Brett Borden - Mathematical methods in physics, engineering and chemistry

1. Chapter 3
2. Chapter 5
3. Chapter 6
4. Chapter 7
5. Chapter 8

1. Chapter 1
2. Chapter 2
3. Chapter 3
4. Chapter 4
5. Chapter 5
6. Chapter 6
7. Chapter 7
8. Chapter 8
9. Chapter 9
10. Chapter 10
11. Chapter 11
12. Appendix A
13. Appendix C
14. Appendix D

BRETT BORDEN AND JAMES LUSCOMBE

Naval Postgraduate School
Monterey, CA, USA

This edition first published 2020
2020 John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication data applied for

ISBN: 9781119579656

Cover Design: Wiley
Cover Image: FrankRamspott/Getty Images

Mathematics is deeply ingrained in physics, in how it's taught and how it's practiced. Courses in mathematical methods of physics are core components of physics curricula, at the advanced undergraduate and beginning graduate levels. In our experience, textbooks that seek to provide a comprehensive coverage of mathematical methods tend not to mesh well with the needs of today's students, who face curricula continually under pressure to squeeze more content into the allotted time. The amount of mathematics one could be called upon to know in a scientific career is daunting hence, the temptation to try to cover it all in textbooks. We have developed, over years of teaching, a set of notes outlining the essentials of the subject, which has turned into this book. Our goal has been to emphasize topics that the majority of students will require in the course of their studies. Not every student will go on to a career in theoretical physics, and thus not every student need be exposed to specialized topics at this point in their education.

Following is a sketch of the contents of this book.

• Linear algebra: What's more important in the education of scientists and engineers, calculus or linear algebra? We opt for the latter. Students are assumed to have had vector calculus (a standard component of Calculus III), a summary of which is provided in .
• Partial differential equations: A pervasive concept in physics is that of fields, the behavior of which in space and time is described by partial differential equations (PDEs). To study physics at the advanced undergraduate level, one must be proficient in solving PDEs and that provides another overarching theme: Boundary value problems. cover separate topics that could form the basis of a followon course or a graduatelevel course appropriate for some instructional programs. Ambitious programs could cover the entire book in one semester.

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• Special functions: We cover the most commonly encountered special functions. The gamma function is treated in . We omit special functions seen only in other courses, e.g. the Laguerre and Hermite polynomials.
• Complex analysis: It's never quite obvious where a chapter on complex analysis should go. We place the theory of analytic functions (). We cover the standard topics of contour integration and Cauchy's theorem. We develop the nonstandard topics of the approximation of integrals (steepest descent and stationary phase) and the analytic signal (Hilbert transform, the PaleyWeiner and Titchmarsh theorems). The latter is important in applications involving signal processing, which many students end up doing in their thesis work, and is natural to include in a discussion on analytic functions.
• Green functions: Inhomogeneous differential equations are naturally solved using the method of Green functions. We illustrate (in ) the Green function method for the inhomogeneous Helmholtz, diffusion, and wave equations.
• Integral equations: Many key equations occur as integral equations in scattering theory, for example, either quantum or electromagnetic. We cover integral equations in , a topic not always included in books at this level. Yet, it's important for students to understand that the separationofvariables method (introduced earlier in the book) is often not realistic it relies on the boundaries of systems having rather simple shapes (spherical, cylindrical, etc.). Numerical solutions of PDEs are often based on integralequation approaches.
• Tensor analysis: The book ends with an introduction to tensors, . Tensors are motivated as a mathematical tool for treating systems featuring anisotropy, and for their fundamental role in establishing covariant equations. We present tensors with sufficient depth so as to provide a foundation for their use in the special theory of relativity. The covariant derivative, developed at the end of the chapter, would be the starting point for more advanced applications.