Shimamoto d multivariable calculus 2019

MULTIVARIABLE CALCULUS Don Shimamoto Multivariable Calculus Don Shimamoto Swarthmore College ii c 2019 Don Shimamoto ISBN: 978-1-7082-4699-0 This work is licensed under the Creative Commons Attribution 4.0 International License To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ Unless noted otherwise, the graphics in this work were created using the software packages: Cinderella, Mathematica, and, in a couple of instances, Adobe Illustrator and Canvas Contents Preface I Preliminaries Rn 1.1 1.2 1.3 1.4 1.5 1.6 1.7 II vii Vector arithmetic Linear transformations The matrix of a linear transformation Matrix multiplication The geometry of the dot product Determinants Exercises for Chapter Vector-valued functions of one variable 23 Paths and curves 2.1 Parametrizations 2.2 Velocity, acceleration, speed, arclength 2.3 Integrals with respect to arclength 2.4 The geometry of curves: tangent and normal vectors 2.5 The cross product 2.6 The geometry of space curves: Frenet vectors 2.7 Curvature and torsion 2.8 The Frenet-Serret formulas 2.9 The classification of space curves 2.10 Exercises for Chapter III Real-valued functions Real-valued functions: preliminaries 3.1 Graphs and level sets 3.2 More surfaces in R3 3.3 The equation of a plane in R3 3.4 Open sets 3.5 Continuity 3 10 11 15 18 25 25 28 30 31 33 38 40 42 42 45 53 iii 55 55 58 60 62 66 iv CONTENTS 3.6 3.7 3.8 3.9 Some properties of continuous functions The Cauchy-Schwarz and triangle inequalities Limits Exercises for Chapter Real-valued functions: differentiation 4.1 The first-order approximation 4.2 Conditions for differentiability 4.3 The mean value theorem 4.4 The C test 4.5 The Little Chain Rule 4.6 Directional derivatives 4.7 ∇f as normal vector 4.8 Higher-order partial derivatives 4.9 Smooth functions 4.10 Max/min: critical points 4.11 Classifying nondegenerate critical points 4.11.1 The second-order approximation 4.11.2 Sums and differences of squares 4.12 Max/min: Lagrange multipliers 4.13 Exercises for Chapter Real-valued functions: integration 5.1 Volume and iterated integrals 5.2 The double integral 5.3 Interpretations of the double integral 5.4 Parametrizations of surfaces 5.4.1 Polar coordinates (r, θ) in R2 5.4.2 Cylindrical coordinates (r, θ, z) in R3 5.4.3 Spherical coordinates (ρ, φ, θ) in R3 5.5 Integrals with respect to surface area 5.6 Triple integrals and beyond 5.7 Exercises for Chapter IV 69 71 72 73 83 83 88 89 90 92 92 94 96 98 98 102 102 104 105 108 121 121 126 131 134 136 137 138 140 144 147 Vector-valued functions Differentiability and the chain rule 6.1 Continuity revisited 6.2 Differentiability revisited 6.3 The chain rule: a conceptual approach 6.4 The chain rule: a computational approach 6.5 Exercises for Chapter 155 157 157 159 161 162 166 Change of variables 173 7.1 Change of variables for double integrals 174 7.2 A word about substitution 177 7.3 Examples: linear changes of variables, symmetry 178 CONTENTS 7.4 7.5 V v Change of variables for n-fold integrals 182 Exercises for Chapter 189 Integrals of vector fields 195 Vector fields 197 8.1 Examples of vector fields 197 8.2 Exercises for Chapter 200 Line integrals 9.1 Definitions and examples 9.2 Another word about substitution 9.3 Conservative fields 9.4 Green’s theorem 9.5 The vector field W 9.6 The converse of the mixed partials theorem 9.7 Exercises for Chapter 203 203 210 211 215 219 222 226 10 Surface integrals 10.1 What the surface integral measures 10.2 The definition of the surface integral 10.3 Stokes’s theorem 10.4 Curl fields 10.5 Gauss’s theorem 10.6 The inverse square field 10.7 A more substitution-friendly notation for surface integrals 10.8 Independence of parametrization 10.9 Exercises for Chapter 10 235 235 240 245 250 253 257 259 261 264 11 Working with differential forms 11.1 Integrals of differential forms 11.2 Derivatives of differential forms 11.3 A look back at the theorems of multivariable calculus 11.4 Exercises for Chapter 11 273 274 275 279 280 Answers to selected exercises 285 Index 307 vi CONTENTS ... second vector, not a scalar multiple of v, and c and d are scalars, the parallelogram law of addition suggests that the combination cv + dw should lie in a “plane” determined by v and w and that,... the determinant (c) det(At ) = det A (d) det(AB) = (det A)(det B) Proof For by determinants, the proofs are easy (a) det x x = det [ xx11 xx22 ] = x1 x2 − x1 x2 = (b) det y x = det [ xy11 xy22 ]... 4.1–4.6, find the indicated matrix products 4.1 AB and BA, where A = −1 and B = 1 −1 3  4.3 AB and BA, where A = 1  4.4 AB and BA, where A = 0 4.2 AB and BA, where A = and B =   0 0 and B =