# Introduction to Analysis in Several Variables (Advanced Calculus)

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Introduction to Analysis in Several Variables
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

§1.1. One variable calculus

2

§1.2. Euclidean spaces

19

§1.3. Vector spaces and linear transformations

25

§1.4. Determinants

35

Chapter 2. Multivariable diﬀerential calculus

45

§2.1. The derivative

46

§2.2. Inverse function and implicit function theorem

66

§2.3. Systems of diﬀerential equations and vector ﬁelds

80

Chapter 3. Multivariable integral calculus and calculus on surfaces 101

§3.1. The Riemann integral in n variables

102

§3.2. Surfaces and surface integrals

135

§3.3. Partitions of unity

167

§3.4. Sard’s theorem

168

§3.5. Morse functions

169

§3.6. The tangent space to a manifold

171

Chapter 4. Diﬀerential forms and the Gauss-Green-Stokes formula 177

§4.1. Diﬀerential forms

179

§4.2. Products and exterior derivatives of forms

186

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viii

Contents

§4.3. The general Stokes formula

190

§4.4. The classical Gauss, Green, and Stokes formulas

196

§4.5. Diﬀerential forms and the change of variable formula

207

Chapter 5. Applications of the Gauss-Green-Stokes formula

213

§5.1. Holomorphic functions and harmonic functions

215

§5.2. Diﬀerential forms, homotopy, and the Lie derivative

230

§5.3. Diﬀerential forms and degree theory

236

Chapter 6. Diﬀerential geometry of surfaces

255

§6.1. Geometry of surfaces I: geodesics

260

§6.2. Geometry of surfaces II: curvature

274

§6.3. Geometry of surfaces III: the Gauss-Bonnet theorem

291

§6.4. Smooth matrix groups

306

§6.5. The derivative of the exponential map

327

§6.6. A spectral mapping theorem

332

Chapter 7. Fourier analysis

335

§7.1. Fourier series

339

§7.2. The Fourier transform

356

§7.3. Poisson summation formulas

380

§7.4. Spherical harmonics

383

§7.5. Fourier series on compact matrix groups

428

§7.6. Isoperimetric inequality

436

Appendix A. Complementary material

439

§A.1. Metric spaces, convergence, and compactness

440

§A.2. Inner product spaces

453

§A.3. Eigenvalues and eigenvectors

459

§A.4. Complements on power series

465

§A.5. The Weierstrass theorem and the Stone-Weierstrass theorem 471

§A.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 advanced calculus, whose aim is to provide a ﬁrm logical foundation for analysis, 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  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 Riemann integral and the fundamental theorem of calculus. This section distills material developed in more detail in the companion text . We have included 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 ∈ 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 ∈ 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 Riemann integral for a class of functions on Rn, and establish basic properties, including a change of variable formula. We then study smooth mdimensional 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 deﬁned 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. Important operations on diﬀerential forms include products and the exterior derivative. A key result of Chapter 4 is a general Stokes formula, an important integral identity that can be seen as a multi-dimensional version of the fundamental theorem of calculus. In §4.4 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 GaussGreen-Stokes theorems from Chapter 4. In §5.1 we use Green’s Theorem to derive fundamental properties of holomorphic functions of a complex variable. Sprinkled throughout earlier sections are some allusions to functions of complex variables, particularly in some of the exercises in §§2.1–2.2. Readers with no previous exposure to complex variables might wish to return to these exercises after getting through §5.1. 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 §5.2 we deﬁne the notion of smoothly homotopic maps and consider the behavior of closed diﬀerential forms under pull back by smoothly homotopic maps. This material is then applied in §5.3, which introduces degree theory and derives some interesting consequences. Key results include the Brouwer ﬁxed point theorem, the existence of critical points of smooth vector 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 §6.2 to material on degree theory from §5.3, 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 §7.4.
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 §7.2 treats the Fourier transform for functions on Rn. Section 7.3 introduces 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 §§7.1–7.3 bears on topics rather diﬀerent from the geometrical 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 §3.1. 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 Riemannian 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). Section 7.4 also makes use of the Gauss-Green-Stokes formula, and applications to harmonic functions, from §§4.4 and 5.1. We believe the reader will gain a good appreciation of the utility of unifying geometrical concepts with those aspects of Fourier analysis developed in the ﬁrst part of Chapter 7.
We complement §7.4 with a brief discussion of Fourier series on compact matrix groups, in §7.5.
Section 7.6 deals with the purely geometric problem of showing that, among smoothly bounded planar domains Ω ⊂ R2, with ﬁxed area, the disks have the smallest perimeter. This is the 2D isoperimetric inequality. Its placement here is due to the fact that its proof is an application of Fourier series.
The text ends with a collection of appendices, some giving further background material, others providing complements to results of the main text.

xii

Preface

Appendix A.1 covers some basic notions of metric spaces and compactness, used from time to time throughout the text, such as in the study of the Riemann integral and in the proof of the fundamental existence theorem for ODE. As is the case with §1.1, this appendix distills material developed at a more leisurely pace in , again serving to make this text independent of the ﬁrst one.
Appendices A.2 and A.3 complement results on linear algebra presented in Chapter 1 with some further results. Appendix A.2 treats a general class of inner product spaces, both ﬁnite and inﬁnite dimensional. Treatments in the latter case are relevant to results on Fourier analysis in Chapter 7. Appendix A.3 treats eigenvalues and eigenvectors of linear transformations on ﬁnite dimensional vector spaces, providing results useful in various places, from §2.1 to §6.6.
Appendix A.4 discusses the remainder term in the power series of a function. Appendix A.5 deals with the Weierstrass theorem, on approximating a continuous function by polynomials, and an extension, known as the StoneWeierstrass theorem, a useful tool in analysis, with applications in Sections 5.3, 7.1, and 7.4. Appendix A.6 builds on material on harmonic functions presented in Chapters 5 and 7. Results range from a removable singularity theorem to extensions of Liouville’s theorem.
We point out some distinctive features of this treatment of Advanced Calculus.
1) Applications of the Gauss-Green-Stokes formulas. These formulas form a high point in any advanced calculus course, but we do not want them to be seen as the culmination of the course. Their signiﬁcance arises from their many applications. The ﬁrst application we treat is to the theory of functions of a complex variable, including the Cauchy integral theorem and basic consequences. This basically constitutes a mini-course in complex analysis. (A much more extensive treatment can be found in .) We also derive applications to the study of harmonic functions, in n variables, a study that is closely related to complex analysis when n = 2.
We also apply diﬀerential forms and the Stokes formula to results of a topological ﬂavor, involving a set of tools known as degree theory. We start with a result known as the Brouwer ﬁxed point theorem. We give a short proof, as a direct application of the Stokes formula, thus making this theorem a precursor to degree theory, rather than an application.
2) The unity of analysis and geometry. This starts with calculus on surfaces,

Preface

xiii

computing surface areas and surface integrals, given in terms of the metric tensors these surfaces inherit, but it proceeds much further. There is the question of ﬁnding geodesics, shortest paths, described by certain diﬀerential equations, whose coeﬃcients arise from the metric tensor. Another issue is what makes a curved surface curved. One particular measure is called the Gauss curvature. There are formulas for the integrated Gauss curvature, which in turn make contact with degree theory. Such matters are examples of connections uniting analysis and geometry, pursued in the text.
Other connections arise in the treatment of Fourier analysis. In addition to Fourier analysis on Euclidean space, the text treats Fourier analysis on spheres. Matrix groups, such as rotation groups SO(n), make an appearance, both as tools to study Fourier analysis on spheres, and as further sources of problems in Fourier analysis, thereby expanding the theater on which to bring to bear techniques of advanced calculus developed here.
We follow this preface with a list of some basic notation, of use throughout the text.
Acknowledgment During the preparation of this book, my research has been supported by a number of NSF grants, most recently DMS-1500817.
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