Advanced Calculus

By Avner Friedman

This rigorous two-part treatment advances from functions of one variable to those of several variables. Intended for students who have already completed a one-year course in elementary calculus, it defers the introduction of functions of several variables for as long as possible, and adds clarity and simplicity by avoiding a mixture of heuristic and rigorous arguments.
The first part explores functions of one variable, including numbers and sequences, continuous functions, differentiable functions, integration, and sequences and series of functions. The second part examines functions of several variables: the space of several variables and continuous functions, differentiation, multiple integrals, and line and surface integrals, concluding with a selection of related topics. Complete solutions to the problems appear at the end of the text.

• Language: English
• Publisher: Dover Publications
• Release date: Oct 16, 2012
• ISBN: 9780486137865

Book preview

ADVANCED

CALCULUS

Avner Friedman

The Ohio State University

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright © 1971, 1999 by Avner Friedman

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2007, is an unabridged republication of the work published by Holt, Rinehart and Winston, Inc., New York, in 1971.

International Standard Book Number: eISBN: 978-0-486-13786-5

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

PREFACE

This book is intended for students who have already completed a one year course in elementary calculus. These students are familiar with the powerful methods of calculus in solving a variety of problems arising in physics, engineering, and geometry. They are therefore sufficiently motivated towards a rigorous treatment of the subject.

Many of the ideas of advanced calculus are by no means simple or intuitive. (It took mathematicians two hundred years to develop them!) Furthermore, for most students this is the first substantial course in mathematics taught in a rigorous way. We have therefore attempted to keep this book on a comfortable level. For example, it has been our experience that the student does not feel at ease with calculus in two or more dimensions. We have therefore deferred the introduction of functions of several variables as long as possible. We feel that further clarity and simplicity is added by avoiding mixture of heuristic and rigorous arguments.

The book contains material for more than a year’s course; some choice is thus offered. Our own inclination is to omit parts of several sections throughout Chapters 1–9, and present some of the topics of Chapter 10. Students may be assigned to do independent study based on some sections of Chapter 10; the sections in this chapter are independent of each other.

When we refer to a particular section, say to Theorem 1 of Section 1, it is understood that the reference is to Section 1 of the same chapter. When we wish to refer to Section 1 of a different chapter, say Chapter 2, then we write Section 2.1.

I would like to thank Andrew J. Callegari and Joseph E. D’Atri for reading the manuscript and making some useful comments, and Zeev Schuss for verifying the answers of many of the problems.

A.F.

Tel Aviv, Israel

May 1971

CONTENTS

Preface

part one         FUNCTIONS OF ONE VARIABLE

1    NUMBERS AND SEQUENCES

1.    The Natural Numbers

2.    The Rational Numbers

3.    The Real Numbers

4.    Bounded Sets

5.    Limits of Sequences

6.    Operations with Limits

7.    Monotone Sequences

8.    Bolzano–Weierstrass Theorem

2    CONTINUOUS FUNCTIONS

1.    Definition of Continuous Function

2.    Operations with Continuous Functions

3.    Maximum and Minimum

4.    Intermediate Values

5.    Monotone Functions and Inverse Functions

6.    Semicontinuity

7.    Uniform Continuity

8.    Functions of Bounded Variation

3    DIFFERENTIABLE FUNCTIONS

1.    Definition of the Derivative

2.    Operations with Differentiable Functions

3.    Inverse Functions

4.    Higher Derivatives

5.    The Mean Value Theorem

6.    The Second Mean Value Theorem

7.    L’Hospital’s Rule

8.    Taylor’s Theorem

9.    Computation of Extremum

4    INTEGRATION

1.    The Darboux Integral

2.    The Riemann Integral

3.    Properties of the Integral

4.    Examples of Integrable Functions

5.    The Fundamental Theorem of Calculus

6.    Methods of Integration

7.    Mean Value Theorems

8.    Improper Integrals

5    SEQUENCES AND SERIES OF FUNCTIONS

1.    Series of Numbers: Criteria for Convergence

2.    Series of Numbers: Conditional and Absolute Convergence

3.    Multiplication of Series

4.    Uniform Convergence

5.    Integration and Differentiation of Series

6.    Power Series

7.    Taylor’s Series

8.    Convergences of Improper Integral

part two        FUNCTIONS OF SEVERAL VARIABLES

6    SPACE OF SEVERAL VARIABLES AND CONTINUOUS FUNCTIONS ON IT

1.    Open and Closed Sets in the Plane

2.    Convergence in the Plane

3.    The n-Dimensional Space

4.    Continuous Functions

5.    Limit Superior and Limit Inferior

6.    Vector Valued Functions; Curves

7    DIFFERENTIATION

1.    Partial Derivatives

2.    Differentials: The Chain Rule

3.    The Mean Value Theorem and Taylor’s Theorem

4.    Computation of Extremum

5.    Implicit Function Theorems

6.    Inverse Mappings

7.    Jacobians

8.    Vectors

8    MULTIPLE INTEGRALS

1.    Sets of Content Zero

2.    Double Integrals

3.    Iterated Integrals

4.    Multiple Integrals

5.    Change of Variables in Multiple Integrals

6.    Uniform Convergence

7.    Improper Integrals

8.    Integrals Depending on a Parameter

9.    Improper Integrals Depending on a Parameter

9    LINE AND SURFACE INTEGRALS

1.    Length of Curves

2.    Line Integrals

3.    Integral Along a Curve

4.    Independence of Line Integrals on the Curves

5.    Green’s Theorem

6.    Change of Variables in Double Integrals

7.    Surfaces and Area

8.    Surface Integrals

9.    The Divergence Theorem

10.    Change of Variables in Triple Integrals

11.    Stokes’ Theorem

10    SELECTED TOPICS

1.    Euler’s Summation Formula

2.    Asymptotic Series

3.    Asymptotic Behavior of Integrals Depending on a Parameter

4.    Fourier Series

5.    Lagrange Multipliers

6.    The Divergence Theorem in Rn

7.    Harmonic Functions

8.    Ordinary Differential Equations

9.    The Riemann-Steiltjes Integral

ANSWERS TO PROBLEMS

Index

part one

FUNCTIONS OF ONE VARIABLE

1

NUMBERS AND SEQUENCES

1.   THE NATURAL NUMBERS

The positive integers 1,2, 3, … are called natural numbers. Since we intend to do things rigorously, we cannot be satisfied with our everyday familiarity with these numbers, and we should try to axiomatize their properties. Let us first write down five statements concerning the natural numbers that we feel should be true:

(I) 1 is a natural number.

(II) To every natural number n there is associated in a unique way another natural number n′ called the successor of n.

(III) 1 is not a successor of any natural number.

(IV) If two natural numbers have the same successor, then they are equal.

(V) Let M be a subset of the natural numbers such that: (i) 1 is in M, and (ii) if a natural number is in M, then its successor also is in M. Then M coincides with the set of all the natural numbers.

From now on we consider the statements (I)–(V) to be axioms. They are called the Peano axioms. The natural numbers will be the objects occurring in the Peano axioms. Axiom (V) is called the principle of mathematical induction.

We denote the successor of 1 by 2, the successor of 2 by 3, and so on. Note that 2 ≠ 1. Indeed, if 2 = 1, then 1 is the successor of 1, thus contradicting (III). Note next that 3 ≠ 2. Indeed, if 3 = 2 then, by (IV), 2 = 1, which is false. In general, one can show that all the numbers obtained by taking the successors of 1 any number of times are all different. The proof of this statement, which we shall not give here, is based on induction, that is, on Axiom (V).

We would like to state Axiom (V) in a form more suitable for application:

(V′) Let P(n) be a property regarding the natural number n, for any n. Suppose that (i) P(1) is true, and (ii) if P(n) is true, P(n′) also is true. Then P(n) is true for all n.

If we define M to be the set of all natural numbers for which P(n) is true, then (V′) follows from (V). If, on the other hand, we define P(n) to be the property that n belongs to M, then (V) follows from (V′). Thus (V) and (V′) are equivalent axioms.

The Peano axioms give us objects with which to work. We now proceed to define operations on these objects. There are two operations that we consider: addition (+) and multiplication (·). To any given pair of natural numbers each of these operations corresponds another natural number. The precise definition of this correspondence is given in the following theorem.

THEOREM 1. There exist unique operations + and . with the following properties:

The proof will not be given here. We shall often write mn instead of m · n.

THEOREM 2. The following properties are true for all natural numbers m, n, k:

The proof of Theorem 2 can be given by induction; it is based on the properties (1) and (2).

We state, without proof, another theorem, known as the trichotomy law:

THEOREM 3. Given any natural numbers m and n, one and only one of the following possibilities occurs:

  (i)   m = n.

 (ii)   m = n + x for some natural number x.

(iii)   n = m + y for some natural number y.

If (ii) holds, we write m > n or n < m, and we say that m is larger or greater than n and that n is smaller or less than m. If either (i) or (ii) holds, we write m n or n m, and say that m is larger or equal to n and that n is less than or equal to m.

PROBLEMS

1.   If n > m, then n + k > m + k.

2.   If n > m, m > k, then n > k.

3.   If n + k m + k, then n m.

4.   (n + 1)² = n² + 2n + 1, where n² = n · n.

5.   (n + 1)³ = n³ + 3n² + 3n + 1, where n³ = n² · n.

6.   Prove the associative law (m + n) + k = m + (n + k), by induction on k.

7.   Prove the distributive law (m + n)k = mk + nk, by induction on n.

8.   If m > n, then mk > nk. Conversely, if mk > nk, then m > n.

2.   THE RATIONAL NUMBERS

To every natural number n we correspond a new symbol (–n), or –n, called minus n. We also introduce the symbol 0 called zero. Next we define the operation of addition (– n) + m as follows:

  (i)   If n = m, then (–n) + m = 0.

 (ii)   If n = m + x, x a natural number, then (–n) + m = −x

(iii)   If m = n + y, y a natural number, then (−n) + m = y.

Note, by the trichotomy law, that one and only one of Cases (i), (ii), or (iii) occurs.

Next we define: m + (−n) = (−n) + m, (−n) + (−m) = −(n + m), 0 + m = m + 0 = m, 0 + (−n) = (−n) + 0 = −n, and 0 + 0 = 0.

Multiplication is defined as follows:

The symbols −n are called the negative integers. The integers consist of the natural numbers (also called the positive integers), the negative integers, and 0. We state without proof:

THEOREM 1. For any integers m, n, k, Rules (3), (4), and (5) of Section 1 are true.

If a is a negative integer −n, then we define −a as n. We also define −0 to be 0. Then, for any integer a, a + (− a) = (− a) + a = 0.

Given any two integers a, b, there is a solution x of the equation

Indeed, x = (− a) + b is such a solution, since, by Theorem 1, a + [(− a) + b] = [a + (−a)] + b = 0 + b = b. If y is another solution, then a + x = a + y. Hence (− a) + (a + x) = (− a) + (a + y). Using Theorem 1, we get x = y. Thus Equation (1) has a unique solution.

Notation. We write b + (− a) = b − a = −a + b.

If the solution x of (1) is a positive integer, we write b > a or a < b, and we say that b is larger (or greater) than a and that a is smaller (or less) than b. If x is negative, then the equation b + y = a has the positive solution y = − x. Hence a > b.

We now shall introduce fractions. These are symbols that we write in the form a/b , where a and b are any integers, and b ≠ 0. These symbols are subject to the following definitions:

Note that if a/b = c/d and c/d = e/f, then a/b = e/f.

The last two definitions are acceptable only if we can show that b ≠ 0, d ≠ 0 imply that bd ≠ 0. This, however, can be checked by considering the four possibilities: b positive or negative, d positive or negative.

Definitions (3) and (4) would be most unnatural if it turned out that it is possible to have a/b = a′/b′, c/d = c′/d′ but a/b + c/d is not equal to a′/b′ + c′/d′ [or (a/b) · (c/d) is not equal to (a′/b′) · (c′/d′)]. The following theorem shows that this cannot occur.

THEOREM 2. If a′/b′ = a/b and c′/d′ = c/d, then

Proof. To prove (5) we have to show that

or

But this follows by multiplying the relation a′b = ab′ by dd′, the relation c′d = cd′ by bb′, and adding the resulting equalities. To prove (6) we have to show that

or

But this follows from

A fraction a/b is called negative if either a > 0, b < 0 or a < 0, b > 0. It is called positive if either a > 0, b > 0 or a < 0, b < 0. It is called zero if a = 0. It is easily seen that if a fraction c/d is equal to a fraction a/b, then they are either both positive, or both negative, or both zero.

THEOREM 3. The following properties hold for any fractions a/b, c/d, e/f:

Consider the equation

It has a solution x = (bc − ad)/bd. If x is positive, then we write a/b < c/d or c/d > a/b, and say that c/d is larger than a/b and that a/b is less than c/d. If x is negative, then the equation

has the positive solution y = − x, so that a/b > c/d. Note that c/d is positive (negative) if it is larger (smaller) than zero.

The definition of fractions is very intuitive and is, in fact, suggested by our experience with quotients of integers. There is, however, one disturbing feeling about the concept of fractions, due to the fact that fractions having different forms may be equal to each other. This makes it impossible to speak of the zero fraction (since there are many fractions 0/b taking the role of zero). We also cannot assert that Equation (10) has a unique solution. Similarly, the equation

does not have a unique solution.

To overcome this unpleasant situation, we introduce the concept of a rational number.

DEFINITION. A rational number (a, b) (where a and b are integers, and b ≠ 0) is the class of all the fractions e/f that are equal to a/b.

Addition and multiplication for rational numbers are given by

Theorem 2 shows that these definitions are meaningful. Theorem 3 implies:

THEOREM 4. Rational numbers satisfy the commutative laws for addition and multiplication, the associative laws for addition and multiplication, and the distributive laws.

Let us write the analog of Equation (11) for rational numbers:

This equation has a unique solution x = (bc, ad). We write this solution also in the form (c, d)/(a, b) or (c, d)(a, b)−1.

Let us write the analog of (10) for rational numbers:

This equation also has a unique solution: x = (bc − ad, bd). We write it also as (c, d) − (a, b).

Note that there is a one-to-one correspondence between the integers a and the rational numbers (a, 1). This correspondence a → (a, 1) is preserved under addition and multiplication. Indeed, this follows from the relations

Hence, if we write an integer a in the form (a, 1), we see that the integers can be identified with a subset of the rational numbers.

In what follows we shall adopt the definition of rational numbers as classes of fractions a/b. However, for brevity, we shall write the rational numbers (a, b) usually in the form a/b. When we write a/b = c/d, we mean that (a, b) = (c, d), that is, ad = bc. The rational numbers b/1 will also be written, briefly, as b. In particular, the rational number zero will be denoted by 0.

PROBLEMS

1.   Prove by induction on n that:

(a)   1 + 2 + ··· + n = n(n + 1)/2.

(b)   1 + 2² + ··· + n² = n(n + 1)(2n + 1)/6.

(c)   1 + α + α² + ··· + αn = (αn + ¹ − 1)/(α − 1) where α is rational, α ≠ 1.

2.   If a, b are integers and a · b = 0, then either a = 0 or b = 0.

3.   If x and y are rational numbers and x ≠ 0, y ≠ 0, then x · y ≠ 0.

4.   Prove Theorem 3, using Theorem 1.

5.   Let r, s, t be rational numbers. If r > s, s > t, then r > t.

6.   Let r, s, t be rational numbers. If r > s and t > 0, then rt > st. If r > s and t < 0, then rt < st.

3.   THE REAL NUMBERS

As we already know, if a and b are rational numbers, then the equation a + x = b has a unique rational solution. Similarly, the equation ax = b has a unique rational solution, provided a ≠ 0. However, quadratic equations

with rational a, b, c may not have rational solutions. Such an example is given in the following theorem.

THEOREM 1. The equation x² = 2 has no rational solution.

Proof. Suppose the assertion is false, that is, suppose there is a rational number x such that x² = 2. If x > 0, then we can write x = m/n where m, n are natural numbers without a common factor. If x < 0, then we can write x = − m/n with m, n natural numbers without a common factor. In both cases we get 2 = x² = m²/n². Hence

If m is odd then m² also is odd. Since the right-hand side of (1) is even, we would get a contradiction. Hence m is even, that is, m = 2k where k is a natural number. Substituting this into (1) we find that 2n² = 4k², or n² = 2k². We now can argue as before and conclude that n is an even number. Consequently, m and n have a common factor 2, which is a contradiction.

We shall extend the concept of a number and define real numbers. It will turn out that the equation x² = 2 (and many other quadratic equations) will have a solution that is a real number.

Notation. If a is one of the elements of a set A, then we write: a ∈ A.

Consider two sets A and B of rational numbers, having the following properties:

  (I)   Every rational number belongs either to A or to B.

 (II)   Each of the sets A, B is nonempty.

(III)   Each number in A is less than each number in B.

We then call the pair (A, B) a Dedekind cut, or, briefly, a cut.

There are four possibilities:

(i) The set A contains a largest rational number α, that is, α a for all a ∈ A. The set B does not contain a smallest rational number (that is, a number that is less than or equal to every number in B).

(ii) The set B contains a smallest rational β, but the set A does not contain a largest rational number.

(iii) The set A does not contain a largest rational number, and the set B does not contain a smallest rational number.

(iv) The set A contains a largest rational α, and the set B contains a smallest rational β.

We can immediately exclude the last possibility. Indeed, if (iv) holds, then the number (α + β)/2 is a rational number that does not belong either to A or to B; this contradicts (I).

We would like to exclude in the future all cuts (A, B) for which B has a smallest rational βwhere Ā consists of β and of all the numbers in A, then the new cut will be equivalent to (A, B) in a way similar to that of a fraction a/b being equivalent to a fraction ac/bc. For the sake of convenient reference we state:

DEFINITION. A normalized cut is a cut for which either (i) or (iii) holds. If a cut (A, B) satisfies (ii), then a cut (C, D) is called a normalization of (A, B) if C consists of the smallest rational number of B and of all the rationals in A.

We can now define the concept of a real number.

DEFINITION. A normalized cut (A, B) is called a real number. If (i) holds, then we say that the real number (or the cut) is rational, and if (iii) holds, then we say that the real number (or the cut) is irrational.

We proceed to define addition and multiplication for real numbers.

DEFINITION OF ADDITION. Let (A, B) and (C, D) be real numbers. Let Ē consist of all the rational numbers a + c where a ∈ A, c ∈ Cconsist of all the rational numbers that do not belong to Ē. Let (E, F. The real number (E, F) is called the sum of (A, B) and (C, D). We write

DEFINITION OF MULTIPLICATION. Let (A, B) and (C, D) be real numbers. Consider four cases:

(a) A and C contain nonnegative rationals.

(b) A contains nonnegative rationals, but C does not contain non-negative rationals.

(c) C contains nonnegative rationals, but A does not contain non-negative rationals.

(d) Neither A nor C contains nonnegative rationals.

as follows: If (a) holds, then Ē consists of all the rationals that are less than or equal to the numbers ac where a ∈ A, c ∈ C and a 0, c 0. If (b) holds, then Ē consists of all numbers that are less than or equal to ac where a ∈ A, c ∈ C, and a 0. If (c) holds, then Ē consists of all numbers that are less than or equal to ac where a ∈ A, c ∈ C, and c consists of all the numbers that are larger than or equal to bd where b ∈ B, d ∈ D and b 0, d 0. Denote by (E, F. The real number (E, F) is called the multiplication or the product of (A, B) by (C, D). We write

We can correspond to every rational number α a real rational number (A, B), where A consists of all the rational numbers a α. This correspondence α → (A, B) is preserved by addition and multiplication, that is, if

then

It follows that the real rational numbers are the same, except for notation, as the rational numbers. From now on we shall denote real numbers also by Latin letters. When we speak of a rational number we may either mean a rational number as defined in Section 2 or a real rational number as defined in this section.

The student is certainly familiar with the intuitive concept of a real irrational number as an expanded decimal fraction

How does this concept relate to the above definition of a real number (A, B)? The answer is very simple. Take A to consist of all rational numbers a = a0 · a1a2a3 ··· such that either a0 < r0 or a0 = r0,···, ak = rk, ak + 1 < rk + 1 for some k. In this way we correspond to every irrational r an irrational cut. This correspondence is preserved by addition and multiplication.

We state without proof:

THEOREM 2. The following properties are true for all real numbers r, s, t:

A real number (A, B) is called positive if there exists a positive rational number a in A. It is called negative if it is neither positive nor zero. A number (A, B) is negative if, and only if, there exists a negative rational number in B.

More generally, we write (C, D) > (A, B. We then also write (A, B) < (C, D), and say that (C, D) is larger than (A, B) and that (A, B) is smaller than (C, D).

We write (C, D(A, B) if either (C, D) > (A, B) or (C, D) = (A, B). It is easily seen that (C, D(A, B) if, and only if, d a for all d ∈ D, a ∈ A.

We state without proof:

THEOREM 3. Let r be a real number (A, B). Then r a for all a ∈ A, and r < b for all b ∈ B. If r is irrational, then r > a for all a ∈ A.

Note that r a means: if (C, D) is the rational cut of a, then b c for any b in B and c in C

THEOREM 4. Let s and (A, B) be real numbers. If s > (A, B), then there exists a number such that . If s < (A, B), then there exists a number ā in A such that s < ā.

One can prove that there exists a unique real number x satisfying any equation of the form

We then write x = (C, D) − (A, B). Similarly, it can be proved that if (A, B) ≠ 0, then there exists a unique real solution x of the equation

We denote it by (C, D)/(A, B) or (C, D)(A, B)−1.

The following theorem is called the Archimedean law.

THEOREM 5. Let a and b be two positive real numbers. Then there exists a positive integer n such that na > b.

Proof. Denote by A the set of all the rational numbers x satisfying x < na for some positive integer n. Denote by B the set of all rational numbers not in A. Thus, y ∈ B if, and only if, y na for all positive integers n. If the assertion of the theorem is false, then the set B is nonempty. A is also nonempty since a < 2a, so that a ∈ A. We claim that (A, B) is a cut. To prove it, it remains to show that if x ∈ A, y ∈ B, then x < y. But, clearly, for some positive integer n, x < na y.

Denote by r the real number of the cut (A, B). For any positive integer m, (m + 1)a is in A since (m + 1)a < na if n is the positive integer m + 2. Hence, by Theorem 3, (m + 1)a r, or ma r − a β − a for any β ∈ B. Since m is arbitrary, β ∈ a must belong to B, so that α < β − a for any α ∈ A, β ∈ B. Denote the cut of r ∈ a by (C, D). Then D consists of all the numbers β ∈ α, β ∈ B. From the last inequality and the statement preceding Theorem 3 we then deduce that r − a r, a contradiction.

Applying Theorem

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