Tromp J. A Geometrical Introduction to Tensor Calculus 2025

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Tromp J. A Geometrical Introduction to Tensor Calculus 2025

Textbook in PDF format Tensors are widely used in physics and engineering to describe physical properties that have multiple dimensions and magnitudes. In recent years, they have become increasingly important for data analytics and machine learning, allowing for the representation and processing of data in neural networks and the modeling of complex relationships in multidimensional spaces. This incisive book provides a geometrical understanding of tensors and their calculus from the point of view of a physicist. With a wealth of examples presented in visually engaging boxes, it takes readers through all aspects of geometrical continuum mechanics and the field and dynamic equations of Einstein, Einstein-Cartan, and metric-affine theories of general relativity. A Geometrical Introduction to Tensor Calculus gives graduate students, advanced undergraduates, and researchers a powerful and mathematically elegant tool for comprehending the behavior and applications of tensors across an array of fields. Offers a physicist’s perspective on geometrical tensor calculus Includes dozens of examples that illustrate the geometrical use of tensors in continuum mechanics and general relativity Can serve as the basis for a course in tensor calculus for physicists and engineers List of Examples Preface Acknowledgments Introduction Linear Spaces and Transformations Properties of Linear Spaces Vector Spaces Linear Transformations Differentiable Manifolds Charts and Coordinates Definition Local Coordinate Changes Functions on Manifolds Orientable Manifolds Vectors and One-Forms Vectors Vectors as Tangents to Curves Bases and Coordinates Vector Field Transformations One-Forms Duality Bases Transformations Alternative Perspective Lie Bracket Tensors Definition Operations on Tensors Addition Tensor Product Contraction Transpose of (2,0) and (0,2) Tensors Transpose of a (1,1) Tensor Transformations Tetrad Formalism Pseudotensors Kronecker or Identity Tensor Logarithms and Exponentials of (1,1) Tensors Tensor Densities and Capacities Pseudotensor Densities and Capacities Levi-Civita Density and Capacity Determinant of Rank-2 Tensors Inverse of Rank-2 Tensors Metric Tensor Formulation Geometrical Meaning Norm of Vectors and One-Forms Metric in Tetrads Adjoint of a (1,1) Tensor Tensor Densities and Capacities Revisited Levi-Civita Pseudotensor Kronecker Determinants Rotations Euler Angles Rodrigues’s Formula Maps between Manifolds Maps Maps between Manifolds of Different Dimensions Pullback Pushforward Maps between Manifolds of the Same Dimensions Differentiation on Manifolds Covariant Derivative Formulation Transformation of Connection Coefficients Divergence Parallel Transport Torsion and Curvature Tensors Bianchi Identities Torsion-Free Connection Covariant Derivative of the Metric Tensor Mixed Covariant Derivative in Tetrad Basis Spin Connection Contracted Bianchi Identities Covariant Derivative of Tensor Densities and Capacities Nonmetricity Euler Derivative Lie Derivative Lie Derivative of Vectors Geometrical Interpretation Autonomous Lie Derivative Lie Derivative of One-Forms Lie Derivative of (p, q) Tensors Lie Derivative of Functions Lie Derivative of Metric Tensors Lie Derivative of Levi-Civita Tensor Differential Forms Definition Operations on Forms Addition Exterior Product Interior Product k-Vectors Hodge Dual Volumes Properties Surfaces Exterior Derivative Coordinate-Free Definition Exact Forms Commutativity with Pullback and Pushforward Lie Derivative of a Form Vector- and Tensor-Valued Forms Transformations of Tensor-Valued Forms Operations on Tensor-Valued Forms Connection One-Forms Torsion Two-Forms Exterior Covariant Derivative Covariant Lie Derivative Curvature Two-Forms Commutator of Covariant Lie and Exterior Covariant Derivatives Bianchi Identities Revisited Nonmetricity Revisited Integration of Forms Line Integrals Surface Integrals Volume Integrals Stokes’s Theorem Fundamental Theorem of Calculus Green’s Theorem Gauss’s Theorem Stokes’s Theorem Variational Principles Noether’s Theorem Applications Glossary Bibliography Author Index General Index