Michael e taylor introduction to analysis in several variables advanced calculus (pure and applied undergraduate texts)

I Real functions - FUnctions of several variables - Integral formulas Stokes, Gauss, Green, etc... Further results on harmonic functions Trang 9 This text was produced for the second p

to Analysis

in Several

Variables

Advanced Calculus

2010 Mathematics Subject Classification Primary 26B05, 26BlO, 26B12, 26B15, 26B20

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Library of Congress Cataloging-in-Publication Data

Names: Taylor, Michael E., 1946- author

Title: Introduction to analysis in several variables : advanced calculus / Michael E Taylor Description: Providence, Rhode Island American Mathematical Society, [2020] I Series: Pure

and applied undergraduate texts, 1943-9334 ; volume 46 I Includes bibliographical references and index

Identifiers: LCCN 20200097351 ISBN 9781470456696 (paperback) I ISBN 9781470460167 (ebook) Subjects: LCSH: Calculus I FUnctions of several real variables I Functions of several complex variables I AMS: Real functions - FUnctions of several variables - Continuity and differentia­ tion questions I Real functions - FUnctions of several variables - Implicit function theorems, Jacobians, transformations with several variables I Real functions - FUnctions of several variables - Calculus of vector functions I Real functions - FUnctions of several variables - Integration: length, area, volume I Real functions - FUnctions of several variables - Integral formulas (Stokes, Gauss, Green, etc.)

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10 9 8 7 6 5 4 3 2 1 25 24 23 22 21 20

Preface

Some basic notation

Cha pter 1 Background

2.2 Inverse function and implicit function theorems

2.3 Systems of differential equations and vector fields

Chapter 3 Multivariable integral calculus and calculus on surfaces

3.2 Surfaces and surface integrals

3.3 Partitions of unity

3.4 Sard's theorem

3.5 Morse functions

3.6 The tangent space to a manifold

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

4.1 Differential forms

4.2 Products and exterior derivatives of forms

4.3 The general Stokes formula

v

4.4 The classical Gauss, Green, and Stokes formulas

4.5 Differential forms and the change of variable formula

Chapter 5 Applications of the Gauss-Green-Stokes formula

5.1 Holomorphic functions and harmonic functions

5.2 Differential forms, homotopy, and the Lie derivative

5.3 Differential forms and degree theory

Chapter 6 Differential geometry of surfaces

6.1 Geometry of surfaces I: geodesics

6.2 Geometry of surfaces II: curvature

6.3 Geometry of surfaces III: the Gauss-Bonnet theorem

6.4 Smooth matrix groups

6.5 The derivative of the exponential map

6.6 A spectral mapping theorem

Chapter 7 Fourier analysis

7.1 Fourier series

7.2 The Fourier transform

7.3 Poisson summation formulas

7.4 Spherical harmonics

7.5 Fourier series on compact matrix groups

7.6 Isoperimetric inequality

Appendix A Complementary material

A.1 Metric spaces, convergence, and compactness

A.2 Inner product spaces

A.3 Eigenvalues and eigenvectors

A.4 Complements on power series

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

A.6 Further results on harmonic functions

A.7 Beyond degree theory-introduction to de Rham theory

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 three semesters of calculus and a course in linear algebra The first part trea ts analysis in one variable, and the text [49] 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 [49] We have included it here to facilitate the indepen­dent use of this text Subsequent sections in Chapter 1 present the basic linear algebra background of use for the rest of this text They include rna terial 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 multidimensional differential calculus on domains in n-dimen­sional Euclidean space Itn The first section defines the derivative of a differentiable map F : (') Itm, at a point x E ('), for (') open in Itn, as a linear map from Itn to

Itm, 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 X n systems of differential equations, bringing in the concepts of vector fields and flows on an open set (') E Itn While the emphasis here is on differential calculus, we do make use of integral calculus in one variable, as exposed in Chapter l

Chapter 3 treats multidimensional integral calculus We define the Riemann in­tegral for a class of functions on Itn and establish basic properties, including a change

-vii

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

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 multidimensional 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 Gauss-Green­Stokes 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

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 pullback 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 Jordan­Brouwer separation theorem (in the smooth case), and the study of critical points of a vector field tangent to a compact surface, and connections with the Euler characteris­tic We also show how degree theory yields another proof of the fundamental theorem

of algebra

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 resul ts presen ted in § 7 A

Chapter 7 is devoted to an introduction to multidimensional Fourier analysis Sec­tion 7.1 treats Fourier series on the n-dimensional torus un, 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 mate­

a function on sn-l in terms of eigenfunctions of the Laplace operator !J.s, arising from

over SO(n - 1) Section 704 also makes use of the Gauss-Green-Stokes formula and ap­plications to harmonic functions, from §§4A 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 §7A 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

perimeter This is the two-dimensional isoperimetric inequality Its placement here is due to the fact tha t its proof is an a pplica tion 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 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, Appendix A.1 distills material developed at a more leisurely pace in [49], 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 oflinear transforma tions on finite-dimensional vector spaces, providing results useful in various places, from §2.1 to §6.6

Appendix AA 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 Stone-Weierstrass theorem, a useful tool in analysis, with applications in §§5.3, 7.1, and 704 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 Appendix A.7 in­troduces de Rham cohomology, as an extension of degree theory, developed in Chapter

5

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 [51].) 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, 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, and is pursued in the text

Other connections arise in the treatment of Fourier analysis In addition to Fourier analysis on Euclidean space, the text trea ts Fourier analysis on spheres Matrix groups, such as rotation groups SO(n), make an appearance, both as tools for studying Fourier analysis on spheres and as further sources of problems in Fourier analysis, thereby expanding the theater in which we bring to bear tech­niques of advanced calculus developed here

Acknowledgment

During the preparation of this book, my research has been supported by a number of NSF grants, most recently DMS-1500817

This list of some basic notation will be used throughout the text

R is the set of real numbers

C is the set of complex numbers

Z is the set of integers

Z+ is the set of integers 2: O

N is the set of integers 2: 1 (the natural numbers)

Q is the set of rational numbers

x E IR means x is an element of IR, i.e., x is a real number

(a, b) denotes the set of x E R such that a < x < b

[a, b] denotes the set of x E R such that a :S x :S b

(x E R : a :S x :S b) denotes the set of x in R such that a :S x :S b

[a, b) = {x E R : a :S x < b}and(a, b] = {x E R : a < x :S b}

Z = x - iy if z = x + iy E C, x, Y E R

-xi

o denotes the closure of the set D

f : A + B denotes that the function f takes points in the setA to points in B One also says f maps A to B

x + Xo means the variable x tends to the limit xo

f(x) = O(x) means f(x)/x is bounded Similarly g(E)

lim sup lakl = lim (sup lakl)

k +oo n +oo k'::?n

Background

This first chapter provides background material on one-variable calculus, the geom­etry of n-dimensional Euclidean space, and linear algebra We begin in §1.1 with a presentation of the elements of calculus in one variable We first define the Riemann integral for a class of functions on an interval We then introduce the derivative and

as essentially inverse operations Further results are dealt with in the exercises, such

as the change of variable formula for integrals and the Taylor formula with remainder for power series

uct on Rn gives rise to a norm, hence to a distance function, making Rn a metric space (More general metric spaces are studied in Appendix A.l.) We define the notion of

Appendix A.l

The spaces Rn are special cases of vector spaces, explored in greater generality in

§1.3 We also study linear transformations T : V W between two vector spaces We define the class of finite-dimensional vector spaces, and show that the dimension of such a vector space is well defined If V is a real vector space and dim V = n, then V is isomorphic to Rn Linear transformations from Rn to Rm are given by m X n matrices

invertible if and only if detA '" O In Chapter 2, such linear transformations arise as derivatives of nonlinear maps, and understanding the behavior of these derivatives is basic to many key results in multivariable calculus, both in Chapter 2 and in subsequent chapters

-1

1.1 One-variable calculus

In this brief discussion of one-variable calculus, we introduce the Riemann integral and relate it to the derivative We will define the Riemann integral of a bounded function over an interval I = [a, b 1 on the real line For now, we assume f is real valued To start, we partition I into smaller intervals A partition l' of I is a finite collection of subintervals (Jk : a :S k :S NJ, disjoint except for their endpoints, whose union is I We can order the Jk so thatJk = [Xko xk+d, where

maxsize (1') = max li(Jk)' O�k�N

Icp(f) = � sup f(x)li(h),

To be more precise, if l' and Q are two partitions of I, we say l' refines Q and write

l' >- Q, if l' is formed by partitioning each interval in Q Equivalently, l' >- Q if and

partitions have a common refinement; just take the union of their endpoints, to form a

its lower sum,

(1.1.4 )

Figure 1.1.1 Upper and lower sums

Consequently, if 1) are any two partitions and Q is a common refinement, we have

We will denote the set of Riemann integrable functions on J by :fI (I)

We derive some basic properties of the Riemann integral

Proposition 1.1.1 Iff, g E :fI (I), then f + g E :fI (I), and

Proof If Jk is any subinterval of J, then

sup(f Ik + g):S sup f Ik + sup Ik g and inf(f Ik + g) 2: inf f Ik + inf Ik g,

so, for any partition :P, we have I:p(f + g) :S I:p(f) + I:p(g) Also, using common refine­ments, we can simultaneously approximate I(f) and I(g) by I:p(f) and I:p(g) and the same goes for I(f + g) Thus the characterization (1.1.6) implies I(f + g) :S I (f) + I (g)

A parallel argument implies I (f + g) 2: I (f) + I (g), and the proposition follows D

Next, there is a fair supply of Riemann integrable functions

Proposition 1.1.2 Iff is continuous on J, then f is Riemann integrable

Proof Any continuous function on a compact interval is bounded and uniformly con­tinuous (see Propositions A.1.15 and A.1.16) Let w( 8 ) be a modulus of continuity for

We denote the set of continuous functions on Iby C(I) Thus Proposition 1.1.2 says

C(I) c :fI (I)

The proof of Proposition 1.1.2 provides a criterion on a partition guaranteeing that

11'(f) and I1'(f) are close to h f dx when f is continuous We produce an extension, giving a condition under which 11'(f) and I(f) are close, and I1'(f) and I(f) are close, given f bounded on I Given a partition 1'0 of I, set

I1'(f) 2: Io(f) - 2 � e(I)

one hand those intervals in l' thatare contained in intervals in Q and on the other hand those intervals in l' that are not contained in intervals in Q Each interval of the first type is also an interval in 1', Each interval of the second type gets partitioned, to yield two intervals in 1', Denote by 1',b the collection of such divided intervals By (1.1.12), the lengths of the intervals in 1',b sum to :S e(I)/k It follows that

II1'(f) - I1\(f)I :S � 2Me(J) :S 2Me � ),

JEP1b and similarly II1'(f) - I1\ (f)1 :S 2Me(I)/k Therefore

- 11'(f) :S 11', (f) + Te(l), - 2M

Since also 11\ (f) :S Io(f) and I1\ (f) 2: Io(f), we obtain (1.1.13)

The following consequence is sometimes called Darboux's theorem

(1.1.15) f E :fI (I) = I(f) = v +oo lim k=l � f( IVk )e(Jvk),

for arbitrary IVk E Jvb in which case the limit is h f dx

Proof As before, assume III :S M Pick E = 11k > O Let Q be a partition such that

Icp/f) :S Io(f) + 2MIi (I)E,

Lpv (f) 2: £o(f) - 2MIi (I)£

I(f) :S 11'/f) :S I(f) + [2MIi (I) + l]E,

£(f) 2: £1'v (f) 2: £(f) - [2MIi (I) + 1]£

D

Remark The sums on the right side of (1.1.15) are called Riemann sums, approxi­

Remark A second proof of Proposition 1.1.1 can readily be deduced from Theorem Ll.4

made, the limit on the right side of (loLlS) might exist for a bounded function I that

(1.Ll6) il(x) = 1 if x E Q, il(x) = 0 if x II' Q,

where Q is the set of rational numbers Now every interval J c J of positive length contains points in Q and points notin Q, so for any partition l' ofIwe haveI1' ( il) = Ii (l) and £1' ( il) = 0, hence

(1.Ll7) I( il) = 1i (I), £( il) = O

Note that if 1'v is a partition of J into v equal subintervals, then we could pick each IVk

to be rational, in which case the limit on the right side of (loLlS) would be 1i(I), or we

tively, we could pick half of them to be rational and half to be irrational, and the limit would be 1i (I)/2

Associated to the Riemann integral is a notion of size of a set S, called content If

S is a subset of J, define the characteristic function,

(1.Ll8) Xs(x) = 1 if X E S, 0 if x II' s

We define upper content cont+ and lower content cont- by

(1.Ll9) cont+ (S) = I(Xs), connS) = £(Xs)·

We say S has content, or is contented if these quantities are equal, which happens if and only if XS E :R(I), in which case the common value of cont+(S) and cont-(S) is

J, in terms of the upper content of its set of discontinuities

There is a more sophisticated notion of the size of a subset of J, called Lebesgue measure The key to the construction of Lebesgue measure is to cover a set S by a

countable (either finite or infinite) set of intervals The outer measure of S e J is defined

It is useful to note that h I dx is additive in J, in the following sense

Proposition 1.1.5 Ifa < b < c, I : [a,c] + R, I, = /lla,bl' I, = /llb,cl, then

and, il this holds,

(1.1.25)

may as well consider a partition :P = :P, U :P" where :P, is a partition of [a, b] and :P, is

a partition of [b, c] Then

(1.1.26)

so

(1.1.27)

Since both terms in braces in (1.1.27) are 2: 0, we have equivalence in (1.1.24) Then (1.1.25) follows from (1.1.26) upon taking sufficiently fine partitions D

LetI = [a, b] If f E Jl(I), then f E Jl( [a, x D for all x E [a, b], and we can consider the function

In other words, if f E Jl(I), then g is Lipschitz continuous on I

A function g : (a, b) + R is said to be differentiable at x E (a, b) provided there exists the limit

When such a limit exists, g'(x), also denoted dg/dx, is called the derivative of g at x

Clearly, g is continuous wherever it is differentiable

The next result is part of the fundamental theorem of calculus

Theorem 1.1.6 Iff E C([a, b D, then thefunction g, defined by (1.1.28), is differentiable

at each point x E (a, b), and

h E (0, 0] Thus the desired limit exists as h " O A similar argument treats h /' o D

The next result is the rest of the fundamental theorem of calculus

Theorem 1.1.7 IfG is differentiable and G'(x) is continuous on [a, b], then

Figure 1.1.2 Illustration of the mean value theorem

We have g E C([a, b]), g(a) = 0, and, by Theorem 1.1.6,

g'(x) = G'(x), 'if x E (a, b)

Thus f(x) = g(x) - G(x) is continuous on [a,b], and

We claim that G( a) = -G( a), we have f(x) (1.1.36) impliesf is = -G( a) for all constant on [a, b] Granted this, sincef(a) x E [a, b], so the integral (1.1.35) = is equal g(a)­

to G(x) - G(a) for all x E [a, b] Taking x = b yields (1.1.34) D

The fact that (1.1.36) implies f is constant on [a,b] is a consequence of the fol­ lowing result, known as the mean value theorem This is illustrated in Figure 1.1.2 Theorem 1.1.8 Let f : [a, j3] R be continuous, and assume f is differentiable on

(a, j3) Then 3 5 E (a, j3) such that

g'(O that g( a) = I'(O - K, so it suffices to show thatg'(O = g(j3) Since [a, j3] is compact, g must assume a maximum and a minimum = o for some 5 E (a,j3) Note also

on [a,j3] Since g(a) = g(j3), one of these must be assumed at an interior point, at which g' vanishes

Now, to see that (1.1.36) implies f is constant on [a, b], ifnot, 313 E (a, b] such that

f(j3) '" f(a) Then just apply Theorem 1.1.8 to f on [a, 13], This completes the proof of

The proof is identical to that of Theorem 1.1.6

Proposition 1.1.10 Assume G is differentiable on [a,b] and G' E Jl([a,b]) Then (1.1.34) holds

lessjUndamental than Theorems 1.1.6 and 1.1.7 There are more satisfactory extensions

of the fundamental theorem of calculus, involving the Lebesgue integral, and a more subtle notion of the derivative of a nonsmooth function For this, we can point the reader to [47, Chapters 10-11]'

So far, we have dealt with integration of real valued functions Iff : I + C, we set

f = fl + if, with fj : I + R and say f E Jl(I) if and only if fl and f, are in Jl(I) Then

J f dx = J fl dx + i J f, dx

There are straightforward extensions of Propositions 1.1.5-1.1.10 to complex valued functions Similar comments apply to functions f : I + Rn

If a function G is differentiable on (a, b) and G' is continuous on (a, b), we say G

is a C1 function and write G' E G E C1(( a, b») Inductively, we say G E Ck(( a, b») provided Ck-1((a, b»)

An easy consequence of the definition (1.1.31) of the derivative is that for any real constants a, b, and c,

more general than Proposition 1.1.2, which guarantees Riemann integrability

Proposition 1.1.11 Let f : I II! be a boundedfunction, with I = [a, b] Suppose that the set S of points of discontinuity off has the property

(1.1.39) cont+(S) = o

Then f ell(I)

Proof Say If(x)1 :S M Take E > O As in (1.1.21), take intervals J" ,lN such that

S the interior of this collection of intervals Consider a partition 1'0 of c J, u U J N and L � =, lI(Jk) < £ In fact, fatten each Jk such that S is contained in I, whose intervals include J" ,IN, amongst others, which we label I" ,IK' Now f is continuous on each interval Iv, so, subdividing each Iv as necessary, hence refining 1'0 to a partition 1'" we arrange that supf - inff < E on each such subdivided interval Denote these subdivided intervals I;, ,II It readily follows that

Remark An even better result is that such f is Riemann integrable if and only if

where n-dimensional setting) is established in Proposition 3.1.31 of this text For the one­ m*(S) is defined by (1.1.22) The implication m*(S) = a =} f E :fI(I) (in the dimensional case, see also [49, Proposition 4.2.12]' For the reverse implication f E :fI(I) [47] =} m*(S) = 0, one can see standard books on measure theory, such as [17] and

We give an example of a function to which Proposition 1.1.11 applies, and then an example for which Proposition 1.1.11 fails to apply, though the function is Riemann integrable

Example 1 LetI = [0, 1] Define f : I + R by

f(O) = 0,

f (x) = (-I)j for x E (r(j + l) , rj ], j 2: O

Then If I :S 1 and the set of points of discontinuity of f is

S = {a} u {2-j : j 2: I }

It is easy to see that cont+ S = O Hence f E :fI(I)

See Exercises 19-20 below for a more elaborate example to which Proposition 1.1.11 applies

Example 2 Again I = [0,1]' Define f : I + R by

1 X = -n , III owest terms

Then If I :S 1 and the set of points of discontinuity of f is

S = In Q

As we have seen following (1.1.23), cont+ S = 1, so Proposition 1.1.11 does not apply Nevertheless, it is fairly easy to see directly that

I(f) = I(f) = 0, so f E :fI(I)

In fact, given E > 0, f 2: E only on a finite set, hence

I(f) :S E, 'if E > o

As indicated following (1.1.23), (1.1.40) does apply to this function By contrast, the function 11 in (1.1.16) is discontinuous at each point of I

We mention an alternative characterization ofI(f) and I(f), which can be useful

Given I = [a,b], we say g : I + R is piecewise constant on I (and write g E PK(I»

provided there exists a partition l' = (hl ofI such that g is constant on the interior of each interval Jk Clearly, PK(I) c 3/(I) It is easy to see that if f : I + R is bounded,

Hence, given f : I + R bounded,

f E 3/(I) � for each £ > 0, 3fo, f, E PK(I) such that

In fact, we have the following, which can be used to prove (1.1.43)

Proposition 1.1.12 Let f E 3/(I), and assume If I :S M Let 'P : [-M,M] + R be continuous Then'P ° f E 3/(I)

Proof We proceed in steps

Step 1 We can obtain 'P as a uniform limit on [-M, M] of a sequence 'Pv of continuous, piecewise linear functions Then 'Pv ° f + 'P ° f uniformly on I A uniform limit g of functions 1.1.12 when 'P is continuous and piecewise linear g v E 3/(I) is in 3/(I) (see Exercise 12) So it suffices to prove Proposition

Step 2 Given 'P : [-M, M] + R continuous and piecewise linear, it is an exercise to write 'P = 'P, - 'P" with 'Pj : [-M,M] + R monotone, continuous, and piecewise linear Now 'P, ° f, 'P, ° f E 3/(I) =} 'P ° f E 3/(I)

Step 3 We now demonstrate Proposition 1.1.12 when 'P : [-M, M] + R is monotone and Lipschitz By Step 2, this will suffice So we assume

-M :S x, < x, :S M ==} 'P( x, ) :S 'P( x, ) and 'P( x, ) - 'P(x,) :S L(x, - x,),

for some L < 00 Given £ > 0, pick fa.!, E PK(I), as in (1.1.42) Then

and 'P0fo, 'P0f, j ('P ° f, - 'P E ° PK(I), fo)dx :s L 'P0fo :S 'P0f :S 'P0f" ju, - fo)dx :s L£

2 Let f : I X S -+ R be continuous, where I = [a,b] and S eRn Take "'(Y) =

h f(x, y) dx Show that", is continuous on S

Hint If fj : I -+ R are continuous and If, (x) - f,(x)1 :S 0 on I, then

(1.1.47) If f, dx - f f, dxl :S e(I)o

Hint f : I X Suppose S Yj E S, Yj -+ y E S Let S = {Yj} U {y} This is compact Thus

-+ R is uniformly continuous Hence

If(x,Yj) - f(x,Y)I :S w(IYj -yl), 'ifx E I,

where w( 0) -+ 0 as 0 -+ O

3 With f as in Exercise 2, suppose gj : S -+ R are continuous and a :S go(Y) <

g, (y) :S b Take cAy) = J:o'/;:1 f(x, y) dx Show that", is continuous on S

Hint Make a change of variables, linear in x, to reduce this to Exercise 2

4 hex) Suppose f : (a, b) -+ (c, d) and g : (c, d) -+ R are differentiable Show that

= g(f(x» is differentiable and

h'(x) = g'(f(x» f'(x)

This is the chain rule

Hint Peek at the proof of the chain rule in §2.1

5 If f, and f, are differentiable on (a, b), show that f, (x)f,(x) is differentiable and

: x (Nx)f,(x») = f{(x)f,(x) + Nx)t;(x)

If f,(x) '" 0, 'if x E (a, b), show that f, (x)1 f,(x) is differentiable and

� ( Nx» ) _ f{(x)f,(x) - Nx)f;(x)

dx f,(x) - f,(x)'

6 Let ", : [a,b] + [A,B] be C1 on a neighborhood J of [a, b], with ",'(x) > o for all

x E [a, b] Assume r:p(a) = A, ",(b) = B Show that the identity

(1.1.48) 1 f(y)dy = 1 f(",(t»)",'(t)dt,

for any f E C(I), I = [A, B], follows from the chain rule and the fundamental theorem

of calculus This is the change of variable formula for the one-dimensional integral

Hint Replace b by x, B by ",(x), and differentiate

f E 3/(I) =} f 0 '" E 3/([ a, b]) and (1.1.48) holds (This result contains that of Exercise 1.)

8 Show tha t if f and g are Clan a neighborhood of [a, b], then

(1.1.49) 1 f(s)g'(s)ds = -1 f'(s)g(s)ds + [J(b)g(b) -f(a)g(a)]

This transformation of integrals is called integration by parts

9 Let f : (-a, a) + It be a Cj+l function Show that for x E (-a, a),

This is Taylor's formula with remainder See §2.1 for the multidimensional extension

Hint Use induction If (1.1.50)-(1.1.51) hold for 0 :S j :S k, show that they hold for

j = k + 1, by showing that

(1.1.52) 1x ( x - s f(k+l)(s)ds )k = f(k+l)(O) xk+l + 1x ( )k+l x - s f(k+2)(s)ds

f(k+l)(S) and with appropriate g(s) See Appendix §A.4 for further material on the re­mainder formula Note that another presentation of (1.1.51) is

Hint Apply (1.1.51) with j replaced by j - 1 Add and subtract f(j)(O) to the factor

f(j)(s) in the resulting integrand

11 Given I = [a, b], show that

for all E E (0, (b - a)/2) Prove thatf E 3/U) and that (1.1.56) holds

14 Use the fundamental theorem of calculus to compute

Formulate and demonstrate basic properties of the integral over R of elements of 3/(R)

17 This exercise discusses the integral test for absolute convergence of an infinite se­ries, which goes as follows Let f be a positive, monotonically decreasing, continuous

Prove this

Hint Use

to lakl < 00 = 100 f(x)dx < 00

18 Use the integral test to show that if p > 0,

k=l

Hint Use Exercise 14 to evaluate IN(P) = fIN x-p dx, for P '" -1, and letN + 00 See

if you can show 1;00 X-I dx = 00 without knowing about 10gN

Subhint Show that 1;' X-I dx = f�N X-I dx

In Exercises 19-20, e c R is the Cantor set, defined as follows Take a closed, bounded interval [a, b 1 = eo Let e1 be obtained from eo by deleting the open middle third interval of length (b - a)/3 At the jth stage, ej is a disjoint union of 2j closed intervals, each oflength r j(b -a) Then ej+1 is obtained from ej by deleting the open middle third of each of these 2j intervals We have eo J e1 J J ej J , each a closed subset of [a, b] The Cantor set is e = nj>oej

19 Show that cont+ ej = (2/3)j(b - a), and conclude that

cont+ e = o

20 Define f : [a, b 1 + R as follows We call an interval of length r j (b - a), omitted

in passing from ej_1 to ej, a j-interval Set

f(x) = 0, if X E e, ( -l)j, if X belongs to a j-interval

Show that the set of discontinuities of f is e Hence Proposition 1.1.11 implies f E Jl( [a, b D

21 Generalize Exercise 8 as follows Assume f and g are differentiable on a neighbor­hood of [a, bl and f', g' E Jl([ a, b D Then show that (1.1.49) holds

Hint Use the results of Exercise 11 to show that Cfg), E Jl([a,bD

22 Let f : I + R be bounded, I = [a, b] Show that

ICf) = inf! j fl dx : fl E C(I), fl 2: f J,

I rCf) = sup! ! fa dx : fa E C(I), fa :S fl·

I

Compare (1.1.41) Then show that

f E :R(I) = for each £ > 0, 3fo'/1 E C(I) such that (1.1.62) fo :S f :S fl and f Ul - fo)dx < £

algebraic and metric structures on Rn First, there is addition If x is as in (1.2.1) and also Y = (Yl, '" ,Yn) E Rn, we have

(ax) · (ax) = a'(x · x),

laxl = lal IxL for a E R, x E Rn

in [Rn is

For us, (1.2.9) and (1.2.12) are simply definitions We do not need to depend on a deriva­tion of the Pythagorean theorem via classical Euclidean geometry Significant proper­ties will be derived below, without recourse to a prior theory of Euclidean geometry

A set X equipped with a distance function is called a metric space We consider metric spaces in general in Appendix A.1 Here, we want to show that the Euclidean distance, defined by (1.2.12), satisfies the triangle inequality,

(1.2.13) d(x,y) :S d(x,z) + d(z,y), Vx,y,z E Rn

This in turn is a consequence of the following, also called the triangle inequality Proposition 1.2.1 The norm (1.2.9) on Rn has the property

(1.2.14 ) Ix + yl :S Ixl + Iyl Vx,y E Rn

Proof We compare the squares of the two sides of (1.2.14) First,

(Ixl + Iyl)' = lxi' + 21xl Iyl + lyl2

We see that (1.2.14) holds if and only if x y :S Ixl Iyl Thus the proof of Proposition

Proposition 1.2.2 For all x, y E Rn,

O :S Ix - yl' = (x - y) (x - y) = lxi' + Iyl' - 2x · y,

2x · y :S lxi' + Iyl', Vx,y E Rn

Ifwe replace x by tx and y by t-ly, with t > 0, the left side of (1.2.19) is unchanged, so

we have

(1.2.20) 2x · y :S t'lxl' + t-'Iyl', Vt > o

Now we pick t so that the two terms on the right side of (1.2.20) are equal, namely

(1 2 21) t' = Q:l lxi ' t-' = !:::.! Iyl '

(At this point, note that (1.2.17) is obvious if x = 0 or y = 0, so we will assume that

x '" 0 and y '" 0.) Plugging (1.2.21) into (1.2.20) gives

(1.2.22) X · y :S Ixl · Iyl V x,y E Itn

This is almost (1.2.17) To finish, we can replace x in (1.2.22) by -x = (-l)x, getting

and together (1.2.22) and (1.2.23) gives (1.2.17) D

We now discuss a number of notions and results related to convergence in Itn

First, a sequence of points (Pj) in Itn converges to a limit P E Itn (we write Pj -+ p) if and only if

(1.2.24 ) IPj - pi + 0,

where I I is the Euclidean norm on Itn, defined by (1.2.9), and the meaning of (1.2.24)

is that for every 0 > 0 there exists N such that

If we write Pj = (Plj, · , Pnj) and P = (PI' , Pn), then (1.2.24) is equivalent to

(Plj - PI)' + + (Pnj - Pn)' + 0, as j -+ 00,

which holds if and only if

IPej - Pel + 0 as j -+ 00, for each e E {1, , n}

That is to say, convergence Pj -+ p in Itn is eqivalent to convergence of each compo­nent

A set S c Itn is said to be closed if and only if

(1.2.26) Pj E S, Pj -+ P ==} p E S

The complement Itn \ S of a closed set S is open Alternatively, D c Itn is open if and only if, given q ED, there exists 0 > 0 such that Biq) C D, where

(1.2.27) B£(q) = (p E Itn : Ip - ql < oj,

so q cannot be a limit of a sequence of points in Itn \ D

An important property of Itn is completeness, a property defined as follows A se­quence (Pj) of points in Itn is called a Cauchy sequence if and only if

(1.2.28) IPj - Pkl + 0, as j, k -+ 00

Again we see that (Pj) is Cauchy in Itn if and only if each component is Cauchy in It

It is easy to see that if Pj -+ P for some P E Itn, then (1.2.28) holds The completeness property is the converse

Theorem 1.2.3 If(pj) is a Cauchy sequence in Itn, then it has a limit, i.e., (1.2.24) holds for some p E Itn

Proof Since convergence Pj + P in Itn is equivalent to convergence in It of each component, the result is a consequence of the completeness of It This is proved in

Completeness provides a path to the following key notion of compactness A non­empty set K c Itn is said to be compact if and only if the following property holds (1.2.29) Each infinite sequence (Pj) in K has a subsequence

that converges to a point in K

!tis clear that ifK is compact, then itmustbe closed !tmustalso be bounded, i.e., there exists R < 00 such thatK C BR(O) Indeed, if K is not bounded, there exist Pj E K such that Ipj+ll 2: Ipjl + 1 In such a case, Ipj - Pkl 2: 1 whenever j '" k, so (Pj) cannot have a convergent subsequence The following converse result is the n-dimensional Bolzano-Weierstrass theorem

Proof If K e lt n is closed and bounded, it is a closed subset of some box

(1.2.30) 13 = {(Xl, " " Xn) E Itn : a :S Xk :S b, \f k}

Clearly, every closed subset of a compact set is compact, so it suffices to show that 13 is compact Now, each closed bounded interval [a, b 1 in It is compact, as shown in Ap­pendix A.l, and (by reasoning similar to the proof of Theorem 1.2.3) the compactness

We establish some further properties of compact sets K c Itn, leading to the impor­tant result, Proposition 1.2.8 below A generalization of this result is given in Appendix A.1

Proposition 1.2.5 Let K c Itn be compact Assume Xl J X, J X3 J form a decreasing sequence of closed subsets of K If each Xm '" 0, then nmXm '" O

Proof Pick Xm E Xm If K is compact, (xm) has a convergent subsequence, xmk + y

Since {xmk : k 2: e} c Xm" which is closed, we have y E nmXm D Corollary 1.2.6 Let K c Itn be compact Assume Ul C U, C U3 C form an increasing sequence of open sets in Itn IfumUm J K, then UM J Kfor some M

Before getting to Proposition 1.2.8, we bring in the following Let 0 denote the set

are rational The set on C Itn has the following denseness property: given P E Itn and

£ > 0, there exists q E on such that Ip - ql < £ Let

Note that 0 and on are countable, i.e., they can be put in one-to-one correspondence with N Hence :R is a countable collection of balls The following lemma is left as an exercise for the reader

Lemma 1.2.7 Let 0 c Rn be a nonempty open set Then

(1.2.32) 0 = U{E : E E :R, E cO}

To state the next result, we say that a collection {Ua : a E A} covers K if K c

uaEAUa If each Ua C Rn is open, it is called an open cover of K If 13 c A and

K C UPE1J Up, we say {Up : f3 E 13} is a subcover The following is part of the n­

dimensional Heine-Borel theorem (See Theorem A.1.1O.)

Proposition 1.2.8 If K e Rn is compact, then it has the following property

Proof By Lemma 1.2.7, it suffices to prove the following

Every countable cover {Ej : j E N} of K by open balls

has a finite sub cover

(1.2.34 )

To see this, write :R = {Ej : j E N} Given the cover {Ua}, pass to {Ej : j E J}, where

j E J if and only of Ej is contained in some Ua By (1.2.32), {Ej : j E J} covers K If (1.2.34) holds, we have a subcover {Ee : e E L} for some finite L C J Pick ae E A such that Ee C Ua, The {Ua, : e E L} is the desired finite subcover advertised in (1.2.33) Finally, to prove (1.2.34), we set

and apply Corollary 1.2.6

Exercises

D

1 Jdentifying z = x + iy E C with (x,y) E R' and w = u + iv E C with (u, v) E R',

show that the dot product satisfies

Z · W = Re zw

2 Show that the inequality (1.2.14) implies (1.2.13)

3 Prove Lemma 1.2.7

4 Use Proposition 1.2.8 to prove the following extension of Proposition 1.2.5

closed subsets of K Assume that for each finite set 13 C A, naE1JXa '" 0 Then

n Xa ", 0

Hint Consider Ua = Rn \ Xa'

5 Let K e Rn be compact Show that there exist xo, Xl E K such that

IXol :S lxL VX E K,

IXol = min XEK Ixi IXI I = max XEK

Ixi-1.3 Vector spaces and linear transfonnations

We have seen in §1.2 how Rn is a vector space, with vector operations given by (1.2.2)­(1.2.3) for row vectors, and by (1.2.4)-(1.2.5) for column vectors We could also use complex numbers, replacing Rn by en, and allowing a E e in (1.2.3) and (1.2.5) We will use F to denote R or C

Many other vector spaces arise naturally We define this general notion now A vector space over F is a set V, endowed with two operations, that of vector addition and multiplication by scalars That is, given v, W E V and a E F, then v + w and av

are defined in V Furthermore, the following properties are to hold, for all u, v, W E

V, a, b E F First there are laws for vector addition:

Next there are laws for multiplication by scalars:

(1.3.2) Associative law : a(bv) = (ab )v,

Unit : 1 v = v

Finally there are two distributive laws:

(a + b)u = au + bu

It is easy to see that Rn and en satisfy all these rules We will present a number

of other examples below Let us also note that a number of other simple identities are automatic consequences of the rules given above Here are some, which the reader is invi ted to verify:

Here are some other examples of vector spaces LetI = [a, b] denote an interval in

R, and take a nonnegative integer k Then Ck(I) denotes the set offunctions f : I + F whose derivatives up to order k are continuous We denote by l' the set of polynomials

in x, with coefficients in F We denote by Ji the set of polynomials in x of degree :S k

In these various cases,

Then W inherits the structure of a vector space

Linear transformations and matrices If V and W are vector spaces over F (R or

C), a map

is said to be a linear transformation provided

(1.3.8) T(alvl + a,v,) = alTvl + a,Tv" V aj E F, Vj E V

We also write T E L(V, W) In case V = W, we also use the notation L(V) = L(V, V)

given f E Ck(I), I = [a, b], and the operation of differentiation:

We also have integration:

(1.3.13)

Note also that

J : ck(I) + Ck+I(I), Jf(x) = 1X f(y)dy

(1.3.14) D : 1'k+1 + Ji, J : 1'k + 1'k+l,

where 1'k denotes the space of polynomials in x of degree :S k

Two linear transformations 1) E L(V, W) can be added:

(1.3.15) T, + T, : V + W, (T, + T,)v = T,v + T,v

Also T E L(V, W) can be multiplied by a scalar:

(1.3.16) aT : V + W, (aT)v = a(Tv)

This makes L(V, W) a vector space

We can also compose linear transformations S E L(W,X), T E L(V, W):

and then we have

(1.3.24 ) AB = (dij)' dij = 8=1 � n aitbej'

To establish the identity (1.3.22), we note that it suffices to show the two sides have the same effect on each ej E rk, 1 :S j :S k, where ej is the column vector in rk whose jth entry is 1 and whose other entries are O First note that

(1.3.25)

where the jth column is in E, as one can see via (1.3.10) Similarly, if D denotes the right side of (1.3.22), Dej is the jth column of this matrix, i.e.,

Note thatN(T) is a linear subspace of V and :R(T) is a linear subspace of W If N(T) =

0, we say T is injective; if :R(T) = W, we say T is surjective Note that T is injective if and only if T is one-to-one, i.e.,

(1.3.29)

If T is surjective, we also say T is onto If T is one-to-one and onto, we say it is an

isomorphism In such a case the inverse

(1.3.30) T-' : W + V

is well defined, and it is a linear transformation We also say T is invertible, in such a case

Basis and dimension Given a finite setS = (v" , vk} in a vector space V, the span

of S is the set of vectors in V of the form

(1.3.31)

with Cj arbitrary scalars, ranging over r = R or C This set, denoted Span(S) is a linear subspace of V The set S is said to be linearly dependent if and only if there exist scalars c" , ck-not all zero, such that (1.3.31) vanishes Otherwise we say S is linearly independent

If {v" , Vk} is linearly independent, we say that S is a basis of Span(S) and that

k is the dimension of Span(S) In particular, if this holds and Span(S) = V, we say

k = dim V We also say V has a finite basis, and that V is finite dimensional

By convention, if V has only one element, the zero element, we say V = a and dim V = O

It is easy to see that any finite set S = {v" , Vk} C V has a maximal subset that

is linearly independent and such a subset has the same span as S, so Span(S) has a basis To take a complementary perspective, S will have a minimal subset So with the same span, and any such minimal subset will be a basis of Span(S) Soon we will show

elements (so dim V is well defined) First, let us relate V to rk

So say V has a basis S = {v" , Vk} We define a linear transformation

(1.3.35)

We begin our demonstration tha t dim V is well defined with the following concrete result

Lemma 1.3.1 Ifv" , vk+' are vectors in rk, then they are linearly dependent

Wj = Vj - VkjVk+l, 1 :::; j :::; k,

where Vj = (v'j, , Vkj)t Then the last component of each of the vectors w" , wk

a" , ak-not all zero, such that

so we have

al VI + + akvk = (al Vkl + + akvkk)vk+l,

the desired linear dependence relation on {v" , Vk+rl

With this result in hand, we proceed

D

Proposition 1.3.2 If V has a basis {v" , Vk} with k elements and {w" , we} C V is linearly independent, then e :S k

potheses imply that {A-' w" ,A-'we} is linearly independent in rk, so Lemma 1.3.1

Corollary 1.3.3 If V isfinite dimensional, any two bases of V have the same number of elements If V is isomorphic to W, these spaces have the same dimension