cellar 30, 50, 56, 88general 25reciprocal 28,29,57roof 30,51,89

BasisCartesian 10,40, 82change of 36, 40for 2nd order tensors 19

general 25on a surface 94

Bead 91,94Beltrami's second differential 95Binormal 66Body 81

elastic 99motion of 46,81, 101, 103rigid 45

Buck 84n

Calculus of variations 90Cartesian

coordinates 6, 7tensor notation 28,43

Cayley-Hamilton theorem 44Chadwick InChristoffel symbols 52,57,58,59,64,

80,90,94Classical mechanics 45Codazzi equations 96,97,104Compatibility condition 97Complex variable 69nComponents

Cartesian 10cellar 30contravariant 30

107

108

Components (cont.)covariant 30of V x v, Vv 80of the identity tensor 39physical 51,57,99roof 30,51 ,57

Conservationform 99laws 46of mass 83, 86

Conservative force 46Continuum 22

kinematics of 81mechanics 1, 6n, 46,77,81,83,87,98,

99Contraction 100Coordinate

curve 51,56dependent 88free 2, 46, 49, 78system 2ntransformation 54

Coordinates 2bipolar cylindrical 69, 100Cartesian 2, 6, 7, 65, 79, 84, 99change of 40, 62circular cylindrical 54, 62, 63, 73, 83,

99cylindrical 69elliptic cylindrical 69, 100Eulerian 82Gaussian surface 87, 88, 104general 3-dimensional 54, 55, 79Lagrangian 82oblique Carteian 54, 98, 99orthogonal 60parabolic cylindrical 69, 100plane polar 50referential 82spatial 82spherical 2, 54, 62, 67, 69, 80, 98, 99

Covariantcomponents 30derivat ive 79, 98

Covariantly constant 80, 103Cross product 11, 15

Cartesian components 13cellar components 32determinant for 14

distributive law 12in a general basis 32of V with a vector 77,81roof components 34

Curl 77Curvature 93

Gaussian 91,93,95, 104, 105mean 95, 104, 105of a curve 66, 103

Curved continua 63, 65nCycloid 47n

Descartes 6Determinant 14,23,27,34,37,59,

100Differential geometry 87

extrinsic 87,94intrinsic 87

Direct product 16of V with a vector field 77,81trace of 23

Directed line segment 3Direction 4Directional derivative 72Displacement 5,6, 15,78,98,100Divergence 77

of a tensor 98theorem 78,83,95,96,104

Domain 16n, 24Dot product 7, 15

distributive law 9, 20double 40in a general basis 28of 2nd order tensors 24of vectors 7of V with a vector 77, 80simplification 31

Dyads 16

Eigenvalue problem 23,75nonlinear 75, 76

Eigenvector lOn, 23, 75, 102Einstein 1,27, 91nEisenhart 3nElasticity 15,82,99Ellipsoid 75, 88Energy 47, 65

Index

Index

Equations of motioncontinuum 98Euler's equations of motion 103gross 45ideal gas 99mass point 45

Equilibriumforce 90, 104, 105moment 105

Equivalence class 3Euclidean

geometry 4n, 21space 1, 3, 87

Euler 45nEuler's equations of motion 103Exceptional points 50, 62Exponential representation 43

Field 71nFinite rotation 5,6, 10,41,42Finney 84nFirst fundamental form 89Fluid

flow 71ideal 99incompressible 99mechanics 82

Force 5,6,15,20,45,90,91body 98central 49, 67conservative 46equilibrium 90, 105

Frame 2,36body-fixed 103change of 2, 103fixed 2inertial 2, 45, 103moving 6n

Fourier's law 71Friction 91

Gauss-Bonnet theorem 93Gauss' equations 94, 97, 105General relativity 1,2,60,63,68,87,91Geodesic 89, 90

differential equation for 90, 104triangle 92

Geometric invariant 9, 36Goldstein 45n, 103Gradient 71

of a vector field 77surface 95

Grassmann 13Gross equations of motion 45,101

Head 3Head-to-tail rule 4Heat flux 71, 78Helix 66Hilbert 3nHildebrand 59, 90n

Ideal gas 99Identity tensor 17,80Index

argument 55cellar 28dummy 27, 57nfree 31Greek 88,91raising and lowering 39roof 28

Inertial frame 2,45,65, 102, 103Invariant 2,12,80,100

geometric 36intrinsic 91operators 78scalar 72

Inverseof a tensor 24transformation 81

Jacobian 27, 32; 80, 85matrix 27

Kepler's law 49,54Kinetic energy 46Kronecker delta 29,31,34,89

Lagrange's equations 105Lame coefficients 100

109

110

Lanczos InLaplacian 99

surface 95Law of cosines 8nLinear

functional 15operator 15vector space 5, 23

Mass 45,101center of 44, 101,102density 22, 78, 83, 99, 101

Material derivative 83Matrix 5, IOn

diagonal 60Jacobian 27multiplication 29, 37ntranspose 29

McConnell 80Mean value theorem 97Membrane 63, 104,105Metric tensor 60, 68, 89, 95Metrical properties 87Misner 1, 87nMomentum

flux 22,99linear 45, 101rotational 46, 101, 102

Navier's equations 100Newton's law 2, 45, 55, 58, 69Noll In,23n

Operatorcovariant derivative 79, 95del 76differential 76, 82gradient 77invariant 82linear 15partial derivative 79projection 15replacement 31

Orbit 47,67Origin 4

Orthogonalvectors 8tensor 40

Pais 91nParallelepiped 11, 12,20,25,54Parallelogram

area of 11,21,48rule 5,20

Parametricequation 47, 65, 66, 89, 97form 47,88, 103representation 72

Path derivative 72, 76Permutation

cyclic 12even 33odd 33

Permutation tensor 32,40cellar components 32roof components 34surface 94, 103

Polar decomposition 44Position 4Potential 46, 72Power 46Pressure 99, 105Projection 15Pythagorean theorem 7,8

Quadratic function 75

Range 16n, 24, 81Region

convex 97,98non-convex 98

Reflector 41Relativity theory 6nRiemannian manifold 1, 80Right hand rule 6Rigid body 5, 6, 45, 102

dynamics 15,22motion 2n, 45, 46, 100, 103

Rotations 10Rotator 41,42

Index

Index

Scalarfield 71ntriple product 11

Schwarz inequality 20Serret-Frenet equations 66Shells 2,63,87Soap film 105Solid mechanics 78Span 21Speed 47Spivak 84nStrain-displacement relations 100Strain rate 78Stress

function 72normal 22principal 23shear 22, 72tensor 15,23, 98vector 22

Stress-strain relations 99String 90, 94Struik 66, 93n, 97nSummation convention 27Surface

divergence 95ellipsoidal 89minimal 105non-degenerate 89of revolution 103, 104quadric 87

Tail 3Tensor 15

basis for 35Cartesian basis for 19Cartesian components 17, 18cellar components 34components 34contravariant components 63covariant components 63curvature 95, 103, 104definition 15, 16derivative of 16domain of 16nequality 16field 71n

identity 17,80matrix of components 18metric 60, 68, 89, 95mixed components 35, 63moment of inertia 15, 16n, 102momentum flux 15, 22,99orthogonal 40permutation 32, 34,40physical components 99positive definite 24range of 16nreflector 41roof components 35rotator 41second order 15singular 17skew 17,22strain 15,99strain rate 78stress 15, 23, 98surface metric 89surface permutation 94symmetric 17, 353rd order 40, 65trace of 23transpose of 17

Tetrahedron 11,21, 22Theorema egregium 95Thomas 84nThome 1,87nTorque 11, 15,46Torsion

ofa bar 72ofa curve 66

Trace 23, 100Trajectory 47, 54

length of 47smooth 47

Transformation ofChristoffel symbols 64coordinates 54tensors 37, 63vectors 37, 63

Truesdell In, 2n, 45n

Unitnormal 66, 94

111

112

Unit (cont.)outward normal 92, 104vector field 92tangent 66tensor 17

Vector 3addition 4, 10as an equivalence class 3basis 10,25binormal 66Cartesian components 10cellar components 30contravariant components 30covariant 63covariant components 30definition 3direct product 16dot product 7eigen 23equality 7field 71n,92finite rotat ion 42heat flux 71,78length 4multiplication by a scalar 5normal 66, 94orthogonality 8position 4

projection of 15radius 47reciprocal base 28roof components 30, 57space 5,6,15stress 22triple product 12, 39unit 4unit normal 66unit tangent 66zero 4

Velocity 6,47,65angular 16n, 52field 83fluid 22, 72, 99in rigid body motion 101particle 82radial 52

Viscosity 78Vorticity 71,78

Weingarten's equation 95,97, 105Wheeler 1, 87nWork 15

Zerovector 4tensor 17

Index

Undergraduate Texts in Mathematics(continuedfrom page ii)

Hilton/Holton/Pedersen: MathematicalReflections: In a Room with ManyMirrors.

Iooss/Joseph: Elementary Stabilityand Bifurcation Theory. Secondedition.

Isaac: The Pleasures of Probability .Readings in Mathematics.

James: Topological and UniformSpaces.

Janich: Linear Algebra .Janlch: Topology .Kemeny/Snell: Finite Markov Chains .Kinsey: Topology of Surfaces.Klambauer: Aspects of Calculus .Lang: A First Course in Calculus. Fifth

edition.Lang: Calculus of Several Variables .

Third edition.Lang: Introduction to Linear Algebra .

Second edition.Lang: Linear Algebra. Third edition .Lang: Undergraduate Algebra. Second

edition .Lang: Undergraduate Analysis .Lax/Burstein/Lax: Calculus with

Applications and Computing .Volume 1.

LeCuyer: College Mathematics withAPL.

LidllPilz: Applied Abstract Algebra.Second edition .

Logan: Applied Partial DifferentialEquations.

Macki-Strauss: Introduction to OptimalControl Theory .

MaUtz: Introduction to MathematicalLogic.

MarsdenlWeinstein: Calculus I, II, III.Second edition.

Martin: The Foundations of Geometryand the Non-Euclidean Plane.

Martin: Geometric Constructions .Martin: Transformation Geometry: An

Introduction to Symmetry.Millman/Parker: Geometry: A Metric

Approach with Models . Secondedition .

Moschovakis: Notes on Set Theory .

Owen: A First Course in theMathematical Foundations ofThermodynamics.

Palka: An Introduction to ComplexFunction Theory.

Pedrick: A First Course in Analysis .PeressinilSullivan/Uhl: The Mathematics

ofNonlinear Programming .Prenowitz/Jantosciak: Join Geometries .Priestley: Calculus: A Liberal Art.

Second edition.Protter/Morrey: A First Course in Real

Analysis . Second edition .ProtterlMorrey: Intermediate Calculus .

Second edition.Roman: An Introduction to Coding and

Information Theory.Ross: Elementary Analysis: The Theory

of Calculus.Samuel: Projective Geometry .

Readings in Mathematics.Scharlau/Opolka: From Fermat to

Minkowski .Schiff: The Laplace Transform: Theory

and Applications.Sethuraman: Rings, Fields, and Vector

Spaces: An Approach to GeometricConstructability.

Sigler: Algebra .Silverman/Tate: Rational Points on

Elliptic Curves.Simmonds: A Brief on Tensor Analysis .

Second edition.Singer: Geometry: Plane and Fancy.Singer/Thorpe: Lecture Notes on

Elementary Topology andGeometry .

Smith: Linear Algebra . Third edition.Smith: Primer of Modem Analysis .

Second edition.StantonIWhite: Constructive

Combinatorics .Stillwell: Elements ofAlgebra:

Geometry, Numbers, Equations.Stillwell : Mathematics and Its History .Stillwell: Numbers and Geometry.

Readings in Mathematics.Strayer: Linear Programming and Its

Applications.

Undergraduate Texts in Mathematics

Thorpe: Elementary Topics in DifferentialGeometry.

Toth: Glimpses of Algebra and Geometry.Readings in Mathematics.

Troutman: Variational Calculus andOptimal Control. Second edition.

Valenza: Linear Algebra: An Introductionto Abstract Mathematics .

WhyburnlDuda: Dynamic Topology .Wilson: Much Ado About Calculus .