1st Edition

A Course in Mathematical Methods for Physicists

By Russell L. Herman Copyright 2013
    776 Pages 420 B/W Illustrations
    by CRC Press

    792 Pages
    by CRC Press

    Based on the author’s junior-level undergraduate course, this introductory textbook is designed for a course in mathematical physics. Focusing on the physics of oscillations and waves, A Course in Mathematical Methods for Physicists helps students understand the mathematical techniques needed for their future studies in physics. It takes a bottom-up approach that emphasizes physical applications of the mathematics. 

    The book offers:

    • A quick review of mathematical prerequisites, proceeding to applications of differential equations and linear algebra
    • Classroom-tested explanations of complex and Fourier analysis for trigonometric and special functions
    • Coverage of vector analysis and curvilinear coordinates for solving higher dimensional problems
    • Sections on nonlinear dynamics, variational calculus, numerical solutions of differential equations, and Green's functions

    Introduction and Review
    What Do I Need To Know From Calculus?
    What I Need From My Intro Physics Class?
    Technology and Tables
    Appendix: Dimensional Analysis

    Free Fall and Harmonic Oscillators
    Free Fall
    First Order Differential Equations
    The Simple Harmonic Oscillator
    Second Order Linear Differential Equations
    LRC Circuits
    Damped Oscillations
    Forced Systems
    Cauchy-Euler Equations
    Numerical Solutions of ODEs
    Numerical Applications
    Linear Systems

    Linear Algebra
    Finite Dimensional Vector Spaces
    Linear Transformations
    Eigenvalue Problems
    Matrix Formulation of Planar Systems
    Appendix: Diagonalization and Linear Systems

    Nonlinear Dynamics
    The Logistic Equation
    Autonomous First Order Equations
    Bifurcations for First Order Equations
    Nonlinear Pendulum
    The Stability of Fixed Points in Nonlinear Systems
    Nonlinear Population Models
    Limit Cycles
    Nonautonomous Nonlinear Systems
    Exact Solutions Using Elliptic Functions

    The Harmonics of Vibrating Strings
    Harmonics and Vibrations
    Boundary Value Problems
    Partial Differential Equations
    The 1D Heat Equation
    The 1D Wave Equation
    Introduction to Fourier Series
    Fourier Trigonometric Series
    Fourier Series Over Other Intervals
    Sine and Cosine Series
    Solution of the Heat Equation
    Finite Length Strings
    The Gibbs Phenomenon
    Green’s Functions for 1D Partial Differential Equations
    Derivation of Generic 1D Equations

    Non-sinusoidal Harmonics and Special Functions
    Function Spaces
    Classical Orthogonal Polynomials
    Fourier-Legendre Series
    Gamma Function
    Fourier-Bessel Series
    Sturm-Liouville Eigenvalue Problems
    Nonhomogeneous Boundary Value Problems - Green’s Functions
    Appendix: The Least Squares Approximation
    Appendix: The Fredholm Alternative Theorem

    Complex Representations of Functions
    Complex Representations of Waves
    Complex Numbers
    Complex Valued Functions
    Complex Differentiation
    Complex Integration

    Transform Techniques in Physics
    Complex Exponential Fourier Series
    Exponential Fourier Transform
    The Dirac Delta Function
    Properties of the Fourier Transform
    The Convolution Operation
    The Laplace Transform
    Applications of Laplace Transforms
    The Convolution Theorem
    The Inverse Laplace Transform
    Transforms and Partial Differential Equations

    Vector Analysis and EM Waves
    Vector Analysis
    Electromagnetic Waves
    Curvilinear Coordinates

    Extrema and Variational Calculus
    Stationary and Extreme Values of Functions
    The Calculus of Variations
    Hamilton’s Principle

    Problems in Higher Dimensions
    Vibrations of Rectangular Membranes
    Vibrations of a Kettle Drum
    Laplace’s Equation in 2D
    Three Dimensional Cake Baking
    Laplace’s Equation and Spherical Symmetry
    Schrödinger Equation in Spherical Coordinates
    Solution of the 3D Poisson Equation
    Green’s Functions for Partial Differential Equations

    Review of Sequences and Infinite Series
    Sequences of Real Numbers
    Convergence of Sequences
    Limit Theorems
    Infinite Series
    Convergence Tests
    Sequences of Functions
    Infinite Series of Functions
    Special Series Expansions
    The Order of Sequences and Functions


    Russell L. Herman

    "… a welcome and refreshing addition to a rich body of literature. … should fit into the sophomore or junior year of a typical physics undergraduate curriculum. … Engineers and chemistry majors, too, would benefit from taking such an intermediate-level course, perhaps even more so than from a higher-level one. Our own department at Howard University is considering a mid-level math methods course, and I would definitely recommend this textbook as well suited. A Course in Mathematical Methods for Physicists includes plenty of interesting worked-out examples, many of them quite realistic, and uses them to introduce concepts in a reasonable progression. …
    Although the subject of mathematical methods has inspired many valuable texts, Herman’s approach, motivated by the physics applications, is novel, seldom used by other authors. The myriad well-chosen worked-out examples and other strengths have earned my firm endorsement …"
    —Tristan Hübsch, Howard University, Washington, District of Columbia, USA, from Physics Today