Mathematical Physics: Applied Mathematics for Scientists and Engineers

Mathematical Physics: Applied Mathematics for Scientists and Engineers
	CONTENTS
		1 A Review of Vector and Matrix Algebra Using Subscript/Summation Conventions
			1.1 Notation
			1.2 Vector Operations
		2 Differential and Integral Operations on Vector and Scalar Fields
			2.1 Plotting Scalar and Vector Fields
			2.2 Integral Operators
			2.3 Differential Operations
			2.4 Integral Definitions of the Differential Operators
			2.5 TheTheorems
		3 Curvilinear Coordinate Systems
			3.1 The Position Vector
			3.2 The Cylindrical System
			3.3 The Spherical System
			3.4 General Curvilinear Systems
			3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems
		4 Introduction to Tensors
			4.1 The Conductivity Tensor and Ohm’s Law
			4.2 General Tensor Notation and Terminology
			4.3 Transformations Between Coordinate Systems
			4.4 Tensor Diagonalization
			4.5 Tensor Transformations in Curvilinear Coordinate Systems
			4.6 Pseudo-Objects
		5 The Dirac δ-Function
			5.1 Examples of Singular Functions in Physics
			5.2 Two Definitions of δ(t)
			5.3 6 δ-Functions with Complicated Arguments
			5.4 Integrals and Derivatives of δ(t)
			5.5 Singular Density Functions
			5.6 The Infinitesimal Electric Dipole
			5.7 Riemann Integration and the Dirac δ-Function
		6 Introduction to Complex Variables
			6.1 A Complex Number Refresher
			6.2 Functions of a Complex Variable
			6.3 Derivatives of Complex Functions
			6.4 The Cauchy Integral Theorem
			6.5 Contour Deformation
			6.6 The Cauchy Integral Formula
			6.7 Taylor and Laurent Series
			6.8 The Complex Taylor Series
			6.9 The Complex Laurent Series
			6.10 The Residue Theorem
			6.11 Definite Integrals and Closure
			6.12 Conformal Mapping
		7 Fourier Series
			7.1 The Sine-Cosine Series
			7.2 The Exponential Form of Fourier Series
			7.3 Convergence of Fourier Series
			7.4 The Discrete Fourier Series
		8 Fourier Transforms
			8.1 Fourier Series as T0 → ∞
			8.2 Orthogonality
			8.3 Existence of the Fourier Transform
			8.4 The Fourier Transform Circuit
			8.5 Properties of the Fourier Transform
			8.6 Fourier Transforms-Examples
			8.7 The Sampling Theorem
		9 Laplace Transforms
			9.1 Limits of the Fourier Transform
			9.2 The Modified Fourier Transform
			9.3 The Laplace Transform
			9.4 Laplace Transform Examples
			9.5 Properties of the Laplace Transform
			9.6 The Laplace Transform Circuit
			9.7 Double-Sided or Bilateral Laplace Transforms
		10 Differential Equations
			10.1 Terminology
			10.2 Solutions for First-Order Equations
			10.3 Techniques for Second-Order Equations
			10.4 The Method of Frobenius
			10.5 The Method of Quadrature
			10.6 Fourier and Laplace Transform Solutions
			10.7 Green’s Function Solutions
		11 Solutions to Laplace’s Equation
			11.1 Cartesian Solutions
			11.2 Expansions With Eigenfunctions
			11.3 Cylindrical Solutions
			11.4 Spherical Solutions
		12 Integral Equations
			12.1 Classification of Linear Integral Equations
			12.2 The Connection Between Differential and Integral Equations
			12.3 Methods of Solution
		13 Advanced Topics in Complex Analysis
			13.1 Multivalued Functions
			13.2 The Method of Steepest Descent
		14 Tensors in Non-Orthogonal Coordinate Systems
			14.1 A Brief Review of Tensor Transformations
			14.2 Non-Orthononnal Coordinate Systems
		15 Introduction to Group Theory
			15.1 The Definition of a Group
			15.2 Finite Groups and Their Representations
			15.3 Subgroups, Cosets, Class, and Character
			15.4 Irreducible Matrix Representations
			15.5 Continuous Groups
		Appendix A The Levi-Civita Identity
		Appendix B The Curvilinear Curl
		Appendiv C The Double Integral Identity
		Appendix D Green’s Function Solutions
		Appendix E Pseudovectors and the Mirror Test
		Appendix F Christoffel Symbols and Covariant Derivatives
		Appendix G Calculus of Variations
		Errata List
		Bibliography
		Index