Mathematics of Classical and Quantum Physics

George Gordon Byron, better known as Lord Byron, was an English poet and peer devoted to the romantic movement in England.

• VOLUME ONE1 Vectors in Classical PhysicsIntroduction1.1 Geometric and Algebraic Definitions of a Vector1.2 The Resolution of a Vector into Components1.3 The Scalar Product1.4 Rotation of the Coordinate System: Orthogonal Transformations1.5 The Vector Product1.6 A Vector Treatment of Classical Orbit Theory1.7 Differential Operations on Scalar and Vector Fields*1.8 Cartesian-Tensors2 Calculus of VariationsIntroduction2.1 Some Famous Problems2.2 The Euler-Lagrange Equation2.3 Some Famous Solutions2.4 Isoperimetric Problems - Constraints2.5 Application to Classical Mechanics2.6 Extremization of Multiple Integrals2.7 Invariance Principles and Noether's Theorem3 Vectors and MatricsIntroduction3.1 "Groups, Fields, and Vector Spaces"3.2 Linear Independence3.3 Bases and Dimensionality3.4 Ismorphisms3.5 Linear Transformations3.6 The Inverse of a Linear Transformation3.7 Matrices3.8 Determinants3.9 Similarity Transformations3.10 Eigenvalues and Eigenvectors*3.11 The Kronecker Product4. Vector Spaces in PhysicsIntroduction4.1 The Inner Product4.2 Orthogonality and Completeness4.3 Complete Ortonormal Sets4.4 Self-Adjoint (Hermitian and Symmetric) Transformations4.5 Isometries-Unitary and Orthogonal Transformations4.6 The Eigenvalues and Eigenvectors of Self-Adjoint and Isometric Transformations4.7 Diagonalization4.8 On The Solvability of Linear Equations4.9 Minimum Principles4.10 Normal Modes4.11 Peturbation Theory-Nondegenerate Case4.12 Peturbation Theory-Degenerate Case5. Hilbert Space-Complete Orthonormal Sets of FunctionsIntroduction5.1 Function Space and Hilbert Space5.2 Complete Orthonormal Sets of Functions5.3 The Dirac d-Function5.4 Weirstrass's Theorem: Approximation by Polynomials5.5 Legendre Polynomials5.6 Fourier Series5.7 Fourier Integrals5.8 Sphereical Harmonics and Associated Legendre Functions5.9 Hermite Polynomials5.10 Sturm-Liouville Systems-Orthogaonal Polynomials5.11 A Mathematical Formulation of Quantum MechanicsVOLUME TWO6 Elements and Applications of the Theory of Analytic FunctionsIntroduction6.1 Analytic Functions-The Cauchy-Riemann Conditions6.2 Some Basic Analytic Functions6.3 Complex Integration-The Cauchy-Goursat Theorem6.4 Consequences of Cauchy's Theorem6.5 Hilbert Transforms and the Cauchy Principal Value6.6 An Introduction to Dispersion Relations6.7 The Expansion of an Analytic Function in a Power Series6.8 Residue Theory-Evaluation of Real Definite Integrals and Summation of Series6.9 Applications to Special Functions and Integral Representations7 Green's FunctionIntroduction7.1 A New Way to Solve Differential Equations7.2 Green's Functions and Delta Functions7.3 Green's Functions in One Dimension7.4 Green's Functions in Three Dimensions7.5 Radial Green's Functions7.6 An Application to the Theory of Diffraction7.7 Time-dependent Green's Functions: First Order7.8 The Wave Equation8 Introduction to Integral EquationsIntroduction8.1 Iterative Techniques-Linear Integral Operators8.2 Norms of Operators8.3 Iterative Techniques in a Banach Space8.4 Iterative Techniques for Nonlinear Equations8.5 Separable Kernels8.6 General Kernels of Finite Rank8.7 Completely Continuous Operators9 Integral Equations in Hilbert SpaceIntroduction9.1 Completely Continuous Hermitian Operators9.2 Linear Equations and Peturbation Theory9.3 Finite-Rank Techniques for Eigenvalue Problems9.4 the Fredholm Alternative for Completely Continuous Operators9.5 The Numerical Solutions of Linear Equations9.6 Unitary Transformations10 Introduction to Group TheoryIntroduction10.1 An Inductive Approach10.2 The Symmetric Groups10.3 "Cosets, Classes, and Invariant Subgroups"10.4 Symmetry and Group Representations10.5 Irreducible Representations10.6 "Unitary Representations, Schur's Lemmas, and Orthogonality Relations"10.7 The Determination of Group Representations10.8 Group Theory in Physical ProblemsGeneral BibliographyIndex to Volume OneIndex to Volume Two