Essentials of Math Methods for Physicists

Preface1 Vector Analysis 1.1 Elementary Approach 1.2 Scalar or Dot Product 1.3 Vector or Cross Product 1.4 Triple Scalar Product, Triple Vector Product 1.5 Gradient, ∇ 1.6 Divergence, ∇ 1.7 Curl, ∇x 1.8 Successive Applications of ∇ 1.9 Vector Integration 1.10 Gauss' Theorem 1.11 Stokes' Theorem 1.12 Potential Theory 1.13 Gauss' Law, Poisson's Equation 1.14 Dirac Delta Function2 Vector Analysis in Curved Coordinates and Tensors 2.1 Special Coordinate Systems: Introduction 2.2 Circular Cylinder Coordinates 2.3 Orthogonal Coordinates 2.4 Differential Vector Operators 2.5 Spherical Polar Coordinates 2.6 Tensor Analysis 2.7 Contraction, Direct Product 2.8 Quotient Rule 2.9 Dual Tensors3 Determinants and Matrices 3.1 Determinants 3.2 Matrices 3.3 Orthogonal Matrices 3.4 Hermitian Matrices, Unitary Matrices 3.5 Diagonalization of Matrices4 Group Theory 4.1 Introduction to Group Theory 4.2 Generators of Continuous Groups 4.3 Orbital Angular Momentum 4.4 Homogeneous Lorentz Group5 Infinite Series 5.1 Fundamental Concepts 5.2 Convergence Tests 5.3 Alternating Series 5.4 Algebra of Series 5.5 Series of Functions 5.6 Taylor's Expansion 5.7 Power Series 5.8 Elliptic Integrals 5.9 Bernoulli Numbers, Euler-Maclaurin Formula 5.10 Asymptotic Series6 Functions of a Complex Variable I 389 6.1 Complex Algebra 6.2 Cauchy-Riemann Conditions 6.3 Cauchy's Integral Theorem 6.4 Cauchy's Integral Formula 6.5 Laurent Expansion 6.6 Mapping 6.7 Conformal Mapping7 Functions of a Complex Variable II 7.1 Singularities 7.2 Calculus of Residues 7.3 Method of Steepest Descents8 Differential Equations 8.1 Introduction 8.2 First Order Differential Equations 8.3 Second Order Differential Equations 8.4 Singular Points 8.5 Series Solutions-Frobenius's Method 8.6 A Second Solution 8.7 Numerical Solutions 8.8 Introduction to Partial Differential Equations 8.9 Separation of Variables9 Sturm-Liouville Theory—Orthogonal Functions 9.1 Self-Adjoint ODEs 9.2 Hermitian Operators 9.3 Gram-Schmidt Orthogonalization 9.4 Completeness of Eigenfunctions10 The Gamma Function (Factorial Function) 10.1 Definitions, Simple Properties 10.2 Digamma and Polygamma Functions 10.3 Stirling's Series 10.4 The Incomplete Gamma Function and Related Functions11 Legendre Polynomials and Spherical Harmonics 11.1 Introduction 11.2 Recurrence Relations and Special Properties 11.3 Orthogonality 11.4 Alternate Definitions of Legendre Polynomials 11.5 Associated Legendre Functions12 Bessel Functions 12.1 Bessel Functions of the First Kind Jv(x) 12.2 Neumann Functions, Bessel Functions of the Second Kind 12.3 Asymptotic Expansions 12.4 Spherical Bessel Functions13 Hermite and Laguerre Polynomials 13.1 Hermite Polynomials 13.2 Laguerre Functions14 Fourier Series 14.1 General Properties 14.2 Advantages, Uses of Fourier Series 14.3 Complex Fourier Series 14.4 Properties of Fourier Series15 Integral Transforms 15.1 Introduction, Definitions 15.2 Fourier Transform 15.3 Development of the Inverse Fourier Transforms 15.4 Fourier Transforms-Inversion Theorem 15.5 Fourier Transform of Derivatives 15.6 Convolution Theorem 15.7 Momentum Representation 15.8 Laplace Transforms 15.9 Laplace Transform of Derivatives 15.10 Other Properties 15.11 Convolution or Faltungs Theorem 15.12 Inverse Laplace Transform16 Partial Differential Equations 16.1 Examples of PDEs and Boundary Conditions 16.2 Heat Flow or Diffusion PDE 16.3 Inhomogeneous PDE-Green's Function17 Probability 17.1 Definitions, Simple Properties 17.2 Random Variables 17.3 Binomial Distribution 17.4 Poisson Distribution 17.5 Gauss' Normal Distribution 17.6 Statistics18 Calculus of Variations 18.1 A Dependent and an Independent Variable 18.2 Several Dependent Variables 18.3 Several Independent Variables 18.4 Several Dependent and Independent Variables 18.5 Lagrangian Multipliers: Variation with Constraints 18.6 Rayleigh-Ritz Variational Technique19 Nonlinear Methods and Chaos 19.1 Introduction 19.2 The Logistic Map 19.3 Sensitivity to Initial Conditions and Parameters 19.4 Nonlinear Differential EquationsAppendix: Real Zeros of a FunctionIndex