Tensor Calculus for Physics: A Concise Guide

Description

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. Title Page......Page 3 Copyright Page......Page 4 Contents......Page 7 Preface......Page 11 Acknowledgments......Page 12 1.1 Why Aren’t Tensors Defined by What They Are?......Page 14 1.2 Euclidean Vectors, without Coordinates......Page 16 1.3 Derivatives of Euclidean Vectors with Respect to a Scalar......Page 17 1.4 The Euclidean Gradient......Page 18 1.5 Euclidean Vectors, with Coordinates......Page 19 1.6 Euclidean Vector Operations with and without Coordinates......Page 23 1.7 Transformation Coefficients as Partial Derivatives......Page 30 1.8 What Is a Theory of Relativity?......Page 32 1.9 Vectors Represented as Matrices......Page 35 1.10 Discussion Questions and Exercises......Page 41 2.1 The Electric Susceptibility Tensor......Page 44 2.2 The Inertia Tensor......Page 45 2.3 The Electric Quadrupole Tensor......Page 48 2.4 The Electromagnetic Stress Tensor......Page 49 2.5 Transformations of Two-Index Tensors......Page 53 2.6 Finding Eigenvectors and Eigenvalues......Page 57 2.8 More Than Two Indices......Page 61 2.9 Integration Measures and Tensor Densities......Page 62 2.10 Discussion Questions and Exercises......Page 63 3.1 The Distinction between Distance and Coordinate Displacement......Page 72 3.2 Relative Motion......Page 74 3.3 Upper and Lower Indices......Page 81 3.4 Converting between Vectors and Duals......Page 86 3.5 Contravariant, Covariant, and “Ordinary” Vectors......Page 89 3.6 Tensor Algebra......Page 92 3.7 Tensor Densities Revisited......Page 93 3.8 Discussion Questions and Exercises......Page 100 4.1 Signs of Trouble......Page 107 4.2 The Affine Connection......Page 109 4.3 The Newtonian Limit......Page 111 4.4 Transformation of the Affine Connection......Page 113 4.5 The Covariant Derivative......Page 115 4.6 Relation of the Affine Connection to the Metric Tensor......Page 117 4.7 Divergence, Curl, and Laplacian with Covariant Derivatives......Page 119 4.8 Disccussion Questions and Exercises......Page 123 5.1 What Is Curvature?......Page 128 5.2 The Riemann Tensor......Page 131 5.3 Measuring Curvature......Page 133 5.4 Linearity in the Second Derivative......Page 138 5.5 Discussion Questions and Exercises......Page 140 6.1 Covariant Electrodynamics......Page 146 6.2 General Covariance and Gravitation......Page 152 6.3 Discussion Questions and Exercises......Page 158 Chapter 7. Tensors and Manifolds......Page 164 7.1 Tangent Spaces, Charts, and Manifolds......Page 166 7.2 Metrics on Manifolds and Their Tangent Spaces......Page 170 7.3 Dual Basis Vectors......Page 171 7.4 Derivatives of Basis Vectors and the Affine Connection......Page 176 7.5 Discussion Questions and Exercises......Page 181 8.1 Tensors as Multilinear Forms......Page 184 8.2 1-Forms and Their Extensions......Page 188 8.3 Exterior Products and Differential Forms......Page 199 8.4 The Exterior Derivative......Page 204 8.5 An Application to Physics: Maxwell’s Equations......Page 208 8.6 Integrals of Differential Forms......Page 209 8.7 Discussion Questions and Exercises......Page 213 Appendix A: Common Coordinate Systems......Page 218 Appendix B: Theorem of Alternatives......Page 220 Appendix C: Abstract Vector Spaces......Page 221 Bibliography......Page 222 Index......Page 227