Contents Preface vii 1 A Preview of Applications and Techniques 1 1.1 What Is a Partial Differential Equation?2 1.2 Solving and Interpreting a Partial Differential Equation 7 2 Fourier Series 17 2.1 Periodic Functions 18 2.2 Fourier Series 26 2.3 Fourier Series of Functions with Arbitrary Periods 38 2.4 Half-Range Expansions:The Cosine and Sine Series 50 2.5 Mean Square Approximation and Parseval's Identity 53 2.6 Complex Form of Fourier Series 60 2.7 Forced Oscillations 69 Supplement on Convergence 2.8 Proof of the Fourier Series Representation Theorem 77 2.9 Uniform Convergence and Fourier Series 85 2.10 Dirichlet Test and Convergence of Fourier Series 94 3 Partial Differential Equations in Rectangular Coordinates 103 3.1 Partial Differential Equations in Physics and Engineering 104 3.2 Modeling:Vibrating Strings and the Wave Equation 109 3.3 Solution of the One Dimensional Wave Equation: The Method of Separation of Variables 114 3.4 D'Alembert's Method 126 3.5 The One Dimensional Heat Equation 135 3.6 Heat Conduction in Bars:Varying the Boundary Conditions 146