Schaum's Outline of Advanced Calculus

442 Pages·2002·7.776 MB·English

by  Robert C. Wrede, Murray Spiegel

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
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DOI: 10.1036/0071398341
Akeyingredientinlearningmathematicsisproblemsolving.Thisisthestrength,andnodoubt
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
fundamentalnotionsofthecalculus.Animportantobjectiveofthissecondeditionhasbeento
modernize terminology andconcepts, so that the interrelationships becomeclearer. 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.
IthankthestaffofMcGraw-Hill.Formereditor,GlennMott,suggestedthatItakeonthe
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.
ROBERT C. WREDE
iii
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CHAPTER 1 NUMBERS 1
Sets.Realnumbers.Decimalrepresentationofrealnumbers.Geometric
representationofrealnumbers.Operationswithrealnumbers.Inequal-
ities.Absolutevalueofrealnumbers.Exponentsandroots.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 2 SEQUENCES 23
Definition of a sequence. Limit of a sequence. Theorems on limits of
sequences.Infinity.Bounded,monotonicsequences.Leastupperbound
and greatest lower bound of a sequence. Limit superior, limit inferior.
Nested intervals. Cauchy’s convergence criterion. Infinite series.
CHAPTER 3 FUNCTIONS, 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-handlimits.Theoremsonlimits.Infinity.Speciallimits.Continuity.
Right-andleft-handcontinuity.Continuityinaninterval.Theoremson
continuity. Piecewise continuity. Uniform continuity.
CHAPTER 4 DERIVATIVES 65
Theconceptanddefinitionofaderivative.Right-andleft-handderiva-
tives. Differentiability in an interval. Piecewise differentiability. Differ-
entials. The differentiation of composite functions. Implicit
differentiation.Rulesfordifferentiation.Derivativesofelementaryfunc-
tions. Higher order derivatives. Mean value theorems. L’Hospital’s
rules. Applications.
v
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vi CONTENTS
CHAPTER 5 INTEGRALS 90
Introductionofthedefiniteintegral.Measurezero.Propertiesofdefinite
integrals. Mean value theorems for integrals. Connecting integral and
differentialcalculus.Thefundamentaltheoremofthecalculus.General-
ization of the limits of integration. Change of variable of integration.
Integrals of elementary functions. Special methods of integration.
Improperintegrals.Numericalmethodsforevaluatingdefiniteintegrals.
Applications. Arc length. Area. Volumes of revolution.
CHAPTER 6 PARTIAL 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. Differentials. Theorems on differentials. Differentiation
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 7 VECTORS 150
Vectors. Geometric properties. Algebraic properties of vectors. Linear
independence and linear dependence of a set of vectors. Unit vectors.
Rectangular(orthogonalunit)vectors.Componentsofavector.Dotor
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 involving r. Vec-
tor interpretation of Jacobians, Orthogonal curvilinear coordinates.
Gradient, divergence, curl, and Laplacian in orthogonal curvilinear
coordinates. Special curvilinear coordinates.
CHAPTER 8 APPLICATIONS OF PARTIAL DERIVATIVES 183
Applicationstogeometry.Directionalderivatives.Differentiationunder
the integral sign. Integration under the integral sign. Maxima and
minima. Method of Lagrange multipliers for maxima and minima.
Applications to errors.
CHAPTER 9 MULTIPLE INTEGRALS 207
Double integrals. Iterated integrals. Triple integrals. Transformations
of multiple integrals. The differential element of area in polar
coordinates, differential elements of area in cylindrical and spherical
coordinates.
CONTENTS vii
CHAPTER 10 LINE INTEGRALS, SURFACE INTEGRALS, AND
INTEGRAL THEOREMS 229
Line integrals. Evaluation of line integrals for plane curves. Properties
oflineintegralsexpressedforplanecurves.Simpleclosedcurves,simply
and multiply connected regions. Green’s theorem in the plane. Condi-
tionsforalineintegraltobeindependentofthepath.Surfaceintegrals.
The divergence theorem. Stoke’s theorem.
CHAPTER 11 INFINITE SERIES 265
Definitionsofinfiniteseriesandtheirconvergenceanddivergence.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
onuniformlyconvergentseries.Powerseries.Theoremsonpowerseries.
Operations with power series. Expansion of functions in power series.
Taylor’stheorem.Someimportantpowerseries.Specialtopics.Taylor’s
theorem (for two variables).
CHAPTER 12 IMPROPER 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 13 FOURIER SERIES 336
Periodicfunctions.Fourierseries.Orthogonalityconditionsforthesine
and cosine functions. Dirichlet conditions. Odd and even functions.
HalfrangeFouriersineorcosineseries.Parseval’sidentity.Differentia-
tion and integration of Fourier series. Complex notation for Fourier
series. Boundary-value problems. Orthogonal functions.
viii CONTENTS
CHAPTER 14 FOURIER INTEGRALS 363
The Fourier integral. Equivalent forms of Fourier’s integral theorem.
Fourier transforms.
CHAPTER 15 GAMMA AND BETA FUNCTIONS 375
Thegammafunction.Tableofvaluesandgraphofthegammafunction.
The beta function. Dirichlet integrals.
CHAPTER 16 FUNCTIONS OF A COMPLEX VARIABLE 392
Functions. Limits and continuity. Derivatives. Cauchy-Riemann equa-
tions.Integrals.Cauchy’stheorem.Cauchy’sintegralformulas.Taylor’s
series. Singular points. Poles. Laurent’s series. Branches and branch
points. Residues. Residue theorem. Evaluation of definite integrals.
INDEX 425
Numbers
Mathematicshasitsownlanguagewithnumbersasthealphabet. Thelanguageisgivenstructure
withtheaidofconnectivesymbols,rulesofoperation,andarigorousmodeofthought(logic). These
concepts,whichpreviouslywereexploredinelementarymathematicscoursessuchasgeometry,algebra,
and calculus, are reviewed in the following paragraphs.
SETS
Fundamental in mathematics is the concept of a set, class, or collection of objects having specified
characteristics. For example, we speak of the set of all university professors, the set of all letters
A;B;C;D;...;Z of the English alphabet, and so on. The individual objects of the set are called
members or elements. Any part of a set is called a subset of the given set, e.g., A, B, C is a subset of
A;B;C;D;...;Z. The set consisting of no elements is called the empty set or null set.
REAL NUMBERS
The following types of numbers are already familiar to the student:
1. Natural numbers 1;2;3;4;...; also called positive integers, are used in counting members of a
set. Thesymbolsvariedwiththetimes,e.g.,theRomansusedI,II,III,IV, . . . Thesumaþb
and product a(cid:1)b or ab of any two natural numbers a and b is also a natural number. This is
often expressed by saying that the set of natural numbers is closed under the operations of
addition and multiplication, or satisfies the closure property with respect to these operations.
2. Negativeintegersandzerodenotedby(cid:2)1;(cid:2)2;(cid:2)3;...and0,respectively,arosetopermitsolu-
tionsofequationssuchasxþb¼a,whereaandbareanynaturalnumbers. Thisleadstothe
operation of subtraction, or inverse of addition, and we write x¼a(cid:2)b.
The set of positive and negative integers and zero is called the set of integers.
3. Rationalnumbersorfractionssuchas2,(cid:2)5, . . .arosetopermitsolutionsofequationssuchas
3 4
bx¼aforallintegersaandb,whereb6¼0. Thisleadstotheoperationofdivision,orinverseof
multiplication, and we write x¼a=b or a(cid:3)b where a is the numerator and b the denominator.
Thesetofintegersisasubsetoftherationalnumbers,sinceintegerscorrespondtorational
numbers where b¼1.
pffiffiffi
4. Irrational numbers such as 2 and (cid:1) are numbers which are not rational, i.e., they cannot be
expressed as a=b (called the quotient of a and b), where a and b are integers and b6¼0.
The set of rational and irrational numbers is called the set of real numbers.
1
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