Essentials of Math Methods for Physicists
Published on 11. September 2013
960 pages
978-1-4832-2562-3 (ISBN)
Description
Essentials of Math Methods for Physicists aims to guide the student in learning the mathematical language used by physicists by leading them through worked examples and then practicing problems. The pedagogy is that of introducing concepts, designing and refining methods and practice them repeatedly in physics examples and problems. Geometric and algebraic approaches and methods are included and are more or less emphasized in a variety of settings to accommodate different learning styles of students. Comprised of 19 chapters, this book begins with an introduction to the basic concepts of vector algebra and vector analysis and their application to classical mechanics and electrodynamics. The next chapter deals with the extension of vector algebra and analysis to curved orthogonal coordinates, again with applications from classical mechanics and electrodynamics. These chapters lay the foundations for differential equations, variational calculus, and nonlinear analysisin later discussions. High school algebra of one or two linear equations is also extended to determinants and matrix solutions of general systems of linear equations, eigenvalues and eigenvectors, and linear transformations in real and complex vector spaces. The book also considers probability and statistics as well as special functions and Fourier series. Historical remarks are included that describe some physicists and mathematicians who introduced the ideas and methods that were perfected by later generations to the tools routinely used today. This monograph is intended to help undergraduate students prepare for the level of mathematics expected in more advanced undergraduate physics and engineering courses.
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Content
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.
