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"Ich kann es nun einmal nicht lassen, in diesem Drama von Mathematik und Physik-die sich im Dunkeln befruchten, aber von Angesicht zu Angesicht so gerne einander verkennen und vedeugnen-die RoUe des (wie ich geniigsam erfuhr, oft unerwiinschten) Boten zu spielen." Hermann Weyl It is said that mathematics is the language of Nature. If so, then physics is its poetry. Nature started to whisper into our ears when Egyptians and Babylonians were compelled to invent and use mathematics in their day-to-day activities. The faint geometric and arithmetical pidgin of over four thousand years ago, suitable for rudimentary conversations with nature as applied to simple landscaping, has turned into a sophisticated language in which the heart of matter is articulated. The interplay between mathematics and physics needs no emphasis. What may need to be emphasized is that mathematics is not merely a tool with which the presentation of physics is facilitated, but the only medium in which physics can survive.
Just as language is the means by which humans can express their thoughts and without which they lose their unique identity, mathematics is the only language through which physics can express itself and without which.it loses its identity.
Contents
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'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'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.- 25.4Inner 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.
