Shanghai Lectures on Multivariable Analysis William G. Faris October 18, 2016 ii Contents 1 Differentiation 1 1.1 Fixed point iteration (single variable) . 2 1.2 The implicit function theorem (single variable) . 5 1.3 Linear algebra review (norms) . 7 1.4 Linear algebra review (eigenvalues) . 12 1.5 Differentiation (multivariable) . 16 1.6 Fixed point iteration (multivariable) . 26 1.7 The implicit function theorem (multivariable) . 28 1.8 Second order partial derivatives . 32 1.9 Problems . 34 2 Integration 43 2.1 The Riemann integral . 44 2.2 Jordan content . 48 2.3 Approximation of Riemann integrals . 48 2.4 Fubini's theorem . 50 2.5 Uniform convergence . 54 2.6 Dominated convergence . 55 2.7 Differentiating a parameterized integral . 58 2.8 Approximate delta functions . 60 2.9 Linear algebra (determinants) . 61 2.10 Change of variables . 64 2.11 Problems . 65 3 Differential Forms 71 3.1 Coordinates . 72 3.2 Scalar fields . 73 3.3 Vector fields . 74 3.4 Fluid velocity and the advective derivative . 78 3.5 Differential 1-forms . 79 3.6 Polar coordinates . 82 3.7 Integrating factors and canonical forms . 84 3.8 The second differential . 86 3.9 Regular surfaces . 88 iii iv CONTENTS 3.10 Lagrange multipliers . 91 3.11 Differential k-forms . 92 3.12 The exterior derivative . 94 3.13 The Poincar´elemma . 98 3.14 Substitution and pullback . 100 3.15 Pullback of a differential form . 102 3.16 Pushforward of a vector field . 104 3.17 Orientation . 106 3.18 Integration of top-dimension differential forms . 108 3.19 Integration of forms over singular surfaces . 109 3.20 Stokes' theorem for chains of singular surfaces . 110 3.21 Classical versions of Stokes' theorem . 113 3.22 Picturing Stokes' theorem . 115 3.23 Electric and magnetic fields . 119 4 The Metric Tensor 127 4.1 The interior product . 128 4.2 Volume . 129 4.3 The divergence theorem . 130 4.4 Conservation laws . 132 4.5 The metric . 135 4.6 Twisted forms . 138 4.7 The gradient and divergence and the Laplace operator . 139 4.8 Orthogonal coordinates and normalized bases . 141 4.9 Linear algebra (the Levi-Civita permutation symbol) . 144 4.10 Linear algebra (volume and area) . 145 4.11 Surface area . 148 5 Measure Zero 159 5.1 Outer content and outer measure . 160 5.2 The set of discontinuity of a function . 162 5.3 Lebesgue's theorem on Riemann integrability . 163 5.4 Almost everywhere . 165 5.5 Mapping sets of measure zero . 166 5.6 Sard's theorem . 167 5.7 Change of variables . 169 5.8 Fiber integration . 169 5.9 Probability . 171 5.10 The co-area formula . 172 5.11 Linear algebra (block matrices) . 175 Mathematical Notation 179 Preface The book This book originated in lectures given in Fall 2014 at NYU Shanghai for an advanced undergraduate course in multivariable analysis. There are chapters on Differentiation, Integration, Differential Forms, The Metric Tensor, together with an optional chapter on Measure Zero. The topics are standard, but the attempt is to present ideas that are often overlooked in this context. The fol- lowing chapter by chapter summary sketches the approach. The explanations in the summary are far from complete; they are only intended to highlight points that are explained in the body of the book. Differentiation This main themes of this chapter are standard. • It begins with fixed point iteration, the basis of the subsequent proofs. • The central object in this chapter is a smooth (that is, sufficiently differen- tiable) numerical function f defined on an open subset U of Rk with values in Rn. Here k is the domain dimension and n is the target dimension. The function is called \numerical" to emphasize that both inputs and outputs involve numbers. (See below for other kinds of functions.) Sometimes a numerical function is denoted by an expression like y 7! f(y). The choice of the variable name y is arbitrary. • When k < n a numerical function f from an open subset U of Rk to Rn can give a explicit (parametric) representation of a k-dimensional surface in Rn. • When k < n a numerical function g from an open subset W of Rn to Rn−k can given an implicit representation of a family of k-dimensional surfaces in W . • The implicit function theorem gives conditions for when an implicit rep- resentation gives rise to an explicit representation. v vi CONTENTS • When k = n the function f defines a transformation from an open subset U of Rn to an open set V in Rn. • The inverse function theorem gives conditions that ensure that the trans- formation has an inverse transformation. • In the passive interpretation the smooth transformation f : U ! V has a smooth inverse: the points in U and the points in V give alternative numerical descriptions of the same situation. • In the active interpretation the smooth transformation f : U ! U de- scribes a change of state: the state described by y in U is mapped into the new state described by f(y) in U. The transformation may be iterated. Integration This chapter is about the Riemann integral for functions of several variables. There are several interesting results. • The Fubini theorem for Riemann integrals deals with iterated integrals. • The dominated convergence theorem for Riemann integrals is a result about pointwise convergence. The setting is a sequence of Riemann in- tegrable functions defined on a fixed bounded set with a common bound on their values. The sequence of functions is assumed to converge (in some sense) to another Riemann integrable function. It is elementary to prove that if the functions converge uniformly, then the integrals converge. The dominated convergence theorem says that if the functions converge pointwise, then the integrals converge. The direct proof of the dominated convergence involves a somewhat complicated construction, but the result is spectacularly simple and useful. • There is a treatment of approximate delta functions. • The change of variables formula has an elegant proof via approximate delta functions and the dominated convergence theorem. Differential Forms This chapter and the next are the heart of the book. The central idea is ge- ometrical: differential forms are intrinsic expressions of change, and thus the basic mechanism of calculation with differential forms is equally simple for ev- ery possible choice of coordinate system. (Of course for modeling a particular system one coordinate system may be more convenient than another.) The fun- damental result is Stokes' theorem, which is the natural generalization of the fundamental theorem of calculus. Both the fundamental theorem in one dimen- sion and Stokes' theorem in higher dimensions make no reference to notions of length and area; they simply describe the cumulative effect of small changes. CONTENTS vii Because of this intrinsic nature, the expression of Stoke's theorem is the same in every coordinate system. • Example: Here is a simple example from physics that illustrates how nat- ural it is to have a free choice of coordinate system. Consider an idea gas with pressure P , volume V , and temperature T . The number of gas par- ticles N is assumed constant. Each of the quantities P; V; T is a function of the state of the gas; these functions are related by the ideal gas law PV = NkT: (The constant k transforms temperature units to energy units.) The dif- ferential form of this relation is P dV + V dP = Nk dT: This equation is a precise description of the how these variables change for a small change in the state of the gas. A typical use of the fundamental theorem of calculus is the calculation of work done by the system during a change of state where the temperature is constant. This is obtained by integrating −P dV along states of constant temperature T = T0. Suppose that in this process the volume changes from V0 to V1 and the pressure changes from P0 to P1. Then −P dV = −NkT0 dV=V = −NkT0 d log(V ) has integral NkT0 log(V0=V1). But nothing depends on using volume as an independent variable. On the curve where T = T0 there is a relation P dV + V dP = 0. So on this curve the form is also equal to V dP = NkT0 dP=P = NkT0 log(P ) with integral NkT0 log(P1=P0). Since P0V0 = P1V1 = NkT0, this is the same result. • While many applications of differential forms have nothing to do with length or area, others make essential use of these ideas. For instance, the length of a curve in the plane is obtained by integrating p(dx=dt)2 + (dy=dt)2 dt = p(dx=du)2 + (dy=du)2 du: The motion along the curve may be described either by the t coordinate or the u coordinate, but the length of the curve between two points is a num- ber that does not depend on this choice. In practice such integrals involv- ing square roots of sums of squares can be awkward. Sometimes it helps to use a different coordinate system in the plane. For instance, in polar coordinates the same form has the expression p(dr=dt)2 + r2(dθ=dt)2 dt. The theory of differential forms gives a systematic way of dealing with all such coordinate changes. viii CONTENTS • The chapter introduces a fundamental organizing principle of mathemat- ical modeling. It applies in geometry and in most applications of mathe- matics, and it deserves a serious discussion. In the approach of this book it is based on the elementary concept of an n-dimensional manifold patch. This term is used here for a differentiable manifold M modeled on some open subset of Rn.