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Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts—these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry.
Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the "inertia tensor," and in electricity and magnetism, they encounter the "polarization tensor." However, this piecemeal approach can set students up for misconceptions when they have to learn about tensors in more advanced physics and mathematics studies (e.g., while enrolled in a graduate-level general relativity course or when studying non-Euclidean geometries in a higher mathematics class).
Dwight E. Neuenschwander's Tensor Calculus for Physics is a bottom-up approach that emphasizes motivations before providing definitions. Using a clear, step-by-step approach, the book strives to embed the logic of tensors in contexts that demonstrate why that logic is worth pursuing. It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.
Why Aren’t Tensors Defined by What They Are?
Euclidean Vectors, without Coordinates
Derivatives of Euclidean Vectors with Respect to a Scalar
The Euclidean Gradient
Euclidean Vectors, with Coordinates
Euclidean Vector Operations with and without Coordinates
Transformation Coefficients as Partial Derivatives
What Is a Theory of Relativity?
Vectors Represented as Matrices
Discussion Questions and Exercises
The Electric Susceptibility Tensor
The Inertia Tensor
The Electric Quadrupole Tensor
The Electromagnetic Stress Tensor
Transformations of Two-Index Tensors
Finding Eigenvectors and Eigenvalues
More Than Two Indices
Integration Measures and Tensor Densities
Discussion Questions and Exercises
The Distinction between Distance and Coordinate Displacement
Relative Motion
Upper and Lower Indices
Converting between Vectors and Duals
Contravariant, Covariant, and “Ordinary” Vectors
Tensor Algebra
Tensor Densities Revisited
Discussion Questions and Exercises
Signs of Trouble
The Affine Connection
The Newtonian Limit
Transformation of the Affine Connection
The Covariant Derivative
Relation of the Affine Connection to the Metric Tensor
Divergence, Curl, and Laplacian with Covariant Derivatives
Disccussion Questions and Exercises
What Is Curvature?
The Riemann Tensor
Measuring Curvature
Linearity in the Second Derivative
Discussion Questions and Exercises
Covariant Electrodynamics
General Covariance and Gravitation
Discussion Questions and Exercises
Tensors and Manifolds
Tangent Spaces, Charts, and Manifolds
Metrics on Manifolds and Their Tangent Spaces
Dual Basis Vectors
Derivatives of Basis Vectors and the Affine Connection
Discussion Questions and Exercises
Tensors as Multilinear Forms
-Forms and Their Extensions
Exterior Products and Differential Forms
The Exterior Derivative
An Application to Physics: Maxwell’s Equations
Integrals of Differential Forms
Discussion Questions and Exercises
Appendix A: Common Coordinate Systems
Appendix B: Theorem of Alternatives
Appendix C: Abstract Vector Spaces