Introduction to Analysis in Several Variables (Advanced Calculus) Michael Taylor Math. Dept., UNC E-mail address: [email protected] 2010 Mathematics Subject Classification. 26B05, 26B10, 26B12, 26B15, 26B20 Key words and phrases. real numbers, complex numbers, Euclidean space, metric spaces, compact spaces, Cauchy sequences, continuous function, power series, derivative, mean value theorem, Riemann integral, fundamental theorem of calculus, arclength, exponential function, logarithm, trigonometric functions, Euler's formula, multiple integrals, surfaces, surface area, differential forms, Stokes theorem, degree, Riemannian manifold, metric tensor, geodesics, curvature, Gauss-Bonnet theorem, Fourier analysis Contents Preface ix Some basic notation xv Chapter 1. Background 1 x1.1. One variable calculus 2 x1.2. Euclidean spaces 19 x1.3. Vector spaces and linear transformations 25 x1.4. Determinants 35 Chapter 2. Multivariable differential calculus 45 x2.1. The derivative 46 x2.2. Inverse function and implicit function theorem 66 x2.3. Systems of differential equations and vector fields 80 Chapter 3. Multivariable integral calculus and calculus on surfaces 101 x3.1. The Riemann integral in n variables 102 x3.2. Surfaces and surface integrals 135 x3.3. Partitions of unity 167 x3.4. Sard's theorem 168 x3.5. Morse functions 169 x3.6. The tangent space to a manifold 171 Chapter 4. Differential forms and the Gauss-Green-Stokes formula 177 x4.1. Differential forms 179 x4.2. Products and exterior derivatives of forms 186 vii viii Contents x4.3. The general Stokes formula 190 x4.4. The classical Gauss, Green, and Stokes formulas 196 x4.5. Differential forms and the change of variable formula 207 Chapter 5. Applications of the Gauss-Green-Stokes formula 213 x5.1. Holomorphic functions and harmonic functions 215 x5.2. Differential forms, homotopy, and the Lie derivative 230 x5.3. Differential forms and degree theory 236 Chapter 6. Differential geometry of surfaces 255 x6.1. Geometry of surfaces I: geodesics 260 x6.2. Geometry of surfaces II: curvature 274 x6.3. Geometry of surfaces III: the Gauss-Bonnet theorem 291 x6.4. Smooth matrix groups 306 x6.5. The derivative of the exponential map 327 x6.6. A spectral mapping theorem 332 Chapter 7. Fourier analysis 335 x7.1. Fourier series 339 x7.2. The Fourier transform 356 x7.3. Poisson summation formulas 380 x7.4. Spherical harmonics 383 x7.5. Fourier series on compact matrix groups 428 x7.6. Isoperimetric inequality 436 Appendix A. Complementary material 439 xA.1. Metric spaces, convergence, and compactness 440 xA.2. Inner product spaces 453 xA.3. Eigenvalues and eigenvectors 459 xA.4. Complements on power series 465 xA.5. The Weierstrass theorem and the Stone-Weierstrass theorem 471 xA.6. Further results on harmonic functions 475 Bibliography 481 Index 485 Preface This text was produced for the second part of a two-part sequence on ad- vanced calculus, whose aim is to provide a firm logical foundation for analy- sis, for students who have had 3 semesters of calculus and a course in linear algebra. The first part treats analysis in one variable, and the text [44] was written to cover that material. The text at hand treats analysis in several variables. These two texts can be used as companions, but they are written so that they can be used independently, if desired. Chapter 1 treats background needed for multivariable analysis. The first section gives a brief treatment of one-variable calculus, including the Rie- mann integral and the fundamental theorem of calculus. This section distills material developed in more detail in the companion text [44]. We have in- cluded it here to facilitate the independent use of this text. Subsequent sections in Chapter 1 present basic linear algebra background of use for the rest of the text. They include material on n-dimensional Euclidean spaces and other vector spaces, on linear transformations on such spaces, and on determinants of such linear transformations. Chapter 2 develops multi-dimensional differential calculus on domains in n-dimensional Euclidean space Rn. The first section defines the derivative of a differentiable map F : O! Rm, at a point x 2 O, for O open in Rn, as a linear map from Rn to Rm, and establishes basic properties, such as the chain rule. The next section deals with the Inverse Function Theorem, giving a condition for such a map to have a differentiable inverse, when n = m. The third section treats n × n systems of differential equations, bringing in the concepts of vector fields and flows on an open set O 2 Rn. While the emphasis here is on differential calculus, we do make use of integral calculus in one variable, as exposed in Chapter 1. ix x Preface Chapter 3 treats multi-dimensional integral calculus. We define the Rie- mann integral for a class of functions on Rn, and establish basic proper- ties, including a change of variable formula. We then study smooth m- dimensional surfaces in Rn, and extend the Riemann integral to a class of functions on such surfaces. Going further, we abstract the notion of surface to that of a manifold, and study a class of manifolds known as Riemannian manifolds. These possess an object known as a \metric tensor." We also define the Riemann integral for a class of functions on such manifolds. The change of variable formula is instrumental in this extension of the integral. In Chapter 4 we introduce a further class of objects that can be de- fined on surfaces, differential forms. A k-form can be integrated over a k-dimensional surface, endowed with an extra structure, an \orientation." Again the change of variable formula plays a role in establishing this. Im- portant operations on differential forms include products and the exterior derivative. A key result of Chapter 4 is a general Stokes formula, an impor- tant integral identity that can be seen as a multi-dimensional version of the fundamental theorem of calculus. In x4.4 we specialize this general Stokes formula to classical cases, known as theorems of Gauss, Green, and Stokes. A concluding section of Chapter 4 makes use of material on differential forms to give another proof of the change of variable formula for the integral, much different from the proof given in Chapter 3. Chapter 5 is devoted to several applications of the material on the Gauss- Green-Stokes theorems from Chapter 4. In x5.1 we use Green's Theorem to derive fundamental properties of holomorphic functions of a complex vari- able. Sprinkled throughout earlier sections are some allusions to functions of complex variables, particularly in some of the exercises in xx2.1{2.2. Read- ers with no previous exposure to complex variables might wish to return to these exercises after getting through x5.1. In this section, we also discuss some results on the closely related study of harmonic functions. One result is Liouville's Theorem, stating that a bounded harmonic function on all of Rn must be constant. When specialized to holomorphic functions on C = R2; this yields a proof of the Fundamental Theorem of Algebra. In x5.2 we define the notion of smoothly homotopic maps and consider the behavior of closed differential forms under pull back by smoothly homo- topic maps. This material is then applied in x5.3, which introduces degree theory and derives some interesting consequences. Key results include the Brouwer fixed point theorem, the existence of critical points of smooth vec- tor fields tangent to an even-dimensional sphere, and the Jordan-Brouwer separation theorem (in the smooth case). We also show how degree theory yields another proof of the fundamental theorem of algebra. Preface xi Chapter 6 applies results of Chapters 2{5 to the study of the geometry of surfaces (and more generally of Riemannian manifolds). Section 6.1 studies geodesics, which are locally length-minimizing curves. Section 6.2 studies curvature. Several varieties of curvature arise, including Gauss curvature and Riemann curvature, and it is of great interest to understand the relations between them. Section 6.3 ties the curvature study of x6.2 to material on degree theory from x5.3, in a result known as the Gauss-Bonnet theorem. Section 6.4 studies smooth matrix groups, which are smooth surfaces in M(n; F) that are also groups. These carry left and right invariant metric tensors, with important consequences for the application of such groups to other aspects of analysis, including results presented in x7.4. Chapter 7 is devoted to an introduction to multi-dimensional Fourier analysis. Section 7.1 treats Fourier series on the n-dimensional torus Tn, and x7.2 treats the Fourier transform for functions on Rn. Section 7.3 intro- duces a topic that ties the first two together, known as Poisson's summation formula. We apply this formula to establish a classical result of Riemann, his functional equation for the Riemann zeta function. The material in xx7.1{7.3 bears on topics rather different from the geo- metrical material emphasized in the latter part of Chapter 3 and in Chapters 4{6. In fact, this part of Chapter 7 could be tackled right after one gets through x3.1. On the other hand, the last three sections of Chapter 7 make strong contact with this geometrical material. Section 7.4 treats Fourier analysis on the sphere Sn−1, which involves expanding a function on Sn−1 in terms of eigenfunctions of the Laplace operator ∆S, arising from the Rie- mannian metric on Sn−1. This study of course includes integrating functions over Sn−1. It also brings in the matrix group SO(n), introduced in Chapter 3, which acts on each eigenspace Vk of ∆S, and its subgroup SO(n − 1), and makes use of integrals over SO(n − 1).