An essential guide to understanding tensors through physics.
Understanding tensors is essential for physics students who encounter phenomena where direction matters. A jet stream rushing overhead can trigger vertical convection that leads to thunderstorms. An unbalanced car wheel spinning around a horizontal axis produces a wobble in the vertical plane. Astronauts orbiting Earth observe an electrostatic field as a magnetic one. In all these cases, tensors offer a language that captures directional relationships with precision. In the second edition of Tensor Calculus for Physics, Dwight E. Neuenschwander provides an accessible guide that shows how tensor logic arises naturally from physical problems.
Tensors' true elegance lies in how they transform: when coordinates change from one system to another, tensors follow the same rules, allowing physical laws to retain their form across perspectives. Students are often introduced to tensors piecemeal through the inertia tensor in classical mechanics or the polarization tensor in electricity and magnetism. While useful, this fragmented approach does not prepare the student for tensor features such as affine connections, dual basis vectors, and covariant derivatives they will encounter in advanced studies such as general relativity, continuum mechanics, or non-Euclidean geometry. This concise guide builds from the ground up, providing a clear, step-by-step progression that embeds tensors in contexts where their power becomes self-evident.
This extensively revised second edition incorporates more illustrative examples and carefully designed homework problems to strengthen understanding. Now accompanied by a solutions manual, this edition is an ideal resource for courses in general relativity, covariant electrodynamics, continuum mechanics, fluid dynamics, materials science, and any discipline where tensors illuminate the structure of physical reality.
Table of Contents:
Contents
Preface
Acknowledgements
1. Tensors Need Context
1.1. Why Aren't Tensors Defined by What They Are?
1.2. Euclidean Vectors, without Coordinates
1.3. Derivatives of Euclidean Vectors with Respect to a Scalar
1.4. The Euclidean Gradient
1.5. Euclidean Vectors, with Coordinates
1.6. Euclidean Vector Operations with and without Coordinates
1.7. Transformation Coefficients as Partial Derivatives
1.8. What Is a Theory of Relativity?
1.9. Vectors Represented as Matrices
1.10. Discussion Questions and Exercises
2. Two-Index Tensors
2.1. The Electric Susceptibility Tensor
2.2. The Inertia Tensor
2.3. The Electric Quadrupole Tensor
2.4. The Electromagnetic Stress Tensor
2.5. Transformations of Two-Index Tensors
2.6. Finding Eigenvectors and Eigenvalues
2.7. Two-Index Tensor Components as Products of Vector Components
2.8. More Than Two Indices
2.9. Integration Measures and Tensor Densities
2.10. Discussion Questions and Exercises
3. The Metric Tensor
3.1. The Distinction between Distance and Coordinate Displacement
3.2. Relative Motion
3.3. Upper and Lower Indices
3.4. Converting between Vectors and Duals
3.5. Contravariant, Covariant, and "Ordinary" Vectors
3.6. Tensor Algebra
3.7. Tensor Densities Revisited
3.8. Discussion Questions and Exercises
4. Derivatives of Tensors
4.1. Signs of Trouble
4.2. The Affine Connection
4.3. The Newtonian Limit
4.4. Transformation of the Affine Connection
4.5. The Covariant Derivative
4.6. Relation of the Affine Connection to the Metric Tensor
4.7. Divergence, Curl, and Laplacian with Covariant Derivatives
4.8. Discussion Questions and Exercises
5. Curvature
5.1. What Is Curvature?
5.2. The Riemann Tensor
5.3. Measuring Curvature
5.4. Linearity in the Second Derivative
5.5. Discussion Questions and Exercises
6. Covariance Applications
6.1. Covariant Electrodynamics
6.2. General Covariance and Gravitation
6.3. Discussion Questions and Exercises
7. Tensors and Manifolds
7.1. Tangent Spaces, Charts, and Manifolds
7.2. Metrics on Manifolds and Their Tangent Spaces
7.3. Dual Basis Vectors
7.4. Derivatives of Basis Vectors and the Affine Connection
7.5. Discussion Questions and Exercises
8. Getting Acquainted with Differential Forms
8.1. Tensors as Multilinear Forms
8.2. 1-Forms (or Covariant Vectors Re-Imagined) and Their Extensions
8.3. Exterior Products and Differential Forms
8.4. The Exterior Derivative
8.5. An Application to Physics: Maxwell's Equations
8.6. Integrals of Differential Forms
8.7. Discussion Questions and Exercises
Appendix A: Common Coordinate Systems
Appendix B: The Theorem of Alternatives
Appendix C: Abstract Vector Spaces
Bibliography
Index
About the Author :
Dwight E. Neuenschwander is an emeritus professor of physics at Southern Nazarene University. He is the author of Emmy Noether's Wonderful Theorem and How to Involve Undergraduates in Research: A Field Guide for Faculty.
