Mathematical Techniques and Physical Applications

PrefaceComment on Notation1. Vector Analysis 1.1. Scalars, Tensors, and Vectors 1.2. Scalar, Vector, and Tensor Fields 1.3. Vector Components, Unit Vectors, Right-Handed Cartesian Axes 1.4. Vector Sums and Products A. Vector Addition B. Scalar Product of Vectors C. Vector Product D. Matrix Notation for Scalar and Vector Products E. Triple Scalar and Vector Products F. Reciprocal Systems of Vectors 1.5. Derivatives of a Vector or Vector Field A. Gradient Operation and Nabla Operator B. Potential Vectors, Line Integrals C. Divergence of a Vector Field D. Curl of a Vector Field E. Important Results of Vector Analysis 1.6. Integral Theorems A. Gauss Divergence Theorem B. Green's Theorem C. Stokes' Curl Theorem D. Field Discontinuities, Surface Curl, and Divergence 1.7. Dyadic Formalism A. Vector Product and Lie Algebras 1.8. Orthogonal Curvilinear Coordinates A. Divergence, Gradient, and Curl B. Common Coordinate Systems 1.9. Uses of Vector Analysis A. Energy Density of an Electromagnetic Field and the Pointing Vector B. Electromagnetic Waves C. Convective Derivative Term in Fluid Theory D. Electric and Magnetic Dipoles 1.10. Further Examples Involving the Vector Product A. The Frenet Formulas of Differential Geometry B. The Vector Potential in Magnetic Theory C. Additivity of Angular Momentum D. The Description of Fluid Flow Using a Rotating Coordinate System2. Matrices 2.1. Simultaneous Linear Equations and Matrix Algebra 2.2. Some Common Types of Matrix 2.3. Inverse of a Matrix, Determinant A. Singular Matrices as Divisors of Zero 2.4. Theorems Concerning Matrix Products A. Reversal Rule B. Determinant of a Product Matrix C. Triple Products, Families of Equivalent Matrices 2.5. Eigenvectors and Eigenvalues of a Matrix A. Sum Rules B. Eigenvalues and Eigencolumns of Hermitian and Unitary Matrices C. Linear Independence D. Inhomogeneous Problem: An Orthogonality Criterion for Hermitian Matrices 2.6. Matrices as Representations of Linear Operators A. Invariant Spaces, Basis Vectors B. Change of Basis C. Invariants Associated with a Matrix 2.1. Application of Matrix Theory to Physical Problems A. Jacobian of a Transformation B. Homogeneous Strain C. Normal Modes D. Lorentz Transformations E. Matrices in Quantum Mechanics F. Dirac Relativistic Theory of the Electron G. Slater Determinantal Wavefunction for Electrons H. Density Matrix in Quantum Statistical Mechanics I. Continuous Matrices, Dirac Delta Function J. Green's Functions K. Hilbert Spaces L. Schmidt Orthogonalization Procedure M. Indefinite Metric Formalism N. Metric Spaces O. Schwartz Inequality3. Tensor Analysis 3.1. Cartesian Tensors A. Scalars and Vectors B. Second-Rank Cartesian Tensors as Operators C. Higher Rank Cartesian Tensors, Summation Convention D. Symmetric and Antisymmetric Tensors E. Contraction of Tensors, Inner Products F. Comment on Translational Transformations G. Spherical Tensors H. Transformations in Four Dimensions I. Principal Axes 3.2. Tensors in Nonorthogonal Frames A. Covariant and Contravariant Vector Components B. Second-Rank Tensors C. Line Element, Fundamental Tensor D. Raising and Lowering of Indices, The Tensor gij 3.3. General Tensors A. Covariant and Contravariant Vectors B. Fundamental Tensor gij and Its Symmetry C. Summary D. Eddington's Theorem E. The "Curling" Operation 3.4. The Christoffel Symbols A. Length of a Vector and the Angle between Two Vectors 3.5. Length of a Curve, Geodesies 3.6. Covariant Derivatives A. Alternative Approach to the Covariant Derivative 3.7. The Determinant