Mathematical Physics

0 Mathematical Preliminaries.- 0.1 Sets.- 0.2 Maps.- 0.3 Metric Spaces.- 0.4 Cardinality.- 0.5 Mathematical Induction.- 0.6 Problems.- I Finite-Dimensional Vector Spaces.- 1 Vectors and Transformations.- 1.1 Vector Spaces.- 1.2 Inner Product.- 1.3 Linear Transformations.- 1.4 Algebras.- 1.5 Problems.- 2 Operator Algebra.- 2.1 Algebra of L(V).- 2.2 Derivatives of Functions of Operators.- 2.3 Conjugation of Operators.- 2.4 Hermitian and Unitary Operators.- 2.5 Projection Operators.- 2.6 Operators in Numerical Analysis.- 2.7 Problems.- 3 Matrices: Operator Representations.- 3.1 Matrices.- 3.2 Operations on Matrices.- 3.3 Orthonormal Bases.- 3.4 Change of Basis and Similarity Transformation.- 3.5 The Determinant.- 3.6 The Trace.- 3.7 Problems.- 4 Spectral Decomposition.- 4.1 Direct Sums.- 4.2 Invariant Subspaces.- 4.3 Eigenvalues and Eigenvectors.- 4.4 Spectral Decomposition.- 4.5 Functions of Operators.- 4.6 Polar Decomposition.- 4.7 Real Vector Spaces.- 4.8 Problems.- II Infinite-Dimensional Vector Spaces.- 5 Hilbert Spaces.- 5.1 The Question of Convergence.- 5.2 The Space of Square-Integrable Functions.- 5.3 Problems.- 6 Generalized Functions.- 6.1 Continuous Index.- 6.2 Generalized Functions.- 6.3 Problems.- 7 Classical Orthogonal Polynomials.- 7.1 General Properties.- 7.2 Classification.- 7.3 Recurrence Relations.- 7.4 Examples of Classical Orthogonal Polynomials.- 7.5 Expansion in Terms of Orthogonal Polynomials.- 7.6 Generating Functions.- 7.7 Problems.- 8 Fourier Analysis.- 8.1 Fourier Series.- 8.2 The Fourier Transform.- 8.3 Problems.- III Complex Analysis.- 9 Complex Calculus.- 9.1 Complex Functions.- 9.2 Analytic Functions.- 9.3 Conformal Maps.- 9.4 Integration of Complex Functions.- 9.5 Derivatives as Integrals.- 9.6 Taylor and Laurent Series.- 9.7 Problems.- 10 Calculus of Residues.- 10.1 Residues.- 10.2 Classification of Isolated Singularities.- 10.3 Evaluation of Definite Integrals.- 10.4 Problems.- 11 Complex Analysis: Advanced Topics.- 11.1 Meromorphic Functions.- 11.2 Multivalued Functions.- 11.3 Analytic Continuation.- 11.4 The Gamma and Beta Functions.- 11.5 Method of Steepest Descent.- 11.6 Problems.- IV Differential Equations.- 12 Separation of Variables in Spherical Coordinates.- 12.1 PDEs of Mathematical Physics.- 12.2 Separation of the Angular Part of the Laplacian.- 12.3 Construction of Eigenvalues of L2.- 12.4 Eigenvectors of L2: Spherical Harmonics.- 12.5 Problems.- 13 Second-Order Linear Differential Equations.- 13.1 General Properties of ODEs.- 13.2 Existence and Uniqueness for First-Order DEs.- 13.3 General Properties of SOLDEs.- 13.4 The Wronskian.- 13.5 Adjoint Differential Operators.- 13.6 Power-Series Solutions of SOLDEs.- 13.7 SOLDEs with Constant Coefficients.- 13.8 The WKB Method.- 13.9 Numerical Solutions of DEs.- 13.10 Problems.- 14 Complex Analysis of SOLDEs.- 14.1 Analytic Properties of Complex DEs.- 14.2 Complex SOLDEs.- 14.3 Fuchsian Differential Equations.- 14.4 The Hypergeometric Function.- 14.5 Confluent Hypergeometric Functions.- 14.6 Problems.- 15 Integral Transforms and Differential Equations.- 15.1 Integral Representation of the Hypergeometric Function.- 15.2 Integral Representation of the Confluent Hypergeometric Function.- 15.3 Integral Representation of Bessel Functions.- 15.4 Asymptotic Behavior of Bessel Functions.- 15.5 Problems.- V Operators on Hilbert Spaces.- 16 An Introduction to Operator Theory.- 16.1 From Abstract to Integral and Differential Operators.- 16.2 Bounded Operators in Hilbert Spaces.- 16.3 Spectra of Linear Operators.- 16.4 Compact Sets.- 16.5 Compact Operators.- 16.6 Spectrum of Compact Operators.- 16.7 Spectral Theorem for Compact Operators.- 16.8 Resolvents.- 16.9 Problems.- 17 Integral Equations.- 17.1 Classification.- 17.2 Fredholm Integral Equations.- 17.3 Problems.- 18 Sturm—Liouville Systems: Formalism.- 18.1 Unbounded Operators with Compact Resolvent.- 18.2 Sturm—Liouville Systems and SOLDEs.- 18.3 Other Properties of Sturm—Liouville Systems.- 18.4 Problems.- 19 Sturm—Liouville Systems: Examples.- 19.1 Expansions in Terms of Eigenfunctions.- 19.2 Separation in Cartesian Coordinates.- 19.3 Separation in Cylindrical Coordinates.- 19.4 Separation in Spherical Coordinates.- 19.5 Problems.- VI Green’s Functions.- 20 Green’s Functions in One Dimension.- 20.1 Calculation of Some Green’s Functions.- 20.2 Formal Considerations.- 20.3 Green’s Functions for SOLDOs.- 20.4 Eigenfunction Expansion of Green’s Functions.- 20.5 Problems.- 21 Multidimensional Green’s Functions: Formalism.- 21.1 Properties of Partial Differential Equations.- 21.2 Multidimensional GFs and Delta Functions.- 21.3 Formal Development.- 21.4 Integral Equations and GFs.- 21.5 Perturbation Theory.- 21.6 Problems.- 22 Multidimensional Green’s Functions: Applications.- 22.1 Elliptic Equations.- 22.2 Parabolic Equations.- 22.3 Hyperbolic Equations.- 22.4 The Fourier Transform Technique.- 22.5 The Eigenfunction Expansion Technique.- 22.6 Problems.- VII Groups and Manifolds.- 23 Group Theory.- 23.1 Groups.- 23.2 Subgroups.- 23.3 Group Action.- 23.4 The Symmetric Group Sn.- 23.5 Problems.- 24 Group Representation Theory.- 24.1 Definitions and Examples.- 24.2 Orthogonality Properties.- 24.3 Analysis of Representations.- 24.4 Group Algebra.- 24.5 Relationship of Characters to Those of a Subgroup.- 24.6 Irreducible Basis Functions.- 24.7 Tensor Product of Representations.- 24.8 Representations of the Symmetric Group.- 24.9 Problems.- 25 Algebra of Tensors.- 25.1 Multilinear Mappings.- 25.2 Symmetries of Tensors.- 25.3 Exterior Algebra.- 25.4 Inner Product Revisited.- 25.5 The Hodge Star Operator.- 25.6 Problems.- 26 Analysis of Tensors.- 26.1 Differentiable Manifolds.- 26.2 Curves and Tangent Vectors.- 26.3 Differential of a Map.- 26.4 Tensor Fields on Manifolds.- 26.5 Exterior Calculus.- 26.6 Symplectic Geometry.- 26.7 Problems.- VIII Lie Groups and Their Applications.- 27 Lie Groups and Lie Algebras.- 27.1 Lie Groups and Their Algebras.- 27.2 An Outline of Lie Algebra Theory.- 27.3 Representation of Compact Lie Groups.- 27.4 Representation of the General Linear Group.- 27.5 Representation of Lie Algebras.- 27.6 Problems.- 28 Differential Geometry.- 28.1 Vector Fields and Curvature.- 28.2 Riemannian Manifolds.- 28.3 Covariant Derivative and Geodesics.- 28.4 Isometries and Killing Vector Fields.- 28.5 Geodesic Deviation and Curvature.- 28.6 General Theory of Relativity.- 28.7 Problems.- 29 Lie Groups and Differential Equations.- 29.1 Symmetries of Algebraic Equations.- 29.2 Symmetry Groups of Differential Equations.- 29.3 The Central Theorems.- 29.4 Application to Some Known PDEs.- 29.5 Application to ODEs.- 29.6 Problems.- 30 Calculus of Variations, Symmetries, and Conservation Laws.- 30.1 The Calculus of Variations.- 30.2 Symmetry Groups of Variational Problems.- 30.3 Conservation Laws and Noether’s Theorem.- 30.4 Application to Classical Field Theory.- 30.5 Problems.