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ISBN 10: 3319011944
ISBN 13: 978-3319011943
Author: Sadri Hassani
The goal of this book is to expose the reader to the indispensable role that mathematics plays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multi-dimensional Green’s functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fibre bundles and their applications to differential geometry and gauge theories.
This second edition is a substantial revision with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras, fibre bundles, and gauge theories. The spirit of the first edition, namely the balance between rigour and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the “unreasonable effectiveness of mathematics” in modern physics.
Mathematical Physics A Modern Introduction to Its Foundations 2nd Table of contents:
Chapter 1: Mathematical Preliminaries
1.1 Sets
1.1.1 Equivalence Relations
1.2 Maps
1.3 Metric Spaces
1.4 Cardinality
1.5 Mathematical Induction
1.6 Problems
Part I: Finite-Dimensional Vector Spaces
Chapter 2: Vectors and Linear Maps
2.1 Vector Spaces
2.1.1 Subspaces
2.1.2 Factor Space
2.1.3 Direct Sums
2.1.4 Tensor Product of Vector Spaces
2.2 Inner Product
2.2.1 Orthogonality
2.2.2 The Gram-Schmidt Process
2.2.3 The Schwarz Inequality
2.2.4 Length of a Vector
2.3 Linear Maps
2.3.1 Kernel of a Linear Map
2.3.2 Linear Isomorphism
2.4 Complex Structures
2.5 Linear Functionals
2.6 Multilinear Maps
2.6.1 Determinant of a Linear Operator
2.6.2 Classical Adjoint
2.7 Problems
Chapter 3: Algebras
3.1 From Vector Space to Algebra
3.1.1 General Properties
3.1.2 Homomorphisms
3.2 Ideals
3.2.1 Factor Algebras
3.3 Total Matrix Algebra
3.4 Derivation of an Algebra
3.5 Decomposition of Algebras
3.5.1 The Radical
3.5.2 Semi-simple Algebras
3.5.3 Classification of Simple Algebras
3.6 Polynomial Algebra
3.7 Problems
Chapter 4: Operator Algebra
4.1 Algebra of `39`42`”613A“45`47`”603AEnd(V)
4.1.1 Polynomials of Operators
4.1.2 Functions of Operators
4.1.3 Commutators
4.2 Derivatives of Operators
4.3 Conjugation of Operators
4.3.1 Hermitian Operators
4.3.2 Unitary Operators
4.4 Idempotents
4.4.1 Projection Operators
4.5 Representation of Algebras
4.6 Problems
Chapter 5: Matrices
5.1 Representing Vectors and Operators
5.2 Operations on Matrices
5.3 Orthonormal Bases
5.4 Change of Basis
5.5 Determinant of a Matrix
5.5.1 Matrix of the Classical Adjoint
5.5.2 Inverse of a Matrix
Algorithm for Calculating the Inverse of a Matrix
Rank of a Matrix
5.5.3 Dual Determinant Function
5.6 The Trace
5.7 Problems
Chapter 6: Spectral Decomposition
6.1 Invariant Subspaces
6.2 Eigenvalues and Eigenvectors
6.3 Upper-Triangular Representations
6.4 Complex Spectral Decomposition
6.4.1 Simultaneous Diagonalization
6.5 Functions of Operators
6.6 Real Spectral Decomposition
6.6.1 The Case of Symmetric Operators
6.6.2 The Case of Real Normal Operators
6.7 Polar Decomposition
6.8 Problems
Part II: Infinite-Dimensional Vector Spaces
Chapter 7: Hilbert Spaces
7.1 The Question of Convergence
7.2 The Space of Square-Integrable Functions
7.2.1 Orthogonal Polynomials
7.2.2 Orthogonal Polynomials and Least Squares
7.3 Continuous Index
7.4 Generalized Functions
7.5 Problems
Chapter 8: Classical Orthogonal Polynomials
8.1 General Properties
8.2 Classification
8.3 Recurrence Relations
8.4 Details of Specific Examples
8.4.1 Hermite Polynomials
8.4.2 Laguerre Polynomials
8.4.3 Legendre Polynomials
8.4.4 Other Classical Orthogonal Polynomials
Jacobi Polynomials, Pnµ,nu(x)
Gegenbauer Polynomials, Cnlambda(x)
Chebyshev Polynomials of the First Kind, Tn(x)
Chebyshev Polynomials of the Second Kind, Un(x)
8.5 Expansion in Terms of Orthogonal Polynomials
8.6 Generating Functions
8.7 Problems
Chapter 9: Fourier Analysis
9.1 Fourier Series
9.1.1 The Gibbs Phenomenon
9.1.2 Fourier Series in Higher Dimensions
9.2 Fourier Transform
9.2.1 Fourier Transforms and Derivatives
9.2.2 The Discrete Fourier Transform
9.2.3 Fourier Transform of a Distribution
9.3 Problems
Part III: Complex Analysis
Chapter 10: Complex Calculus
10.1 Complex Functions
10.2 Analytic Functions
10.3 Conformal Maps
10.4 Integration of Complex Functions
10.5 Derivatives as Integrals
10.6 Infinite Complex Series
10.6.1 Properties of Series
10.6.2 Taylor and Laurent Series
10.7 Problems
Chapter 11: Calculus of Residues
11.1 Residues
11.2 Classification of Isolated Singularities
11.3 Evaluation of Definite Integrals
11.3.1 Integrals of Rational Functions
11.3.2 Products of Rational and Trigonometric Functions
11.3.3 Functions of Trigonometric Functions
11.3.4 Some Other Integrals
11.3.5 Principal Value of an Integral
11.4 Problems
Chapter 12: Advanced Topics
12.1 Meromorphic Functions
12.2 Multivalued Functions
12.2.1 Riemann Surfaces
12.3 Analytic Continuation
12.3.1 The Schwarz Reflection Principle
12.3.2 Dispersion Relations
12.4 The Gamma and Beta Functions
12.5 Method of Steepest Descent
12.6 Problems
Part IV: Differential Equations
Chapter 13: Separation of Variables in Spherical Coordinates
13.1 PDEs of Mathematical Physics
13.2 Separation of the Angular Part
13.3 Construction of Eigenvalues of L2
13.4 Eigenvectors of L2: Spherical Harmonics
13.4.1 Expansion of Angular Functions
13.4.2 Addition Theorem for Spherical Harmonics
13.5 Problems
Chapter 14: Second-Order Linear Differential Equations
14.1 General Properties of ODEs
14.2 Existence/Uniqueness for First-Order DEs
14.3 General Properties of SOLDEs
14.4 The Wronskian
14.4.1 A Second Solution to the HSOLDE
14.4.2 The General Solution to an ISOLDE
14.4.3 Separation and Comparison Theorems
14.5 Adjoint Differential Operators
14.6 Power-Series Solutions of SOLDEs
14.6.1 Frobenius Method of Undetermined Coefficients
14.6.2 Quantum Harmonic Oscillator
14.7 SOLDEs with Constant Coefficients
14.8 The WKB Method
14.8.1 Classical Limit of the Schrödinger Equation
14.9 Problems
Chapter 15: Complex Analysis of SOLDEs
15.1 Analytic Properties of Complex DEs
15.1.1 Complex FOLDEs
15.1.2 The Circuit Matrix
15.2 Complex SOLDEs
15.3 Fuchsian Differential Equations
15.4 The Hypergeometric Function
15.5 Confluent Hypergeometric Functions
15.5.1 Hydrogen-Like Atoms
15.5.2 Bessel Functions
15.6 Problems
Chapter 16: Integral Transforms and Differential Equations
16.1 Integral Representation of the Hypergeometric Function
16.1.1 Integral Representation of the Confluent Hypergeometric Function
16.2 Integral Representation of Bessel Functions
16.2.1 Asymptotic Behavior of Bessel Functions
16.3 Problems
Part V: Operators on Hilbert Spaces
Chapter 17: Introductory Operator Theory
17.1 From Abstract to Integral and Differential Operators
17.2 Bounded Operators in Hilbert Spaces
17.2.1 Adjoints of Bounded Operators
17.3 Spectra of Linear Operators
17.4 Compact Sets
17.4.1 Compactness and Infinite Sequences
17.5 Compact Operators
17.5.1 Spectrum of Compact Operators
17.6 Spectral Theorem for Compact Operators
17.6.1 Compact Hermitian Operator
17.6.2 Compact Normal Operator
17.7 Resolvents
17.8 Problems
Chapter 18: Integral Equations
18.1 Classification
18.2 Fredholm Integral Equations
18.2.1 Hermitian Kernel
18.2.2 Degenerate Kernels
18.3 Problems
Chapter 19: Sturm-Liouville Systems
19.1 Compact-Resolvent Unbounded Operators
19.2 Sturm-Liouville Systems and SOLDEs
19.3 Asymptotic Behavior
19.3.1 Large Eigenvalues
19.3.2 Large Argument
19.4 Expansions in Terms of Eigenfunctions
19.5 Separation in Cartesian Coordinates
19.5.1 Rectangular Conducting Box
19.5.2 Heat Conduction in a Rectangular Plate
19.5.3 Quantum Particle in a Box
19.5.4 Wave Guides
19.6 Separation in Cylindrical Coordinates
19.6.1 Conducting Cylindrical Can
19.6.2 Cylindrical Wave Guide
19.6.3 Current Distribution in a Circular Wire
19.7 Separation in Spherical Coordinates
19.7.1 Radial Part of Laplace’s Equation
19.7.2 Helmholtz Equation in Spherical Coordinates
19.7.3 Quantum Particle in a Hard Sphere
19.7.4 Plane Wave Expansion
19.8 Problems
Part VI: Green’s Functions
Chapter 20: Green’s Functions in One Dimension
20.1 Calculation of Some Green’s Functions
20.2 Formal Considerations
20.2.1 Second-Order Linear DOs
20.2.2 Self-adjoint SOLDOs
20.3 Green’s Functions for SOLDOs
20.3.1 Properties of Green’s Functions
20.3.2 Construction and Uniqueness of Green’s Functions
20.3.3 Inhomogeneous BCs
20.4 Eigenfunction Expansion
20.5 Problems
Chapter 21: Multidimensional Green’s Functions: Formalism
21.1 Properties of Partial Differential Equations
21.1.1 Characteristic Hypersurfaces
21.1.2 Second-Order PDEs in m Dimensions
21.2 Multidimensional GFs and Delta Functions
21.2.1 Spherical Coordinates in m Dimensions
21.2.2 Green’s Function for the Laplacian
21.3 Formal Development
21.3.1 General Properties
21.3.2 Fundamental (Singular) Solutions
21.4 Integral Equations and GFs
21.5 Perturbation Theory
21.5.1 The Nondegenerate Case
21.5.2 The Degenerate Case
21.6 Problems
Chapter 22: Multidimensional Green’s Functions: Applications
22.1 Elliptic Equations
22.1.1 The Dirichlet Boundary Value Problem
22.1.2 The Neumann Boundary Value Problem
22.2 Parabolic Equations
22.3 Hyperbolic Equations
22.4 The Fourier Transform Technique
22.4.1 GF for the m-Dimensional Laplacian
22.4.2 GF for the m-Dimensional Helmholtz Operator
22.4.3 GF for the m-Dimensional Diffusion Operator
22.4.4 GF for the m-Dimensional Wave Equation
22.5 The Eigenfunction Expansion Technique
22.6 Problems
Part VII: Groups and Their Representations
Chapter 23: Group Theory
23.1 Groups
23.2 Subgroups
23.2.1 Direct Products
23.3 Group Action
23.4 The Symmetric Group Sn
23.5 Problems
Chapter 24: Representation of Groups
24.1 Definitions and Examples
24.2 Irreducible Representations
24.3 Orthogonality Properties
24.4 Analysis of Representations
24.5 Group Algebra
24.5.1 Group Algebra and Representations
24.6 Relationship of Characters to Those of a Subgroup
24.7 Irreducible Basis Functions
24.8 Tensor Product of Representations
24.8.1 Clebsch-Gordan Decomposition
24.8.2 Irreducible Tensor Operators
24.9 Problems
Chapter 25: Representations of the Symmetric Group
25.1 Analytic Construction
25.2 Graphical Construction
25.3 Graphical Construction of Characters
25.4 Young Operators
25.5 Products of Representations of Sn
25.6 Problems
Part VIII: Tensors and Manifolds
Chapter 26: Tensors
26.1 Tensors as Multilinear Maps
26.2 Symmetries of Tensors
26.3 Exterior Algebra
26.3.1 Orientation
26.4 Symplectic Vector Spaces
26.5 Inner Product Revisited
26.5.1 Subspaces
26.5.2 Orthonormal Basis
26.5.3 Inner Product on Lambdap(V,U)
26.6 The Hodge Star Operator
26.7 Problems
Chapter 27: Clifford Algebras
27.1 Construction of Clifford Algebras
27.1.1 The Dirac Equation
27.2 General Properties of the Clifford Algebra
27.2.1 Homomorphism with Other Algebras
27.2.2 The Canonical Element
27.2.3 Center and Anticenter
27.2.4 Isomorphisms
27.3 General Classification of Clifford Algebras
27.4 The Clifford Algebras Cµnu(R)
27.4.1 Classification of Cn0(R) and C0n(R)
27.4.2 Classification of Cµnu(R)
27.4.3 The Algebra C31(R)
27.5 Problems
Chapter 28: Analysis of Tensors
28.1 Differentiable Manifolds
28.2 Curves and Tangent Vectors
28.3 Differential of a Map
28.4 Tensor Fields on Manifolds
28.4.1 Vector Fields
28.4.2 Tensor Fields
28.5 Exterior Calculus
28.6 Integration on Manifolds
28.7 Symplectic Geometry
28.8 Problems
Part IX: Lie Groups and Their Applications
Chapter 29: Lie Groups and Lie Algebras
29.1 Lie Groups and Their Algebras
29.1.1 Group Action
29.1.2 Lie Algebra of a Lie Group
29.1.3 Invariant Forms
29.1.4 Infinitesimal Action
29.1.5 Integration on Lie Groups
29.2 An Outline of Lie Algebra Theory
29.2.1 The Lie Algebras o(p,n-p) and p(p,n-p)
29.2.2 Operations on Lie Algebras
29.3 Problems
Chapter 30: Representation of Lie Groups and Lie Algebras
30.1 Representation of Compact Lie Groups
30.2 Representation of the General Linear Group
30.3 Representation of Lie Algebras
30.3.1 Representation of Subgroups of GL(V)
30.3.2 Casimir Operators
30.3.3 Representation of so(3) and so(3,1)
30.3.4 Representation of the Poincaré Algebra
30.4 Problems
Chapter 31: Representation of Clifford Algebras
31.1 The Clifford Group
31.2 Spinors
31.2.1 Pauli Spin Matrices and Spinors
31.2.2 Spinors for Cµnu(R)
31.2.3 C31(R) Revisited
31.3 Problems
Chapter 32: Lie Groups and Differential Equations
32.1 Symmetries of Algebraic Equations
32.2 Symmetry Groups of Differential Equations
32.2.1 Prolongation of Functions
32.2.2 Prolongation of Groups
32.2.3 Prolongation of Vector Fields
32.3 The Central Theorems
32.4 Application to Some Known PDEs
32.4.1 The Heat Equation
32.4.2 The Wave Equation
32.5 Application to ODEs
32.5.1 First-Order ODEs
32.5.2 Higher-Order ODEs
32.5.3 DEs with Multiparameter Symmetries
32.6 Problems
Chapter 33: Calculus of Variations, Symmetries, and Conservation Laws
33.1 The Calculus of Variations
33.1.1 Derivative for Hilbert Spaces
33.1.2 Functional Derivative
33.1.3 Variational Problems
33.1.4 Divergence and Null Lagrangians
33.2 Symmetry Groups of Variational Problems
33.3 Conservation Laws and Noether’s Theorem
33.4 Application to Classical Field Theory
33.5 Problems
Part X: Fiber Bundles
Chapter 34: Fiber Bundles and Connections
34.1 Principal Fiber Bundles
34.1.1 Associated Bundles
34.2 Connections in a PFB
34.2.1 Local Expression for a Connection
34.2.2 Parallelism
34.3 Curvature Form
34.3.1 Flat Connections
34.3.2 Matrix Structure Group
34.4 Problems
Chapter 35: Gauge Theories
35.1 Gauge Potentials and Fields
35.1.1 Particle Fields
35.1.2 Gauge Transformation
35.2 Gauge-Invariant Lagrangians
35.3 Construction of Gauge-Invariant Lagrangians
35.4 Local Equations
35.5 Problems
Chapter 36: Differential Geometry
36.1 Connections in a Vector Bundle
36.2 Linear Connections
36.2.1 Covariant Derivative of Tensor Fields
36.2.2 From Forms on P to Tensor Fields on M
36.2.3 Component Expressions
36.2.4 General Basis
36.3 Geodesics
36.3.1 Riemann Normal Coordinates
36.4 Problems
Chapter 37: Riemannian Geometry
37.1 The Metric Connection
37.1.1 Orthogonal Bases
37.2 Isometries and Killing Vector Fields
37.3 Geodesic Deviation and Curvature
37.3.1 Newtonian Gravity
37.4 General Theory of Relativity
37.4.1 Einstein’s Equation
37.4.2 Static Spherically Symmetric Solutions
37.4.3 Schwarzschild Geodesics
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