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 www.wiley-vch.de/textbooks/.Content: Chapter 1 A Review of Vector and Matrix Algebra Using Subscript/Summation Conventions (pages 1–17): Chapter 2 Differential and Integral Operations on Vector and Scalar Fields (pages 18–43): Chapter 3 Curvilinear Coordinate Systems (pages 44–66): Chapter 4 Introduction to Tensors (pages 67–99): Chapter 5 The Dirac ??Function (pages 100–134): Chapter 6 Introduction to Complex Variables (pages 135–218): Chapter 7 Fourier Series (pages 219–249): Chapter 8 Fourier Transforms (pages 250–302): Chapter 9 Laplace Transforms (pages 303–338): Chapter 10 Differential Equations (pages 339–423): Chapter 11 Solutions to Laplace's Equation (pages 424–490): Chapter 12 Integral Equations (pages 491–508): Chapter 13 Advanced Topics in Complex Analysis (pages 509–561): Chapter 14 Tensors in Non?Orthogonal Coordinate Systems (pages 562–596): Chapter 15 Introduction to Group Theory (pages 597–638): Mathematical Physics: Applied Mathematics for Scientists and Engineers......Page 2 CONTENTS......Page 10 1.1 Notation......Page 16 1.2 Vector Operations......Page 20 2.1 Plotting Scalar and Vector Fields......Page 33 2.2 Integral Operators......Page 35 2.3 Differential Operations......Page 38 2.4 Integral Definitions of the Differential Operators......Page 49 2.5 TheTheorems......Page 50 3.1 The Position Vector......Page 59 3.2 The Cylindrical System......Page 60 3.3 The Spherical System......Page 63 3.4 General Curvilinear Systems......Page 64 3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems......Page 73 4.1 The Conductivity Tensor and Ohm’s Law......Page 82 4.3 Transformations Between Coordinate Systems......Page 86 4.4 Tensor Diagonalization......Page 93 4.5 Tensor Transformations in Curvilinear Coordinate Systems......Page 99 4.6 Pseudo-Objects......Page 101 5.1 Examples of Singular Functions in Physics......Page 115 5.2 Two Definitions of δ(t)......Page 118 5.3 6 δ-Functions with Complicated Arguments......Page 123 5.4 Integrals and Derivatives of δ(t)......Page 126 5.5 Singular Density Functions......Page 129 5.6 The Infinitesimal Electric Dipole......Page 136 5.7 Riemann Integration and the Dirac δ-Function......Page 140 6.1 A Complex Number Refresher......Page 150 6.2 Functions of a Complex Variable......Page 153 6.3 Derivatives of Complex Functions......Page 155 6.4 The Cauchy Integral Theorem......Page 159 6.5 Contour Deformation......Page 161 6.6 The Cauchy Integral Formula......Page 162 6.7 Taylor and Laurent Series......Page 165 6.8 The Complex Taylor Series......Page 168 6.9 The Complex Laurent Series......Page 174 6.10 The Residue Theorem......Page 186 6.11 Definite Integrals and Closure......Page 190 6.12 Conformal Mapping......Page 204 7.1 The Sine-Cosine Series......Page 234 7.2 The Exponential Form of Fourier Series......Page 242 7.3 Convergence of Fourier Series......Page 246 7.4 The Discrete Fourier Series......Page 249 8.1 Fourier Series as T0 → ∞......Page 265 8.2 Orthogonality......Page 268 8.3 Existence of the Fourier Transform......Page 269 8.4 The Fourier Transform Circuit......Page 271 8.5 Properties of the Fourier Transform......Page 273 8.6 Fourier Transforms-Examples......Page 282 8.7 The Sampling Theorem......Page 305 9.1 Limits of the Fourier Transform......Page 318 9.2 The Modified Fourier Transform......Page 321 9.3 The Laplace Transform......Page 328 9.4 Laplace Transform Examples......Page 329 9.5 Properties of the Laplace Transform......Page 333 9.6 The Laplace Transform Circuit......Page 342 9.7 Double-Sided or Bilateral Laplace Transforms......Page 346 10.1 Terminology......Page 354 10.2 Solutions for First-Order Equations......Page 357 10.3 Techniques for Second-Order Equations......Page 362 10.4 The Method of Frobenius......Page 369 10.5 The Method of Quadrature......Page 373 10.6 Fourier and Laplace Transform Solutions......Page 381 10.7 Green’s Function Solutions......Page 391 11.1 Cartesian Solutions......Page 439 11.2 Expansions With Eigenfunctions......Page 448 11.3 Cylindrical Solutions......Page 456 11.4 Spherical Solutions......Page 473 12 Integral Equations......Page 506 12.1 Classification of Linear Integral Equations......Page 507 12.2 The Connection Between Differential and Integral Equations......Page 508 12.3 Methods of Solution......Page 513 13.1 Multivalued Functions......Page 524 13.2 The Method of Steepest Descent......Page 557 14.1 A Brief Review of Tensor Transformations......Page 577 14.2 Non-Orthononnal Coordinate Systems......Page 579 15.1 The Definition of a Group......Page 612 15.2 Finite Groups and Their Representations......Page 613 15.3 Subgroups, Cosets, Class, and Character......Page 622 15.4 Irreducible Matrix Representations......Page 627 15.5 Continuous Groups......Page 645 Appendix A The Levi-Civita Identity......Page 654 Appendix B The Curvilinear Curl......Page 656 Appendiv C The Double Integral Identity......Page 660 Appendix D Green’s Function Solutions......Page 662 Appendix E Pseudovectors and the Mirror Test......Page 668 Appendix F Christoffel Symbols and Covariant Derivatives......Page 670 Appendix G Calculus of Variations......Page 676 Errata List......Page 680 Bibliography......Page 686 Index......Page 688

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. Solutions to the odd-numbered exercises are available for lecturers at (http://www.wiley-vch.de/textbooks/) www.wiley-vch.de/textbooks/ . 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/. 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 Show moreseries, 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