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1 Vector Analysis 1 1.1 Definitions, Elementary Approach 1 1.2 Rotation of the Coordinate Axes 7 1.3 Scalar or Dot Product 12 1.4 Vector or Cross Product 18 1.5 Triple Scalar Product, Triple Vector Product 25 1.6 Gradient, V 32 1.7 Divergence, V 38 1.8 Curl, Vx 43 1.9 Successive Applications of V 49 1.10 Vector Integration 54 1.11 Gauss' Theorem 60 1.12 Stokes' Theorem 64 1.13 Potential Theory 68 1.14 Gauss' Law, Poisson's Equation 79 1.15 Dirac Delta Function 83 1.16 Helmholtz's Theorem 95 Additional Readings 101 2 Vector Analysis in Curved Coordinates and Tensors 103 2.1 Orthogonal Coordinates in R3 103 2.2 Differential Vector Operators 110 2.3 Special Coordinate Systems: Introduction 114 2.4 Circular Cylinder Coordinates 115 2.5 Spherical Polar Coordinates 123 ν 2.6 Tensor Analysis 133 2.7 Contraction, Direct Product 139 2.8 Quotient Rule 141 2.9 Pseudotensors, Dual Tensors 142 2.10 General Tensors 151 2.11 Tensor Derivative Operators 160 Additional Readings 163 3 Determinants and Matrices 165 3.1 Determinants 165 3.2 Matrices 176 3.3 Orthogonal Matrices 195 3.4 Hermitian Matrices, Unitary Matrices 208 3.5 Diagonalization of Matrices 215 3.6 Normal Matrices 231 Additional Readings 239 4 Group Theory 241 4.1 Introduction to Group Theory 241 4.2 Generators of Continuous Groups 246 4.3 Orbital Angular Momentum 261 4.4 Angular Momentum Coupling 266 4.5 Homogeneous Lorentz Group 278 4.6 Lorentz Covariance of Maxwell's Equations 283 4.7 Discrete Groups 291 4.8 Differential Forms 304 Additional Readings 319 5 Infinite Series 321 5.1 Fundamental Concepts 321 5.2 Convergence Tests 325 5.3 Alternating Series 339 5.4 Algebra of Series 342 5.5 Series of Functions 348 5.6 Taylor's Expansion 352 5.7 Power Series 363 5.8 Elliptic Integrals 370 5.9 Bernoulli Numbers, Euler-Maclaurin Formula 376 5.10 Asymptotic Series 389 5.11 Infinite Products 396 Additional Readings 401 6 Functions of a Complex Variable I Analytic P

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