Tensor Calculus for Physics: A Concise Guide

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.

Table of Contents:
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. Disccussion Questions and Exercises Chapter 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 Chapter 6. Covariance Applications 6.1. 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