Mathematical Physics with Partial Differential Equations, Second Edition, is designed for upper division undergraduate and beginning graduate students taking mathematical physics taught out by math departments. The new edition is based on the success of the first, with a continuing focus on clear presentation, detailed examples, mathematical rigor and a careful selection of topics. It presents the familiar classical topics and methods of mathematical physics with more extensive coverage of the three most important partial differential equations in the field of mathematical physics—the heat equation, the wave equation and Laplace’s equation.
The book presents the most common techniques of solving these equations, and their derivations are developed in detail for a deeper understanding of mathematical applications. Unlike many physics-leaning mathematical physics books on the market, this work is heavily rooted in math, making the book more appealing for students wanting to progress in mathematical physics, with particularly deep coverage of Green’s functions, the Fourier transform, and the Laplace transform. A salient characteristic is the focus on fewer topics but at a far more rigorous level of detail than comparable undergraduate-facing textbooks. The depth of some of these topics, such as the Dirac-delta distribution, is not matched elsewhere.
New features in this edition include: novel and illustrative examples from physics including the 1-dimensional quantum mechanical oscillator, the hydrogen atom and the rigid rotor model; chapter-length discussion of relevant functions, including the Hermite polynomials, Legendre polynomials, Laguerre polynomials and Bessel functions; and all-new focus on complex examples only solvable by multiple methods.
1. 3.1. Introduction - p. 153 2. 3.2. Construction of Green's Function Using the Dirac-Delta Function - p. 154 3. 3.3. Green's Function Using Variation of Parameters - p. 162 4. 3.4. Construction of Green's Function From Eigenfunctions - p. 166 5. 3.5. More General Boundary Conditions - p. 169 6. 3.6. The Fredholm Alternative (or, What If 0 Is an Eigenvalue?) - p. 171 7. 3.7. Green's Function for the Laplacian in Higher Dimensions - p. 178 8. Chapter 4. Fourier Series - p. 187 9. 4.1. Introduction - p. 187 10. 4.2. Basic Definitions - p. 188 11. 4.3. Methods of Convergence of Fourier Series - p. 194 12. 4.4. The Exponential Form of Fourier Series - p. 207 13. 4.5. Fourier Sine and Cosine Series - p. 212 14. 4.6. Double Fourier Series - p. 215 15. Chapter 5. Three Important Equations - p. 217 16. Chapter 6. Sturm–Liouville Theory - p. 237 17. Chapter 7. Using Generating Functions to Solve Specialized Differential Equations - p. 257 18. Chapter 8. Separation of Variables in Cartesian Coordinates - p. 275 19. 8.1. Introduction - p. 275 20. 8.2. Solving Laplace's Equation on a Rectangle - p. 275 21. 8.3. Laplace's Equation on a Cube - p. 283 22. 8.4. Solving the Wave Equation in One Dimension by Separation of Variables - p. 287 23. 8.5. Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables - p. 294 24. 8.6. Solving the Heat Equation in One Dimension Using Separation of Variables - p. 297 25. 8.7. Steady State of the Heat Equation - p. 303 26. 8.8. Checking the Validity of the Solution - p. 310 27. Chapter 9. Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables - p. 315 28. Chapter 10. Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables - p. 337 29. 10.1. An Example Where Legendre Equations Arise - p. 337 30. 10.2. The Solution to Bessel's Equation in Spherical Coordinates - p. 340 31. 10.3. Legendre's Equation and Its Solutions - p. 345 32. 10.4. Associated Legendre Functions - p. 349 33. 10.5. Laplace's Equation in Spherical Coordinates - p. 353 34. 10.6. Rigid Rotor - p. 356 35. 10.7. One Dimension Quantum Mechanical Oscillator - p. 360 36. 10.8. The Hydrogen Atom - p. 368 37. Chapter 11. The Fourier Transform - p. 375 38. 11.1. Introduction - p. 375 39. 11.2. The Fourier Transform as a Decomposition - p. 376 40. 11.3. The Fourier Transform From Fourier Series - p. 377 41. 11.4. Some Properties of the Fourier Transform - p. 379 42. 11.5. Solving Partial Differential Equations Using the Fourier Transform - p. 383 43. 11.6. The Spectrum of the Negative Laplacian in One Dimension - p. 391 44. 11.7. The Fourier Transform in Three Dimensions - p. 395 45. Chapter 12. The Laplace Transform - p. 401 46. 12.1. Introduction - p. 401 47. 12.2. Properties of the Laplace Transform - p. 402 48. 12.3. Solving Differential Equations Using the Laplace Transform - p. 406 49. 12.4. Solving the Heat Equation Using the Laplace Transform - p. 411 50. 12.5. The Wave Equation and the Laplace Transform - p. 418 51. Chapter 13. Solving PDEs With Green's Functions - p. 427 52. 13.1. Solving the Heat Equation Using Green's Function - p. 427 53. 13.2. The Method of Images - p. 431 54. 13.3. Green's Function for the Wave Equation - p. 442 55. 13.4. Green's Function and Poisson's Equation - p. 451 56. Appendix 1 - p. 457 57. Appendix 2 - p. 463 58. Appendix 3 - p. 467 59. Bibliography - p. 473 60. Index - p. 475
Mathematical Physics with Partial Differential Equations, Second Edition, is designed for upper division undergraduate and beginning graduate students taking mathematical physics taught out by math departments. The new edition is based on the success of the first, with a continuing focus on clear presentation, detailed examples, mathematical rigor and a careful selection of topics. It presents the familiar classical topics and methods of mathematical physics with more extensive coverage of the three most important partial differential equations in the field of mathematical physics—the heat equation, the wave equation and Laplace’s equation.
The book presents the most common techniques of solving these equations, and their derivations are developed in detail for a deeper understanding of mathematical applications. Unlike many physics-leaning mathematical physics books on the market, this work is heavily rooted in math, making the book more appealing for students wanting to progress in mathematical physics, with particularly deep coverage of Green’s functions, the Fourier transform, and the Laplace transform. A salient characteristic is the focus on fewer topics but at a far more rigorous level of detail than comparable undergraduate-facing textbooks. The depth of some of these topics, such as the Dirac-delta distribution, is not matched elsewhere.
New features in this edition include: novel and illustrative examples from physics including the 1-dimensional quantum mechanical oscillator, the hydrogen atom and the rigid rotor model; chapter-length discussion of relevant functions, including the Hermite polynomials, Legendre polynomials, Laguerre polynomials and Bessel functions; and all-new focus on complex examples only solvable by multiple methods.
1. 3.1. Introduction - p. 153 2. 3.2. Construction of Green's Function Using the Dirac-Delta Function - p. 154 3. 3.3. Green's Function Using Variation of Parameters - p. 162 4. 3.4. Construction of Green's Function From Eigenfunctions - p. 166 5. 3.5. More General Boundary Conditions - p. 169 6. 3.6. The Fredholm Alternative (or, What If 0 Is an Eigenvalue?) - p. 171 7. 3.7. Green's Function for the Laplacian in Higher Dimensions - p. 178 8. Chapter 4. Fourier Series - p. 187 9. 4.1. Introduction - p. 187 10. 4.2. Basic Definitions - p. 188 11. 4.3. Methods of Convergence of Fourier Series - p. 194 12. 4.4. The Exponential Form of Fourier Series - p. 207 13. 4.5. Fourier Sine and Cosine Series - p. 212 14. 4.6. Double Fourier Series - p. 215 15. Chapter 5. Three Important Equations - p. 217 16. Chapter 6. Sturm–Liouville Theory - p. 237 17. Chapter 7. Using Generating Functions to Solve Specialized Differential Equations - p. 257 18. Chapter 8. Separation of Variables in Cartesian Coordinates - p. 275 19. 8.1. Introduction - p. 275 20. 8.2. Solving Laplace's Equation on a Rectangle - p. 275 21. 8.3. Laplace's Equation on a Cube - p. 283 22. 8.4. Solving the Wave Equation in One Dimension by Separation of Variables - p. 287 23. 8.5. Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables - p. 294 24. 8.6. Solving the Heat Equation in One Dimension Using Separation of Variables - p. 297 25. 8.7. Steady State of the Heat Equation - p. 303 26. 8.8. Checking the Validity of the Solution - p. 310 27. Chapter 9. Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables - p. 315 28. Chapter 10. Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables - p. 337 29. 10.1. An Example Where Legendre Equations Arise - p. 337 30. 10.2. The Solution to Bessel's Equation in Spherical Coordinates - p. 340 31. 10.3. Legendre's Equation and Its Solutions - p. 345 32. 10.4. Associated Legendre Functions - p. 349 33. 10.5. Laplace's Equation in Spherical Coordinates - p. 353 34. 10.6. Rigid Rotor - p. 356 35. 10.7. One Dimension Quantum Mechanical Oscillator - p. 360 36. 10.8. The Hydrogen Atom - p. 368 37. Chapter 11. The Fourier Transform - p. 375 38. 11.1. Introduction - p. 375 39. 11.2. The Fourier Transform as a Decomposition - p. 376 40. 11.3. The Fourier Transform From Fourier Series - p. 377 41. 11.4. Some Properties of the Fourier Transform - p. 379 42. 11.5. Solving Partial Differential Equations Using the Fourier Transform - p. 383 43. 11.6. The Spectrum of the Negative Laplacian in One Dimension - p. 391 44. 11.7. The Fourier Transform in Three Dimensions - p. 395 45. Chapter 12. The Laplace Transform - p. 401 46. 12.1. Introduction - p. 401 47. 12.2. Properties of the Laplace Transform - p. 402 48. 12.3. Solving Differential Equations Using the Laplace Transform - p. 406 49. 12.4. Solving the Heat Equation Using the Laplace Transform - p. 411 50. 12.5. The Wave Equation and the Laplace Transform - p. 418 51. Chapter 13. Solving PDEs With Green's Functions - p. 427 52. 13.1. Solving the Heat Equation Using Green's Function - p. 427 53. 13.2. The Method of Images - p. 431 54. 13.3. Green's Function for the Wave Equation - p. 442 55. 13.4. Green's Function and Poisson's Equation - p. 451 56. Appendix 1 - p. 457 57. Appendix 2 - p. 463 58. Appendix 3 - p. 467 59. Bibliography - p. 473 60. Index - p. 475
