Mathematical Physics Applied Mathematics for Scientists and Engineers

by Kusse, Bruce R.; Westwig, Erik A.

• ISBN13: 9783527406722
• ISBN10: 3527406727
• Edition: 2nd
• Format: Paperback
• Copyright: 2006-03-10
• Publisher: Wiley-VCH

Summary

What sets this volume apart from other mathematics texts is its emphasis on mathematical tools commonly used by scientists and engineers to solve real-world problems. Using a unique approach, it covers intermediate and advanced material in a manner appropriate for undergraduate students. Based on author Bruce Kusse's course at the Department of Applied and Engineering Physics at Cornell University, Mathematical Physics begins with essentials such as vector and tensor algebra, curvilinear coordinate systems, complex variables, Fourier series, Fourier and Laplace transforms, differential and integral equations, and solutions to Laplace's equations. The book moves on to explain complex topics that often fall through the cracks in undergraduate programs, including the Dirac delta-function, multivalued complex functions using branch cuts, branch points and Riemann sheets, contravariant and covariant tensors, and an introduction to group theory. This expanded second edition contains a new appendix on the calculus of variation -- a valuable addition to the already superb collection of topics on offer.This is an ideal text for upper-level undergraduates in physics, applied physics, physical chemistry, biophysics, and all areas of engineering. It allows physics professors to prepare students for a wide range of employment in science and engineering and makes an excellent reference for scientists and engineers in industry. Worked out examples appear throughout the book and exercises follow every chapter. Solutions to the odd-numbered exercises are available for lecturers from the publisher.

Author Biography

Bruce Kusse is Professor of Applied and Engineering Physics at Cornell University, where he has been teaching since 1970. He holds a PhD from the MIT in electrical engineering with a specialty in plasma physics.

Erik Westwig is a software engineer with Palisade Corporation, New Jersey. He holds an MS in applied physics from Cornell University.

Table of Contents

A Review of Vector and Matrix Algebra Using Subscript/Summation Conventions	1	(17)
Notation	1	(4)
Vector Operations	5	(13)
Differential and Integral Operations on Vector and Scalar Fields	18	(26)
Plotting Scalar and Vector Fields	18	(2)
Integral Operators	20	(3)
Differential Operations	23	(11)
Integral Definitions of the Differential Operators	34	(1)
The Theorems	35	(9)
Curvilinear Coordinate Systems	44	(23)
The Position Vector	44	(1)
The Cylindrical System	45	(3)
The Spherical System	48	(1)
General Curvilinear Systems	49	(9)
The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems	58	(9)
Introduction to Tensors	67	(33)
The Conductivity Tensor and Ohm's Law	67	(4)
General Tensor Notation and Terminology	71	(1)
Transformations Between Coordinate Systems	71	(7)
Tensor Diagonalization	78	(6)
Tensor Transformations in Curvilinear Coordinate Systems	84	(2)
Pseudo-Objects	86	(14)
The Dirac δ-Function	100	(35)
Examples of Singular Functions in Physics	100	(3)
Two Definitions of δ(t)	103	(5)
δ-Functions with Complicated Arguments	108	(3)
Integrals and Derivatives of δ(t)	111	(3)
Singular Density Functions	114	(7)
The Infinitesimal Electric Dipole	121	(4)
Riemann Integration and the Dirac δ-Function	125	(10)
Introduction to Complex Variables	135	(84)
A Complex Number Refresher	135	(3)
Functions of a Complex Variable	138	(2)
Derivatives of Complex Functions	140	(4)
The Cauchy Integral Theorem	144	(2)
Contour Deformation	146	(1)
The Cauchy Integral Formula	147	(3)
Taylor and Laurent Series	150	(3)
The Complex Taylor Series	153	(6)
The Complex Laurent Series	159	(12)
The Residue Theorem	171	(4)
Definite Integrals and Closure	175	(14)
Conformal Mapping	189	(30)
Fourier Series	219	(31)
The Sine-Cosine Series	219	(8)
The Exponential Form of Fourier Series	227	(4)
Convergence of Fourier Series	231	(3)
The Discrete Fourier Series	234	(16)
Fourier Transforms	250	(53)
Fourier Series as T0 → ∞	250	(3)
Orthogonality	253	(1)
Existence of the Fourier Transform	254	(2)
The Fourier Transform Circuit	256	(2)
Properties of the Fourier Transform	258	(9)
Fourier Transforms---Examples	267	(23)
The Sampling Theorem	290	(13)
Laplace Transforms	303	(36)
Limits of the Fourier Transform	303	(3)
The Modified Fourier Transform	306	(7)
The Laplace Transform	313	(1)
Laplace Transform Examples	314	(4)
Properties of the Laplace Transform	318	(9)
The Laplace Transform Circuit	327	(4)
Double-Sided or Bilateral Laplace Transforms	331	(8)
Differential Equations	339	(85)
Terminology	339	(3)
Solutions for First-Order Equations	342	(5)
Techniques for Second-Order Equations	347	(7)
The Method of Frobenius	354	(4)
The Method of Quadrature	358	(8)
Fourier and Laplace Transform Solutions	366	(10)
Green's Function Solutions	376	(48)
Solutions to Laplace's Equation	424	(67)
Cartesian Solutions	424	(9)
Expansions With Eigenfunctions	433	(8)
Cylindrical Solutions	441	(17)
Spherical Solutions	458	(33)
Integral Equations	491	(18)
Classification of Linear Integral Equations	492	(1)
The Connection Between Differential and Integral Equations	493	(5)
Methods of Solution	498	(11)
Advanced Topics in Complex Analysis	509	(53)
Multivalued Functions	509	(33)
The Method of Steepest Descent	542	(20)
Tensors in Non-Orthogonal Coordinate Systems	562	(35)
A Brief Review of Tensor Transformations	562	(2)
Non-Orthonormal Coordinate Systems	564	(33)
Introduction to Group Theory	597	(42)
The Definition of a Group	597	(1)
Finite Groups and Their Representations	598	(9)
Subgroups, Cosets, Class, and Character	607	(5)
Irreducible Matrix Representations	612	(18)
Continuous Groups	630	(9)
Appendix A The Levi-Civita Identity	639	(2)
Appendix B The Curvilinear Curl	641	(4)
Appendix C The Double Integral Identity	645	(2)
Appendix D Green's Function Solutions	647	(6)
Appendix E Pseudovectors and the Mirror Test	653	(2)
Appendix F Christoffel Symbols and Covariant Derivatives	655	(6)
Appendix G Calculus of Variations	661	(4)
Errata List	665	(6)
Bibliography	671	(2)
Index	673