iv Contents 3.7 The Two Dimensional Wave and Heat Equations 155 3.8 Laplace's Equation in Rectangular Coordinates 163 3.9 Poisson's Equation:The Method of Eigenfunction Expansions 170 3.10 Neumann and Robin Conditions 180 3.11 The Maximum Principle 187 4 Partial Differential Equations in Polar and Cylindrical Coordinates 193 4.1 The Laplacian in Various Coordinate Systems 194 4.2 Vibrations of a Circular Membrane:Symmetric Case 198 4.3 Vibrations of a Circular Membrane:General Case 207 4.4 Laplace's Equation in Circular Regions 216 4.5 Laplace's Equation in a Cylinder 228 4.6 The Helmholtz and Poisson Equations 231 Supplement on Bessel Functions 4.7 Bessel's Equation and Bessel Functions 237 4.8 Bessel Series Expansions 248 4.9 Integral Formulas and Asymptotics for Bessel Functions 261 5 Partial Differential Equations in Spherical Coordinates 269 5.1 Preview of Problems and Methods 270 5.2 Dirichlet Problems with Symmetry 274 5.3 Spherical Harmonics and the General Dirichlet Problem 281 5.4 The Helmholtz Equation with Applications to the Poisson,Heat, and Wave Equations 291 Supplement on Legendre Functions 5.5 Legendre's Differential Equation 300 5.6 Legendre Polynomials and Legendre Series Expansions 308 5.7 Associated Legendre Functions and Series Expansions 319 801 6 Sturm-Liouville Theory with Engineering Applications 325 6.1 Orthogonal Functions 326 6.2 Sturm-Liouville Theory 333 6.3 The Hanging Chain 346 6.4 Fourth Order Sturm-Liouville Theory 353 6.5 Elastic Vibrations and Buckling of Beams 360 6.6 The Biharmonic Operator 371 6.7 Vibrations of Circular Plates 377 年hu0020s