Mathematical Physics with Partial Differential Equations

1st Edition

Authors: James Kirkwood
Hardcover ISBN: 9780123869111
eBook ISBN: 9780123869944
Imprint: Academic Press
Published Date: 20th January 2012
Page Count: 432
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Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field – the heat equation, the wave equation, and Laplace’s equation. The most common techniques of solving such equations are developed in this book, including Green’s functions, the Fourier transform, and the Laplace transform, which all have applications in mathematics and physics far beyond solving the above equations. The book’s focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book’s rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics.

Key Features

  • Examines in depth both the equations and their methods of solution
  • Presents physical concepts in a mathematical framework
  • Contains detailed mathematical derivations and solutions— reinforcing the material through repetition of both the equations and the techniques
  • Includes several examples solved by multiple methods—highlighting the strengths and weaknesses of various techniques and providing additional practice


Advanced Undergraduate and Graduate Students, Instructors, Academic Researchers in University Mathematics Departments

Table of Contents

  • Preface
  • Chapter 1. Preliminaries
    • 1-1. Self-Adjoint Operators
    • 1-2. Curvilinear Coordinates
    • 1-3. Approximate Identities and the Dirac-δ Function
    • 1-4. The Issue of Convergence
    • 1-5. Some Important Integration Formulas
  • Chapter 2. Vector Calculus
    • 2-1. Vector Integration
    • 2-2. Divergence and Curl
    • 2-3. Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem
  • Chapter 3. Green’s Functions
    • 3-1. Construction of Green’s Function using the Dirac-δ Function
    • 3-2. Construction of Green’s Function using Variation of Parameters
    • 3-3. Construction of Green’s Function from Eigenfunctions
    • 3-4. More General Boundary Conditions
    • 3-5. The Fredholm Alternative (Or, what if 0 is an Eigenvalue?)
    • 3-6. Green’s function for the Laplacian in Higher Dimensions
  • Chapter 4. Fourier Series
    • 4-1. Basic Definitions
    • 4-2. Methods of Convergence of Fourier Series
    • 4-3. The Exponential Form of Fourier Series
    • 4-4. Fourier Sine and Cosine Series
    • 4-5. Double Fourier Series
  • Chapter 5. Three Important Equations
    • 5-1. Laplace’s Equation
    • 5-2. Derivation of the Heat Equation in One Dimension
    • 5-3. Derivation of the Wave equation in One Dimension
    • 5-4. An Explicit Solution of the Wave Equation
    • 5-5. Converting Second-Order PDEs to Standard Form
  • Chapter 6. Sturm-Liouville Theory
    • 6-1. The Self-Adjoint Property of a Sturm-Liouville Equation
    • 6-2. Completeness of Eigenfunctions for Sturm-Liouville Equations
    • 6-3. Uniform Convergence of Fourier Series
  • Chapter 7. Separation of Variables in Cartesian Coordinates
    • 7-1. Solving Laplace’s Equation on a Rectangle
    • 7-2. Laplace’s Equation on a Cube


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© Academic Press 2013
Academic Press
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About the Author

James Kirkwood

Affiliations and Expertise

Professor of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA, USA


"The text presents some of the most important topics and methods of mathematical physics…The book’s rigor is appropriate for readers wanting to continue their study of further areas of mathematical physics."--Zentralblatt MATH 2012-1235-35002