Geometrical Introduction to Tensor Calculus

Jeroen Tromp is the Blair Professor of Geology and Professor of Applied and Computational Mathematics at Princeton University. His books include Theoretical and Computational Seismology and (with F. A. Dahlen) Theoretical Global Seismology (both Princeton).

• List of ExamplesPrefaceAcknowledgments1 Introduction2 Linear Spaces and Transformations2.1 Properties of Linear Spaces2.2 Vector Spaces2.3 Linear Transformations3 Differentiable Manifolds3.1 Charts and Coordinates3.2 Definition3.3 Local Coordinate Changes3.4 Functions on Manifolds3.5 Orientable Manifolds4 Vectors and One-Forms4.1 Vectors4.1.1 Vectors as Tangents to Curves4.1.2 Bases and Coordinates4.1.3 Vector Field4.1.4 Transformations4.2 One-Forms4.2.1 Duality4.2.2 Bases4.2.3 Transformations4.3 Alternative Perspective4.4 Lie Bracket5 Tensors5.1 Definition5.2 Operations on Tensors5.2.1 Addition5.2.2 Tensor Product5.2.3 Contraction5.2.4 Transpose of (2,0) and (0,2) Tensors5.2.5 Transpose of a (1,1) Tensor5.3 Transformations5.3.1 Tetrad Formalism5.3.2 Pseudotensors5.4 Kronecker or Identity Tensor5.5 Logarithms and Exponentials of (1,1) Tensors5.6 Tensor Densities and Capacities5.6.1 Pseudotensor Densities and Capacities5.7 Levi-Civita Density and Capacity5.8 Determinant of Rank-2 Tensors5.9 Inverse of Rank-2 Tensors5.10 Metric Tensor5.10.1 Formulation5.10.2 Geometrical Meaning5.10.3 Norm of Vectors and One-Forms5.10.4 Metric in Tetrads5.11 Adjoint of a (1,1) Tensor5.12 Tensor Densities and Capacities Revisited5.13 Levi-Civita Pseudotensor5.14 Kronecker Determinants5.15 Rotations5.15.1 Euler Angles5.15.2 Rodrigues’s Formula6 Maps between Manifolds6.1 Maps6.2 Maps between Manifolds of Different Dimensions6.2.1 Pullback6.2.2 Pushforward6.3 Maps between Manifolds of the Same Dimensions7 Differentiation on Manifolds7.1 Covariant Derivative7.1.1 Formulation7.1.2 Transformation of Connection Coefficients7.1.3 Divergence7.1.4 Parallel Transport7.1.5 Torsion and Curvature Tensors7.1.6 Bianchi Identities7.1.7 Torsion-Free Connection7.1.8 Covariant Derivative of the Metric Tensor7.1.9 Mixed Covariant Derivative in Tetrad Basis7.1.10 Spin Connection7.1.11 Contracted Bianchi Identities7.1.12 Covariant Derivative of Tensor Densities and Capacities7.1.13 Nonmetricity7.2 Euler Derivative7.3 Lie Derivative7.3.1 Lie Derivative of Vectors7.3.2 Geometrical Interpretation7.3.3 Autonomous Lie Derivative7.3.4 Lie Derivative of One-Forms7.3.5 Lie Derivative of (p, q) Tensors7.3.6 Lie Derivative of Functions7.3.7 Lie Derivative of Metric Tensors7.3.8 Lie Derivative of Levi-Civita Tensor8 Differential Forms8.1 Definition8.2 Operations on Forms8.2.1 Addition8.2.2 Exterior Product8.2.3 Interior Product8.3 k-Vectors8.4 Hodge Dual8.5 Volumes8.5.1 Properties8.6 Surfaces8.7 Exterior Derivative8.7.1 Coordinate-Free Definition8.7.2 Exact Forms8.7.3 Commutativity with Pullback and Pushforward8.8 Lie Derivative of a Form8.9 Vector- and Tensor-Valued Forms8.9.1 Transformations of Tensor-Valued Forms8.9.2 Operations on Tensor-Valued Forms8.9.3 Connection One-Forms8.9.4 Torsion Two-Forms8.9.5 Exterior Covariant Derivative8.9.6 Covariant Lie Derivative8.9.7 Curvature Two-Forms8.9.8 Commutator of Covariant Lie and Exterior Covariant Derivatives8.9.9 Bianchi Identities Revisited8.9.10 Nonmetricity Revisited8.10 Integration of Forms8.10.1 Line Integrals8.10.2 Surface Integrals8.10.3 Volume Integrals8.11 Stokes’s Theorem8.11.1 Fundamental Theorem of Calculus8.11.2 Green’s Theorem8.11.3 Gauss’s Theorem8.11.4 Stokes’s Theorem8.11.5 Variational Principles8.11.6 Noether’s Theorem8.11.7 ApplicationsGlossaryBibliographyAuthor IndexGeneral Index

"A refreshing exploration of both the theory and . . . application of tensor calculus."---Kymani Tieral Keden Armstrong-Williams, Physics Book Reviews