You are here

Tensor Calculus for Physics: A Concise Guide

Dwight E. Neuenschwander
Johns Hopkins University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a very well motivated introduction to tensors, aimed at upper-division undergraduate physics majors. The author begins by admitting that the usual definition of tensors in terms of invariance under change of coordinates is very mysterious, and instead of starting there he works up to it gradually with a series of examples from physics (mechanics, electrodynamics, and special relativity). He starts with vectors, pointing out that coordinate systems are a human invention and that the physical quantities involved must be invariant regardless of the type and position of the coordinate systems. Coordinates are how we measure but Nature doesn’t know anything about them. From this he gets that the physical quantities must be invariant under all kinds of coordinate changes (including but not limited to the usual rotations) and shows how this can be abstracted to define vectors.

He then considers several physical problems in which the equations include quantities that turn out to be rank-2 tensors. The older notation of dyads is covered briefly but not used for anything. There is a great deal on special and general relativity as part of the motivation, but it’s not comprehensive.

The exercises are very strong. They are divided into discussion questions, which often ask “why do we do it this way and not that way?” to test your understanding of the development, and actual exercises, a few of which are drill but most are to prove something or calculate some important formula.

Most of the book works by example and analogy and is not very formal. It develops general rules and properties of tensors, but the emphasis is on specific physical applications. The last two chapters revisit the material from a more formal and mathematical perspective, covering manifolds and differential and multilinear forms.

One weakness of the book, from a more advanced standpoint, is that it only studies tensors that are already known to be tensors; there’s no guidance about discovering or creating tensors. Historically tensors came out of differential geometry, and were already well-understood by the time Einstein needed them for general relativity. His approach was geometric, and he knew from the kind of geometries he was considering that he would need tensors. The present book works more from known formulas where the physical laws have already been discovered, and it is not very geometrical. The material on differential forms gives a different formalism for what has already been covered, and is not approached from geometry.

Bottom line: a very good book for physics students. It’s good for math students too if they have a strong physics background (otherwise the examples are too hard to follow).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

  • Preface
  • Acknowledgments
  • Chapter 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
  • Chapter 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
  • Chapter 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
  • Chapter 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
  • Chapter 5. Curvature
    • 5.l 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
  • Chapter 6. Covariance Applications
    • 6.l Covariant Electrodynamics
    • 6.2 General Covariance and Gravitation
    • 6.3 Discussion Questions and Exercises
  • Chapter 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
  • Chapter 8. Getting Acquainted with Differential Forms
    • 8.1 Tensors as Multilinear Forms
    • 8.2 1-Forms 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: Theorem of Alternatives
  • Appendix C: Abstract Vector Spaces
  • Bibliography
  • Index