Mathematical Physics with Partial Differential Equations

Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field – the heat equation, the wave equation, and Laplace’s equation. The most common techniques of solving such equations are developed in this book, including Green’s functions , the Fourier transform , and the Laplace transform , which all have applications in mathematics and physics far beyond solving the above equations. The book’s focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book’s rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics. Examines in depth both the equations and their methods of solution Presents physical concepts in a mathematical framework Contains detailed mathematical derivations and solutions— reinforcing the material through repetition of both the equations and the techniques Includes several examples solved by multiple methods—highlighting the strengths and weaknesses of various techniques and providing additional practice Front Cover......Page 1 Title Page ......Page 5 Copyright Page......Page 6 Contents......Page 7 Preface......Page 13 1-1 SELF-ADJOINT OPERATORS......Page 15 Fourier Coefficients......Page 19 Exercises......Page 25 1-2 CURVILINEAR COORDINATES......Page 28 Scaling Factors......Page 31 Volume Integrals......Page 32 The Gradient......Page 36 The Laplacian......Page 37 Other Curvilinear Systems......Page 39 Applications......Page 45 An Alternate Approach (Optional)......Page 46 Exercises......Page 47 1-3 APPROXIMATE IDENTITIES AND THE DIRAC-δ FUNCTION......Page 48 Approximate Identities......Page 49 The Dirac-δ Function in Physics......Page 51 Some Calculus for the Dirac-δ Function......Page 54 The Dirac-δ Function in Curvilinear Coordinates......Page 56 Exercises......Page 58 Series of Real Numbers......Page 59 Convergence versus Absolute Convergence......Page 61 Series of Functions......Page 62 Power Series......Page 68 Taylor Series......Page 70 Exercises......Page 74 1-5 SOME IMPORTANT INTEGRATION FORMULAS......Page 78 Other Facts We Will Use Later......Page 82 Another Important Integral......Page 83 Exercises......Page 84 2-1 Vector Integration......Page 87 Path Integrals......Page 88 Line Integrals......Page 91 Surfaces......Page 94 Parameterized Surfaces......Page 96 Integrals of Scalar Functions Over Surfaces......Page 97 Surface Integrals of Vector Functions......Page 99 Exercises......Page 105 Line Integrals......Page 106 2-2 Divergence and Curl......Page 107 Cartesian Coordinate Case......Page 108 Cylindrical Coordinate Case......Page 111 Spherical Coordinate Case......Page 114 The Curl in Cartesian Coordinates......Page 118 The Curl in Cylindrical Coordinates......Page 123 The Curl in Spherical Coordinates......Page 128 2-3 Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem......Page 136 The Divergence (Gauss’) Theorem......Page 141 Stokes’ Theorem......Page 149 An Application of Stokes’ Theorem......Page 154 An Application of the Divergence Theorem......Page 155 Conservative Fields......Page 156 Green’s Theorem Problems......Page 162 Stokes’ Theorem Problems......Page 163 Divergence Theorem Problems......Page 166 Conservative Field Problems......Page 167 Introduction......Page 169 3-1 Construction of Green’s Function Using the Dirac-δ Function......Page 170 3-2 Construction of Green’s Function Using Variation of Parameters......Page 178 3-3 Construction of Green’s Function from Eigenfunctions......Page 182 3-4 More General Boundary Conditions......Page 185 3-5 The Fredholm Alternative (or, What If 0 is an Eigenvalue?)......Page 187 3-6 Green’s function for the Laplacian in higher dimensions......Page 194 Exercises......Page 200 Introduction......Page 201 4-1 Basic Definitions......Page 202 Exercises......Page 205 4-2 Methods of Convergence of Fourier Series......Page 207 Fourier Series on Arbitrary Intervals......Page 213 Exercises......Page 218 4-3 The Exponential Form of Fourier Series......Page 220 Exercises......Page 221 4-4 Fourier Sine and Cosine Series......Page 222 4-5 Double Fourier Series......Page 224 Exercise......Page 226 Introduction......Page 227 5-1 Laplace’s Equation......Page 229 5-2 Derivation of the Heat Equation in One Dimension......Page 230 5-3 Derivation of the Wave equation in One Dimension......Page 232 5-4 An Explicit Solution of the Wave Equation......Page 236 Exercises......Page 241 5-5 Converting Second-order PDEs to Standard Form......Page 242 Exercise......Page 246 Introduction......Page 247 6-1 The Self-Adjoint Property of a Sturm-Liouville Equation......Page 248 Exercises......Page 250 6-2 Completeness of Eigenfunctions for Sturm-Liouville Equations......Page 251 6-3 Uniform Convergence of Fourier Series......Page 259 7-1 Solving Laplace’s Equation on a Rectangle......Page 265 Exercises......Page 270 7-2 Laplace’s Equation on a Cube......Page 272 Exercises......Page 275 7-3 Solving the Wave Equation in One Dimension by Separation of Variables......Page 276 Exercises......Page 281 7-4 Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables......Page 283 7-5 Solving the heat equation in one dimension using separation of variables......Page 285 The Initial Condition Is the Dirac-δ Function......Page 288 Exercises......Page 290 7-6 Steady State of the Heat equation......Page 291 Exercises......Page 295 7-7 Checking the Validity of the Solution......Page 297 An Example Where Bessel Functions Arise......Page 301 8-1 The Solution to Bessel’s Equation in Cylindrical Coordinates......Page 306 Exercises......Page 308 8-2 Solving Laplace’s Equation in Cylindrical Coordinates Using Separation of Variables......Page 309 8-3 The Wave Equation on a Disk (Drum Head Problem)......Page 313 8-4 The Heat Equation on a Disk......Page 317 9-1 An Example Where Legendre Equations Arise......Page 321 9-2 The Solution to Bessel’s Equation in Spherical Coordinates......Page 324 9-3 Legendre’s Equation and Its Solutions......Page 329 Exercises......Page 332 9-4 Associated Legendre Functions......Page 333 9-5 Laplace’s Equation in Spherical Coordinates......Page 336 Exercise......Page 339 Introduction......Page 341 10-1 The Fourier Transform as a Decomposition......Page 342 10-2 The Fourier Transform from the Fourier Series......Page 343 10-3 Some Properties of the Fourier Transform......Page 345 Exercises......Page 348 10-4 Solving Partial Differential Equations Using the Fourier Transform......Page 349 Exercises......Page 355 10-5 The Spectrum of the Negative Laplacian in One Dimension......Page 357 10-6 The Fourier Transform in Three Dimensions......Page 360 Exercise......Page 364 Introduction......Page 365 11-1 Properties of the Laplace Transform......Page 366 11-2 Solving Differential Equations Using the Laplace Transform......Page 370 Exercises......Page 374 11-3 Solving the Heat Equation using the Laplace Transform......Page 375 Exercises......Page 380 11-4 The Wave Equation and the Laplace Transform......Page 382 Exercises......Page 387 12-1 Solving the Heat Equation Using Green’s Function......Page 389 Green’s Function for the Nonhomogeneous Heat Equation......Page 391 Method of Images for a Semi-infinite Interval......Page 393 Method of Images for a Bounded Interval......Page 397 Exercises......Page 403 12-3 Green’s Function for the Wave Equation......Page 404 Exercises......Page 411 12-4 Green’s Function and Poisson’s Equation......Page 412 Exercises......Page 415 CYLINDRICAL COORDINATES......Page 417 THE LAPLACIAN IN SPHERICAL COORDINATES......Page 422 References......Page 427 Index......Page 429 Machine generated contents note: Chapter 1 Prelimininaries- Introduction Chapter 2 Vector Calculus Chapter 3 Green's Functions Chapter 4 Fourier Series Chapter 5 Three Important Equations Chapter 6 Sturm-Liouville Theory Chapter 7 Bessel Equations and Bessel Functions Chapter 8 Legendre Equations and Legendre Polynomials Chapter 9 The Fourier Transform Chapter 10 The Laplace Transform Chapter 11 The Heat Equation Chapter 12 The Wave Equation Chapter 13 Laplace's Equation. Suitable for advanced undergraduate and beginning graduate students taking a course on mathematical physics, this title presents some of the most important topics and methods of mathematical physics. It contains mathematical derivations and solutions - reinforcing the material through repetition of both the equations and the techniques.