A textbook that shows its working.
Most mathematical methods textbooks teach techniques one chapter at a time. They explain how to perform a calculation, but rarely why one method works better than another. Why does that substitution work? Why does a contour close in the upper half-plane? Why must an operator be self-adjoint? Why do Fourier series, Green's functions, Sturm-Liouville theory, and quantum mechanics keep returning to the same ideas?
This book was written to answer those questions.
Every number in this book was computed before it was written. Not quoted, not recalled. Every derivation, worked example, figure, table, and answer was verified by symbolic and numerical computation.
THE 16 CHAPTERS
Vector Analysis. Grad, div, curl, curvilinear coordinates, and the gradient, divergence, and Stokes theorems.
Linear Algebra. Eigenvalues, Hermitian and unitary operators, the spectral theorem, and least squares.
Infinite Series. Convergence, Taylor and Laurent expansions, and asymptotic series.
Complex Variables. Analytic functions, Cauchy's theorem, branch cuts, residues, and conformal mapping.
Residue Calculus. Laurent series, Jordan's lemma, the residue theorem, and contour integration.
Ordinary Differential Equations. Integrating factors, Frobenius, Wronskians, damping, and variation of parameters.
Special Functions. Legendre, Bessel, Hermite, Laguerre, Rodrigues formulas, and spherical harmonics.
Sturm-Liouville Theory. Self-adjoint operators, orthogonality, completeness, and eigenfunction expansions.
Fourier Series. Convergence, Parseval's identity, the Gibbs phenomenon, and spectral decay.
Integral Transforms. Fourier and Laplace transforms, convolution, the Gaussian, and the uncertainty principle.
Partial Differential Equations. Heat, wave, and Laplace equations, separation of variables, and applications to vibrating systems and quantum mechanics.
Green's Functions. Point sources, causal response, Duhamel's principle, and the retarded potential.
Calculus of Variations. Euler-Lagrange equations, the brachistochrone, Rayleigh-Ritz, and Noether's theorem.
Tensors. Metrics, Christoffel symbols, curvature, relativity, and spacetime.
Group Theory. Representations, character tables, symmetry, and selection rules.
Probability. Random walks, Bayes' theorem, entropy, the central limit theorem, and Bell's inequality.
THE CONNECTIONS ARE THE POINT
This is not sixteen independent subjects. Each chapter prepares the next.
- Fourier series become eigenfunction expansions, then the spectral theorem.
- Every classical special function arises from a Sturm-Liouville problem.
- The heat equation is derived from both thermodynamics and probability.
- The Gaussian links Fourier analysis, quantum mechanics, and the central limit theorem.
- The brachistochrone and the geodesic emerge from the same variational principle.
WHAT YOU GET
- 174 fully worked examples with every step shown.
- More than 500 practice problems with a complete answer key.
- Three extensive appendices covering integral tables, transform pairs, special functions, Green's functions, tensor identities, coordinate systems, PDE classifications, and mathematical reference tables.
- A complete glossary, list of symbols, bibliography, and comprehensive index.
WHO THIS BOOK IS FOR
Written for advanced undergraduates, beginning graduate students, engineers, physicists, mathematicians, and scientists. The only prerequisites are calculus and elementary linear algebra.
