| |
| |
| Preface | |
| |
| |
| |
| Describing the Universe | |
| |
| |
| |
| A Universal Language | |
| |
| |
| |
| Scalar and Vector Fields | |
| |
| |
| |
| Curvilinear Coordinates | |
| |
| |
| |
| The Helmholtz Theorem | |
| |
| |
| |
| Vector Spaces | |
| |
| |
| |
| Matrices | |
| |
| |
| Problems | |
| |
| |
| |
| Complex Variables | |
| |
| |
| |
| All About Numbers | |
| |
| |
| |
| Functions of Complex Variables | |
| |
| |
| |
| Complex Series | |
| |
| |
| |
| Complex Numbers and Laplace's Equation | |
| |
| |
| |
| Poles and Zeros | |
| |
| |
| |
| The Residue Theorem | |
| |
| |
| |
| Using the Residue Theorem | |
| |
| |
| |
| Conformal Mapping | |
| |
| |
| |
| The Gamma Function | |
| |
| |
| Problems | |
| |
| |
| |
| Differential Equations | |
| |
| |
| |
| Some Definitions | |
| |
| |
| |
| Common Differential Equations Arising in Physics | |
| |
| |
| |
| Solution of Linear, Ordinary Differential Equations | |
| |
| |
| |
| Numerical Methods | |
| |
| |
| |
| Partial Differential Equations: Separation of Variables | |
| |
| |
| Problems | |
| |
| |
| |
| Fourier Series | |
| |
| |
| |
| Fourier's Theorem | |
| |
| |
| |
| Finding the Coefficients | |
| |
| |
| |
| Fourier Sine and Cosine Series | |
| |
| |
| |
| Use of Fourier Series to Solve Differential Equations | |
| |
| |
| |
| Convergence of Fourier Series | |
| |
| |
| Problems | |
| |
| |
| |
| Laplace Transforms | |
| |
| |
| |
| Definition of the Laplace Transform | |
| |
| |
| |
| Some Basic Properties of the Transform | |
| |
| |
| |
| Use of the Laplace Transform to Solve a Differential Equation | |
| |
| |
| |
| Some Additional Useful Tricks | |
| |
| |
| |
| Convolution | |
| |
| |
| |
| The General Inversion Procedure | |
| |
| |
| |
| Some More Physics | |
| |
| |
| Problems | |
| |
| |
| |
| Generalized Functions in Physics | |
| |
| |
| |
| The Delta Function | |
| |
| |
| |
| Developing a Theory of Distributions | |
| |
| |
| |
| Properties of Distributions | |
| |
| |
| |
| Sequences and Series | |
| |
| |
| |
| Distributions in N Dimensions | |
| |
| |
| |
| Describing Physical Quantities Using Delta Functions | |
| |
| |
| |
| The Green's Function | |
| |
| |
| Problems | |
| |
| |
| |
| Fourier Transforms | |
| |
| |
| |
| Definition of the Fourier Transform | |
| |
| |
| |
| Some Examples | |
| |
| |
| |
| Properties of the Fourier Transform | |
| |
| |
| |
| Causality | |
| |
| |
| |
| Use of Fourier Transforms in the Solution of Partial Differential Equations | |
| |
| |
| |
| Fourier Transforms and Power Spectra | |
| |
| |
| |
| Sine and Cosine Transforms | |
| |
| |
| Problems | |
| |
| |
| |
| Sturm-Liouville Theory | |
| |
| |
| |
| The Sturm-Liouville Problem | |
| |
| |
| |
| Use of Sturm-Liouville Theory in Physics | |
| |
| |
| |
| Problems with Spherical Symmetry: Spherical Harmonics | |
| |
| |
| |
| Problems with Cylindrical Symmetry: Bessel Functions | |
| |
| |
| |
| Spherical Bessel Functions | |
| |
| |
| |
| The Classical Orthogonal Polynomials | |
| |
| |
| Problems | |
| |
| |
| Optional Topics | |
| |
| |
| |
| Tensors | |
| |
| |
| |
| Cartesian Tensors | |
| |
| |
| |
| Inner and Outer Products | |
| |
| |
| |
| Pseudo-tensors and Cross Products | |
| |
| |
| |
| General Tensor Calculus | |
| |
| |
| |
| The Metric Tensor | |
| |
| |
| |
| Contraction | |
| |
| |
| |
| Basis Vectors and Basis Forms | |
| |
| |
| |
| Derivatives | |
| |
| |
| Problems | |
| |
| |
| |
| Group Theory | |
| |
| |
| |
| Definition of a Group | |
| |
| |
| |
| Examples of Groups | |
| |
| |
| |
| Classes | |
| |
| |
| |
| Subgroups | |
| |
| |
| |
| Cyclic Groups | |
| |
| |
| |
| Factor Groups and Direct Product Groups | |
| |
| |
| |
| Isomorphism | |
| |
| |
| |
| Representations | |
| |
| |
| |
| Generators of Groups | |
| |
| |
| |
| Lie Algebras | |
| |
| |
| Problems | |
| |
| |
| |
| Green's Functions | |
| |
| |
| |
| Division-of-Region Method | |
| |
| |
| |
| Expansion in Eigenfunctions | |
| |
| |
| |
| Transform Methods | |
| |
| |
| |
| Extension to N Dimensions | |
| |
| |
| |
| Inhomogeneous Boundary Conditions | |
| |
| |
| |
| Green's Theorem | |
| |
| |
| |
| The Green's Function for Poisson's Equation in a Bounded Region | |
| |
| |
| Problems | |
| |
| |
| |
| Approximate Evaluation of Integrals | |
| |
| |
| |
| The Method of Steepest Descent | |
| |
| |
| |
| The Method of Stationary Phase | |
| |
| |
| Problems | |
| |
| |
| |
| Calculus of Variations | |
| |
| |
| |
| Integral Principles in Physics | |
| |
| |
| |
| The Euler Equation | |
| |
| |
| |
| Variation Subject to Constraints | |
| |
| |
| |
| Extension to Functions of More Than One Variable | |
| |
| |
| Problems | |
| |
| |
| Appendices | |
| |
| |
| |
| Transformation Properties of the Vector Cross Product | |
| |
| |
| |
| Proof of the Helmholtz Theorem | |
| |
| |
| |
| Proof by Induction: The Cauchy Formula | |
| |
| |
| |
| The Mean Value Theorem for Integrals | |
| |
| |
| |
| The Gibbs Phenomenon | |
| |
| |
| |
| The Laplace Transform and Convolution | |
| |
| |
| |
| Proof That P[superscript m subscript l]([mu]) = (-1)[superscript m](1 - [mu superscript 2])[superscript m/2]d[superscript m]/d[mu superscript m]P[subscript l]([mu]) | |
| |
| |
| |
| Proof of the Relation [function of superscript infinity subscript 0] [rho]J[subscript m](k[rho])J[subscript m](k'[rho])d[rho] = 1/k[delta](k - k') | |
| |
| |
| |
| The Error Function | |
| |
| |
| |
| Classification of Partial Differential Equations | |
| |
| |
| |
| The Tangent Function: A Detailed Investigation of Series Expansions | |
| |
| |
| Bibliography | |
| |
| |
| Index | |