Mathematical Physics: Applied Mathematics for Scientists and Engineers (Physics Textbook)

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 (http://www.wiley-vch.de/textbooks/) www.wiley-vch.de/textbooks/ . 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

The second, corrected edition of this well-established mathematical text again puts 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.

The book covers applications in all areas of engineering and the physical science, and features numerous figures and worked-out examples throughout the text. Many end-of-chapter exercises are provides; a free solution manual

is available for lecturers. The topics are organized pedagogically, in the order they will be most easily understood.

From the contents:

• A review of Vector and Matrix Algebra Using Subscript/Summation Conventions

• Differential and Integral Operations on Vector and Scalar Fields

• Curvilinear Coordinate Systems

• Tensors in Orthogonal and Skewed Systems

• The Dirac Function

• Complex Variables

• Fourier Series

• Fourier and Laplace Transforms

• Differential Equations

• Solutions to Laplace's Equation

• Integral Equations

Booknews

Based on Kusse's course at Cornell University, a textbook for upper- level undergraduate students emphasizing the mathematical tools commonly used by scientists and engineers to solve real-world problems. Assumes elementary calculus and keeps the number of formal proofs and theorems to a minimum. Begins with basics such as vector and tensor algebra and curvilinear coordinate systems; then tackles topics more complex than are usually taught at the undergraduate level, such as the Dirac delta-function and branch points and Riemann sheets. Annotation c. by Book News, Inc., Portland, Or.

"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."--Publisher's description