Index [link.springer.com]978-1-4419-8522...Index Equations of motion continuum 98 Euler's equations...
Transcript of Index [link.springer.com]978-1-4419-8522...Index Equations of motion continuum 98 Euler's equations...
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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
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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
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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
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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
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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
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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
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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
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LeCuyer: College Mathematics withAPL.
LidllPilz: Applied Abstract Algebra.Second edition .
Logan: Applied Partial DifferentialEquations.
Macki-Strauss: Introduction to OptimalControl Theory .
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MarsdenlWeinstein: Calculus I, II, III.Second edition.
Martin: The Foundations of Geometryand the Non-Euclidean Plane.
Martin: Geometric Constructions .Martin: Transformation Geometry: An
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Second edition.Roman: An Introduction to Coding and
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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
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Combinatorics .Stillwell: Elements ofAlgebra:
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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 .