10 CONTENTS

30.3 Stoke’s Theorem from Green’s Theorem . . . . . . . . . . . . . . . . . . 59830.3.1 The Normal and the Orientation . . . . . . . . . . . . . . . . . . 60030.3.2 The Mobeus Band . . . . . . . . . . . . . . . . . . . . . . . . . 602

30.4 A General Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 60330.5 Conservative Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 605

30.5.1 Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 60730.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

31 Curvilinear Coordinates 61331.1 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61331.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61631.3 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 61731.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61931.5 Transformation of Coordinates. . . . . . . . . . . . . . . . . . . . . . . . 62131.6 Differentiation and Christoffel Symbols . . . . . . . . . . . . . . . . . . . 62331.7 Gradients and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . 62531.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

32 Measures and Integrals 62932.1 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62932.2 Simple Functions, σ Algebras, Measurability . . . . . . . . . . . . . . . . 63232.3 Measures and Outer Measures . . . . . . . . . . . . . . . . . . . . . . . . 63832.4 Measures from Outer Measures . . . . . . . . . . . . . . . . . . . . . . . 63832.5 Riemann Integrals for Decreasing Functions . . . . . . . . . . . . . . . . 64332.6 Lebesgue Integrals of Nonnegative Functions . . . . . . . . . . . . . . . . 64332.7 Nonnegative Simple Functions . . . . . . . . . . . . . . . . . . . . . . . 64432.8 The Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . 64632.9 The Integral’s Righteous Algebraic Desires . . . . . . . . . . . . . . . . . 64732.10 Integrals of Real Valued Functions . . . . . . . . . . . . . . . . . . . . . 64832.11 Dynkin’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65032.12 Product Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65232.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

33 The Lebesgue Measure and Integral in Rp 65733.1 An Outer Measure on P (R) . . . . . . . . . . . . . . . . . . . . . . . . 65733.2 One Dimensional Lebesgue Measure . . . . . . . . . . . . . . . . . . . . 65833.3 The Lebesgue Integral and Riemann Integral . . . . . . . . . . . . . . . . 65933.4 p Dimensional Lebesgue Measure and Integrals . . . . . . . . . . . . . . 659

33.4.1 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 65933.4.2 p Dimensional Lebesgue Measure and Integrals . . . . . . . . . 660

33.5 Lebesgue Measure and Linear Maps . . . . . . . . . . . . . . . . . . . . 66133.6 Change of Variables for Nonlinear Maps . . . . . . . . . . . . . . . . . . 66333.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666Copyright © 2018, You are welcome to use this, including copying it for use in classes

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