Multivariate Calculus

Introduction 7 Optimisation 9 The Second Derivative Test 9 Constrained Optimisation 17 Lagrangian Multipliers 17 Fields and Forms 25 Definitions Galore 25 Integrating 1-forms (vector fields) over curves. 32 Independence of Parametrisation 36 Conservative Fields/Exact Forms 39 Closed Loops and Conservatism 42 Green's Theorem 49 Motivation 49 Functions as transformations 49 Change of Variables in Integration 52 Spin Fields 54 Green's Theorem (Classical Version) 57 Spin fields and Differential 2-forms 60 The Exterior Derivative 65 For the Pure Mathematicians. 72 Return to the (relatively) mundane. 74 More on Differential Stretching 75 Green's Theorem Again 89 Stokes' Theorem (Classical and Modern) 99 Classical 99 Modern 105 Divergence 118 Fourier Theory 125 Various Kinds of Spaces 125 Function Spaces 130 Applications 134 Fiddly Things 137 Odd and Even Functions 144 Fourier Series 145 Differentiation and Integration of Fourier Series 152 Functions of several variables 153 Partial Differential Equations 157 Introduction 157 The Diffusion Equation 161 Intuitive 161 Saying it in Algebra 164 Laplace's Equation 167 The Wave Equation 171 Schrödinger's Equation 175 The Dirichlet Problem for Laplace's Equation 176 Laplace on Disks 183 Solving the Heat Equation 187 Solving the Wave Equation 193 And in Conclusion.. 196