Multivariate Calculus - M. Adler (2002) WW

Multivariate Calculus - M. Adler (2002) WW - 2C2...

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Unformatted text preview: 2C2 Multivariate Calculus Michael D. Alder November 13, 2002 2 Contents 1 Introduction 5 2 Optimisation 7 2.1 The Second Derivative Test . . . . . . . . . . . . . . . . . . . 7 3 Constrained Optimisation 15 3.1 Lagrangian Multipliers . . . . . . . . . . . . . . . . . . . . . . 15 4 Fields and Forms 23 4.1 Definitions Galore . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Integrating 1-forms (vector fields) over curves. . . . . . . . . . 30 4.3 Independence of Parametrisation . . . . . . . . . . . . . . . . 34 4.4 Conservative Fields/Exact Forms . . . . . . . . . . . . . . . . 37 4.5 Closed Loops and Conservatism . . . . . . . . . . . . . . . . . 40 5 Greens Theorem 47 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1.1 Functions as transformations . . . . . . . . . . . . . . . 47 5.1.2 Change of Variables in Integration . . . . . . . . . . . 50 5.1.3 Spin Fields . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2 Greens Theorem (Classical Version) . . . . . . . . . . . . . . 55 3 4 CONTENTS 5.3 Spin fields and Differential 2-forms . . . . . . . . . . . . . . . 58 5.3.1 The Exterior Derivative . . . . . . . . . . . . . . . . . 63 5.3.2 For the Pure Mathematicians. . . . . . . . . . . . . . . 70 5.3.3 Return to the (relatively) mundane. . . . . . . . . . . . 72 5.4 More on Differential Stretching . . . . . . . . . . . . . . . . . 73 5.5 Greens Theorem Again . . . . . . . . . . . . . . . . . . . . . 87 6 Stokes Theorem (Classical and Modern) 97 6.1 Classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Modern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7 Fourier Theory 123 7.1 Various Kinds of Spaces . . . . . . . . . . . . . . . . . . . . . 123 7.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4 Fiddly Things . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.5 Odd and Even Functions . . . . . . . . . . . . . . . . . . . . 142 7.6 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.7 Differentiation and Integration of Fourier Series . . . . . . . . 150 7.8 Functions of several variables . . . . . . . . . . . . . . . . . . 151 8 Partial Differential Equations 155 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.2 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . 159 8.2.1 Intuitive . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2.2 Saying it in Algebra . . . . . . . . . . . . . . . . . . . 162 8.3 Laplaces Equation . . . . . . . . . . . . . . . . . . . . . . . . 165 CONTENTS 5 8.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 169 8.5 Schr odingers Equation . . . . . . . . . . . . . . . . . . . . . . 173 8.6 The Dirichlet Problem for Laplaces Equation . . . . . . . . . 174The Dirichlet Problem for Laplaces Equation ....
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Multivariate Calculus - M. Adler (2002) WW - 2C2...

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