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Theory and Problems of

ADVANCED

CALCULUS

Second Edition

ROBERT WREDE, Ph.D.

MURRAY R. SPIEGEL, Ph.D.

Former Professor and Chairman of Mathematics

Rensselaer Polytechnic Institute

Hartford Graduate Center

Schaum"s Outline Series

McGRAW-HILL

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Copyright © 2002, 1963 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of

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DOI: 10.1036/0071398341

iii A key ingredient in learning mathematics is problem solving. This is the strength, and no doubt the reason for the longevity of Professor Spiegel"s advanced calculus. His collection of solved and unsolved problems remains a part of this second edition. Advanced calculus is not a single theory. However, the various sub-theories, including vector analysis, infinite series, and special functions, have in common a dependency on the fundamental notions of the calculus. An important objective of this second edition has been to modernize terminology and concepts, so that the interrelationships become clearer. For exam- ple, in keeping with present usage fuctions of a real variable are automatically single valued; differentials are defined as linear functions, and the universal character of vector notation and theory are given greater emphasis. Further explanations have been included and, on occasion, the appropriate terminology to support them. The order of chapters is modestly rearranged to provide what may be a more logical structure. A brief introduction is provided for most chapters. Occasionally, a historical note is included; however, for the most part the purpose of the introductions is to orient the reader to the content of the chapters. I thank the staff of McGraw-Hill. Former editor, Glenn Mott, suggested that I take on the project. Peter McCurdy guided me in the process. Barbara Gilson, Jennifer Chong, and Elizabeth Shannon made valuable contributions to the finished product. Joanne Slike and Maureen Walker accomplished the very difficult task of combining the old with the new and, in the process, corrected my errors. The reviewer, Glenn Ledder, was especially helpful in the choice of material and with comments on various topics.

ROBERTC. WREDE

Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for T erms of Use. ffiffff ff ff v

CHAPTER 1NUMBERS 1

Sets. Real numbers. Decimal representation of real numbers. Geometric representation of real numbers. Operations with real numbers. Inequal- ities. Absolute value of real numbers. Exponents and roots. Logarithms. Axiomatic foundations of the real number system. Point sets, intervals. Countability. Neighborhoods. Limit points. Bounds. Bolzano- Weierstrass theorem. Algebraic and transcendental numbers. The com- plex number system. Polar form of complex numbers. Mathematical induction.

CHAPTER 2SEQUENCES 23

Definition of a sequence. Limit of a sequence. Theorems on limits of sequences. Infinity. Bounded, monotonic sequences. Least upper bound and greatest lower bound of a sequence. Limit superior, limit inferior. Nested intervals. Cauchy"s convergence criterion. Infinite series.

CHAPTER 3FUNCTIONS, LIMITS, AND CONTINUITY 39

Functions. Graph of a function. Bounded functions. Montonic func- tions. Inverse functions. Principal values. Maxima and minima. Types of functions. Transcendental functions. Limits of functions. Right- and left-hand limits. Theorems on limits. Infinity. Special limits. Continuity. Right- and left-hand continuity. Continuity in an interval. Theorems on continuity. Piecewise continuity. Uniform continuity.

CHAPTER 4DERIVATIVES 65

The concept and definition of a derivative. Right- and left-hand deriva- tives. Differentiability in an interval. Piecewise differentiability. Differ- entials. The differentiation of composite functions. Implicit differentiation. Rules for differentiation. Derivatives of elementary func- tions. Higher order derivatives. Mean value theorems. L"Hospital"s rules. Applications. For more information about this title, click here. Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for T?erms of Use.

CHAPTER 5INTEGRALS 90

Introduction of the definite integral. Measure zero. Properties of definite integrals. Mean value theorems for integrals. Connecting integral and diffierential calculus. The fundamental theorem of the calculus. General- ization of the limits of integration. Change of variable of integration. Integrals of elementary functions. Special methods of integration. Improper integrals. Numerical methods for evaluating definite integrals. Applications. Arc length. Area. Volumes of revolution.

CHAPTER 6PARTIAL DERIVATIVES 116

Functions of two or more variables. Three-dimensional rectangular coordinate systems. Neighborhoods. Regions. Limits. Iterated limits. Continuity. Uniform continuity. Partial derivatives. Higher order par- tial derivatives. Diffierentials. Theorems on diffierentials. Diffierentiation of composite functions. Euler"s theorem on homogeneous functions. Implicit functions. Jacobians. Partial derivatives using Jacobians. The- orems on Jacobians. Transformation. Curvilinear coordinates. Mean value theorems.

CHAPTER 7VECTORS 150

Vectors. Geometric properties. Algebraic properties of vectors. Linear independence and linear dependence of a set of vectors. Unit vectors. Rectangular (orthogonal unit) vectors. Components of a vector. Dot or scalar product. Cross or vector product. Triple products. Axiomatic approach to vector analysis. Vector functions. Limits, continuity, and derivatives of vector functions. Geometric interpretation of a vector derivative. Gradient, divergence, and curl. Formulas involvingr.Vec- tor interpretation of Jacobians, Orthogonal curvilinear coordinates. Gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates. Special curvilinear coordinates.

CHAPTER 8APPLICATIONS OF PARTIAL DERIVATIVES 183

Applications to geometry. Directional derivatives. Diffierentiation under the integral sign. Integration under the integral sign. Maxima and minima. Method of Lagrange multipliers for maxima and minima.

Applications to errors.

CHAPTER 9MULTIPLE INTEGRALS 207

Double integrals. Iterated integrals. Triple integrals. Transformations of multiple integrals. The diffierential element of area in polar coordinates, diffierential elements of area in cylindrical and spherical coordinates. viCONTENTS

CHAPTER 10LINE INTEGRALS, SURFACE INTEGRALS, AND

INTEGRAL THEOREMS 229

Line integrals. Evaluation of line integrals for plane curves. Properties of line integrals expressed for plane curves. Simple closed curves, simply and multiply connected regions. Green"s theorem in the plane. Condi- tions for a line integral to be independent of the path. Surface integrals.

The divergence theorem. Stoke"s theorem.

CHAPTER 11INFINITE SERIES 265

Definitions of infinite series and their convergence and divergence. Fun- damental facts concerning infinite series. Special series. Tests for con- vergence and divergence of series of constants. Theorems on absolutely convergent series. Infinite sequences and series of functions, uniform convergence. Special tests for uniform convergence of series. Theorems on uniformly convergent series. Power series. Theorems on power series. Operations with power series. Expansion of functions in power series. Taylor"s theorem. Some important power series. Special topics. Taylor"s theorem (for two variables).

CHAPTER 12IMPROPER INTEGRALS 306

Definition of an improper integral. Improper integrals of the first kind (unbounded intervals). Convergence or divergence of improper integrals of the first kind. Special improper integers of the first kind. Convergence tests for improper integrals of the first kind. Improper integrals of the second kind. Cauchy principal value. Special improper integrals of the second kind. Convergence tests for improper integrals of the second kind. Improper integrals of the third kind. Improper integrals containing a parameter, uniform convergence. Special tests for uniform convergence of integrals. Theorems on uniformly conver- gent integrals. Evaluation of definite integrals. Laplace transforms. Linearity. Convergence. Application. Improper multiple integrals.

CHAPTER 13FOURIER SERIES 336

Periodic functions. Fourier series. Orthogonality conditions for the sine and cosine functions. Dirichlet conditions. Odd and even functions. Half range Fourier sine or cosine series. Parseval"s identity. Diffierentia- tion and integration of Fourier series. Complex notation for Fourier series. Boundary-value problems. Orthogonal functions.

CONTENTSvii

CHAPTER 14FOURIER INTEGRALS 363

The Fourier integral. Equivalent forms of Fourier"s integral theorem.

Fourier transforms.

CHAPTER 15GAMMA AND BETA FUNCTIONS 375

The gamma function. Table of values and graph of the gamma function.

The beta function. Dirichlet integrals.

CHAPTER 16FUNCTIONS OF A COMPLEX VARIABLE 392

Functions. Limits and continuity. Derivatives. Cauchy-Riemann equa- tions. Integrals. Cauchy"s theorem. Cauchy"s integral formulas. Taylor"s series. Singular points. Poles. Laurent"s series. Branches and branch points. Residues. Residue theorem. Evaluation of definite integrals.

INDEX 425

viiiCONTENTS 1

Numbers

Mathematics has its own language with numbers as the alphabet. The language is given structure

with the aid of connective symbols, rules of operation, and a rigorous mode of thought (logic). These

concepts, which previously were explored in elementary mathematics courses such as geometry, algebra,

and calculus, are reviewed in the following paragraphs. SETS Fundamental in mathematics is the concept of aset,class,orcollectionof objects having specified

characteristics. For example, we speak of the set of all university professors, the set of all letters

A;B;C;D;...;Zof the English alphabet, and so on. The individual objects of the set are called membersorelements. Any part of a set is called asubsetof the given set, e.g.,A,B,Cis a subset of A;B;C;D;...;Z. The set consisting of no elements is called theempty setornull set.

REAL NUMBERS

The following types of numbers are already familiar to the student:

1.Natural numbers1;2;3;4;...;also calledpositive integers, are used in counting members of a

set. The symbols varied with the times, e.g., the Romans used I, II, III, IV, . . . Thesumaþb andproductaborabof any two natural numbersaandbis also a natural number. This is often expressed by saying that the set of natural numbers isclosedunder the operations of additionandmultiplication,orsatisfies theclosure propertywith respect to these operations.

2.Negative integers and zerodenoted by1;2;3;...and 0, respectively, arose to permit solu-

tions of equations such asxþb¼a, whereaandbare any natural numbers. This leads to the operation ofsubtraction,orinverse of addition,andwewritex¼ab. The set of positive and negative integers and zero is called the set ofintegers.

3.Rational numbersorfractionssuch as

23
5 4 ,...arose to permit solutions of equations such as bx¼afor all integersaandb, whereb6¼0. This leads to the operation ofdivision,orinverse of multiplication,andwe writex¼a=borabwhereais thenumeratorandbthedenominator. The set of integers is a subset of the rational numbers, since integers correspond to rational numbers whereb¼1.

4.Irrational numberssuch as???2pandffiare numbers which are not rational, i.e., they cannot be

expressed asa=b(called thequotientofaandb), whereaandbare integers andb6¼0. The set of rational and irrational numbers is called the set ofreal numbers. Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for T erms of Use.

DECIMAL REPRESENTATION OF REAL NUMBERS

Any real number can be expressed indecimal form,e.g., 17=10¼1:7, 9=100¼0:09,

1=6¼0:16666....Inthe case of a rational number the decimal exapnsion either terminates, or if it

does not terminate, one or a group of digits in the expansion will ultimately repeat, as for example, in

1 7 ¼0:142857142857142....Inthe case of an irrational number such asffiffiffi2p¼1:41423...or ?¼3:14159...no such repetition can occur. We can always consider a decimal expansion as unending,

e.g., 1.375 is the same as 1.37500000 . . . or 1.3749999 . . . . To indicate recurring decimals we some-

times place dots over the repeating cycle of digits, e.g., 1 7

¼0:_11_44_22_88_55_77,

19 6

¼3:1_66.

The decimal system uses the ten digits 0;1;2;...;9. (These symbols were the gift of the Hindus.

They were in use in India by 600

A.D.and then in ensuing centuries were transmitted to the western world by Arab traders.) It is possible to design number systems with fewer or more digits, e.g. thebinary systemuses only two digits 0 and 1 (see Problems 32 and 33).

GEOMETRIC REPRESENTATION OF REAL NUMBERS

The geometric representation of real numbers as points on a line called thereal axis,asinthe figure below, is also well known to the student. For each real number there corresponds one and only one

point on the line and conversely, i.e., there is aone-to-one(see Fig. 1-1)correspondencebetween the set of

real numbers and the set of points on the line. Because of this we often use point and number interchangeably. (The interchangeability of point and number is by no means self-evident; in fact, axioms supporting the relation of geometry and numbers are necessary. The Cantor...Dedekind Theorem is fundamental.)

The set of real numbers to the right of 0 is called the set ofpositive numbers;the set to the left of 0 is

the set ofnegative numbers,while 0 itself is neither positive nor negative. (Both the horizontal position of the line and the placement of positive and negative numbers to the right and left, respectively, are conventions.) Between any two rational numbers (or irrational numbers) on the line there are infinitely many

rational (and irrational) numbers. This leads us to call the set of rational (or irrational) numbers an

everywhere denseset.

OPERATIONS WITH REAL NUMBERS

Ifa,b,cbelong to the setRof real numbers, then:

1.aþbandabbelong toRClosure law

2.aþb¼bþaCommutative law of addition

3.aþðbþcÞ¼ðaþbÞþcAssociative law of addition

4.ab¼baCommutative law of multiplication

5.aðbcÞ¼ðabÞcAssociative law of multiplication

6.aðbþcÞ¼abþacDistributive law

0iscalled theidentity with respect to addition,1is called theidentity with respect to multi-

plication.2

NUMBERS [CHAP. 1

_5_4_3_2_10 1 3 4 521 2 4

3__ppe2

Fig. 1-1

8. For anyathere is a numberxinRsuch thatxþa¼0.

xis called theinverse of a with respect to additionand is denoted by?a.

9. For anya6¼0there is a numberxinRsuch thatax¼1.

xis called theinverse of a with respect to multiplicationand is denoted bya ?1 or 1=a. Convention: For convenience, operations called subtraction and division are defined by a?b¼aþð?bÞand a b

¼ab

?1 ,respectively. These enable us to operate according to the usual rules of algebra. In general any set, such asR, whose members satisfy the above is called a“eld.

INEQUALITIES

Ifa?bis a nonnegative number, we say thataisgreater than or equal to borbisless than or equal to a,and write, respectively,aAborb%a.Ifthere is no possibility thata¼b,wewritea>borbbif the point on the real axis corresponding toalies to the right of the point corresponding tob. EXAMPLES.3<5or5>3;?2?2;x@3means thatxis a real number which may be 3 or less than 3.

Ifa,b;andcare any given real numbers, then:

1. Eithera>b,a¼bora

2. Ifa>bandb>c,thena>cLaw of transitivity

3. Ifa>b,thenaþc>bþc

4. Ifa>bandc>0, thenac>bc

5. Ifa>bandc<0, thenac

ABSOLUTE VALUE OF REAL NUMBERS

The absolute value of a real numbera,denoted byjaj,isdefined asaifa>0,?aifa<0, and 0 if a¼0.

EXAMPLES.j?5j¼5,jþ2j¼2,j?

3 4 j¼ 3 4 ,j?2pj¼2p,j0j¼0.

3.ja?bjAjaj?jbj

The distance between any two points (real numbers)aandbon the real axis isja?bj¼jb?aj.

EXPONENTS AND ROOTS

The productaffia...aof a real numberaby itselfptimes is denoted bya p ,wherepis called the exponentandais called thebase.The following rules hold: 1.a p ffia q ¼a pþq

3.ða

p r ¼a pr 2. aquotesdbs_dbs29.pdfusesText_35

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