Mathematical Physics: Applied Mathematics for Scientists and Engineers

✍ Scribed by Bruce R. Kusse, Erik A. Westwig

• Publisher: Wiley-VCH
• Year: 2006
• Tongue: English
• Leaves: 699
• Edition: 2
• Category: Library

✦ Synopsis

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 at www.wiley-vch.de/textbooks/.

✦ Table of Contents

Mathematical Physics: Applied Mathematics for Scientists and Engineers
CONTENTS
1 A Review of Vector and Matrix Algebra Using Subscript/Summation Conventions
1.1 Notation
1.2 Vector Operations
2 Differential and Integral Operations on Vector and Scalar Fields
2.1 Plotting Scalar and Vector Fields
2.2 Integral Operators
2.3 Differential Operations
2.4 Integral Definitions of the Differential Operators
2.5 TheTheorems
3 Curvilinear Coordinate Systems
3.1 The Position Vector
3.2 The Cylindrical System
3.3 The Spherical System
3.4 General Curvilinear Systems
3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems
4 Introduction to Tensors
4.1 The Conductivity Tensor and Ohm’s Law
4.2 General Tensor Notation and Terminology
4.3 Transformations Between Coordinate Systems
4.4 Tensor Diagonalization
4.5 Tensor Transformations in Curvilinear Coordinate Systems
4.6 Pseudo-Objects
5 The Dirac δ-Function
5.1 Examples of Singular Functions in Physics
5.2 Two Definitions of δ(t)
5.3 6 δ-Functions with Complicated Arguments
5.4 Integrals and Derivatives of δ(t)
5.5 Singular Density Functions
5.6 The Infinitesimal Electric Dipole
5.7 Riemann Integration and the Dirac δ-Function
6 Introduction to Complex Variables
6.1 A Complex Number Refresher
6.2 Functions of a Complex Variable
6.3 Derivatives of Complex Functions
6.4 The Cauchy Integral Theorem
6.5 Contour Deformation
6.6 The Cauchy Integral Formula
6.7 Taylor and Laurent Series
6.8 The Complex Taylor Series
6.9 The Complex Laurent Series
6.10 The Residue Theorem
6.11 Definite Integrals and Closure
6.12 Conformal Mapping
7 Fourier Series
7.1 The Sine-Cosine Series
7.2 The Exponential Form of Fourier Series
7.3 Convergence of Fourier Series
7.4 The Discrete Fourier Series
8 Fourier Transforms
8.1 Fourier Series as T0 → ∞
8.2 Orthogonality
8.3 Existence of the Fourier Transform
8.4 The Fourier Transform Circuit
8.5 Properties of the Fourier Transform
8.6 Fourier Transforms-Examples
8.7 The Sampling Theorem
9 Laplace Transforms
9.1 Limits of the Fourier Transform
9.2 The Modified Fourier Transform
9.3 The Laplace Transform
9.4 Laplace Transform Examples
9.5 Properties of the Laplace Transform
9.6 The Laplace Transform Circuit
9.7 Double-Sided or Bilateral Laplace Transforms
10 Differential Equations
10.1 Terminology
10.2 Solutions for First-Order Equations
10.3 Techniques for Second-Order Equations
10.4 The Method of Frobenius
10.5 The Method of Quadrature
10.6 Fourier and Laplace Transform Solutions
10.7 Green’s Function Solutions
11 Solutions to Laplace’s Equation
11.1 Cartesian Solutions
11.2 Expansions With Eigenfunctions
11.3 Cylindrical Solutions
11.4 Spherical Solutions
12 Integral Equations
12.1 Classification of Linear Integral Equations
12.2 The Connection Between Differential and Integral Equations
12.3 Methods of Solution
13 Advanced Topics in Complex Analysis
13.1 Multivalued Functions
13.2 The Method of Steepest Descent
14 Tensors in Non-Orthogonal Coordinate Systems
14.1 A Brief Review of Tensor Transformations
14.2 Non-Orthononnal Coordinate Systems
15 Introduction to Group Theory
15.1 The Definition of a Group
15.2 Finite Groups and Their Representations
15.3 Subgroups, Cosets, Class, and Character
15.4 Irreducible Matrix Representations
15.5 Continuous Groups
Appendix A The Levi-Civita Identity
Appendix B The Curvilinear Curl
Appendiv C The Double Integral Identity
Appendix D Green’s Function Solutions
Appendix E Pseudovectors and the Mirror Test
Appendix F Christoffel Symbols and Covariant Derivatives
Appendix G Calculus of Variations
Errata List
Bibliography
Index