Table Of ContentSCHAUM’S
outlines
Advanced Calculus
Third Edition
Robert Wrede, Ph.D.
Professor Emeritus of Mathematics
San Jose State University
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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Preface to the
Third Edition
The many problems and solutions provided by the late Professor Spiegel remain invaluable to students as
they seek to master the intricacies of the calculus and related fields of mathematics. These remain an integral
part of this manuscript. In this third edition, clarifications have been provided. In addition, the continuation
of the interrelationships and the significance of concepts, begun in the second edition, have been extended.
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Preface to the
Second Edition
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 cal-
culus. An important objective of this second edition has been to modernize terminology and concepts, so that
the interrelationships become clearer. For example, in keeping with present usage functions 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 oc-
casion, 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.
ROBERT C. WREDE
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Contents
Chapter 1 NUMBERS 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 Complex Number
System. Polar Form of Complex Numbers. Mathematical Induction.
Chapter 2 SEQUENCES 25
Definition of a Sequence. Limit of a Sequence. Theorems on Limits of Se-
quences. 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 3 FUNCTIONS, LIMITS, AND CONTINUITY 43
Functions. Graph of a Function. Bounded Functions. Montonic Functions.
Inverse Functions, Principal Values. Maxima and Minima. Types of Func-
tions. 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 4 DERIVATIVES 71
The Concept and Definition of a Derivative. Right- and Left-Hand Deriva-
tives. Differentiability in an Interval. Piecewise Differentiability. Differen-
tials. The Differentiation of Composite Functions. Implicit Differentiation.
Rules for Differentiation. Derivatives of Elementary Functions. Higher-
Order Derivatives. Mean Value Theorems. L’Hospital’s Rules. Applica-
tions.
Chapter 5 INTEGRALS 97
Introduction of the Definite Integral. Measure Zero. Properties of Definite
Integrals. Mean Value Theorems for Integrals. Connecting Integral and Dif-
ferential Calculus. The Fundamental Theorem of the Calculus. Generaliza-
tion of the Limits of Integration. Change of Variable of Integration.
Integrals of Elementary Functions. Special Methods of Integration. Im-
proper Integrals. Numerical Methods for Evaluating Definite Integrals. Ap-
plications. Arc Length. Area. Volumes of Revolution.
vii
viii Contents
Chapter 6 PARTIAL DERIVATIVES 125
Functions of Two or More Variables. Neighborhoods. Regions. Limits. Iter-
ated Limits. Continuity. Uniform Continuity. Partial Derivatives. Higher-
Order Partial Derivatives. Differentials. Theorems on Differentials.
Differentiation of Composite Functions. Euler’s Theorem on Homogeneous
Functions. Implicit Functions. Jacobians. Partial Derivatives Using Jacobi-
ans. Theorems on Jacobians. Transformations. Curvilinear Coordinates.
Mean Value Theorems.
Chapter 7 VECTORS 161
Vectors. Geometric Properties of Vectors. Algebraic Properties of Vectors.
Linear Independence and Linear Dependence of a Set of Vectors. Unit Vec-
tors. Rectangular (Orthogonal) Unit Vectors. Components of a Vector. Dot,
Scalar, or Inner Product. Cross or Vector Product. Triple Products. Axiom-
atic 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 ∇. Vector
Interpretation of Jacobians and Orthogonal Curvilinear Coordinates. Gra-
dient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordi-
nates. Special Curvilinear Coordinates.
Chapter 8 APPLICATIONS OF PARTIAL DERIVATIVES 195
Applications To Geometry. Directional Derivatives. Differentiation Under
the Integral Sign. Integration Under the Integral Sign. Maxima and Min-
ima. Method of Lagrange Multipliers for Maxima and Minima. Applica-
tions To Errors.
Chapter 9 MULTIPLE INTEGRALS 221
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.
Chapter 10 LINE INTEGRALS, SURFACE INTEGRALS, AND
INTEGRAL THEOREMS 243
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. Conditions
for a Line Integral To Be Independent of the Path. Surface Integrals. The
Divergence Theorem. Stokes’s Theorem.
Chapter 11 INFINITE SERIES 279
Definitions of Infinite Series and Their Convergence and Divergence. Fun-
damental Facts Concerning Infinite Series. Special Series. Tests for Conver-
gence 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. Tay-
lor’s Theorem. Some Important Power Series. Special Topics. Taylor’s Theo-
rem (For Two Variables).
Contents ix
Chapter 12 IMPROPER INTEGRALS 321
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 Convergent Integrals. Evaluation of
Definite Integrals. Laplace Transforms. Linearity. Convergence. Applica-
tion. Improper Multiple Integrals.
Chapter 13 FOURIER SERIES 349
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. Differentiation
and Integration of Fourier Series. Complex Notation for Fourier Series.
Boundary-Value Problems. Orthogonal Functions.
Chapter 14 FOURIER INTEGRALS 377
The Fourier Integral. Equivalent Forms of Fourier’s Integral Theorem.
Fourier Transforms.
Chapter 15 GAMMA AND BETA FUNCTIONS 389
The Gamma Function. Table of Values and Graph of the Gamma Func-
tion. The Beta Function. Dirichlet Integrals.
Chapter 16 FUNCTIONS OF A COMPLEX VARIABLE 405
Functions. Limits and Continuity. Derivatives. Cauchy-Riemann Equations.
Integrals. Cauchy’s Theorem. Cauchy’s Integral Formulas. Taylor’s Series.
Singular Points. Poles. Laurent’s Series. Branches and Branch Points. Resi-
dues. Residue Theorem. Evaluation of Definite Integrals.
INDEX 437
