Sách Theory and Problems of ADVANCED CALCULUS Second Edition - ROBERT WREDE, Ph.D. MURRAY R. SPIEGEL, Ph.

Theory and Problems of ADVANCED CALCULUS (442 pages)
Second Edition
ROBERT WREDE, Ph.D.
MURRAY R. SPIEGEL, Ph.D.


Preface
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 example,
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.
ROBERT C. WREDE



Contents
CHAPTER 1 NUMBERS 1
Sets. Real numbers. Decimal representation of real numbers. Geometric
representation of real numbers. Operations with real numbers. Inequalities.
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 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 3 FUNCTIONS, LIMITS, AND CONTINUITY 39
Functions. Graph of a function. Bounded functions. Montonic functions.
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 4 DERIVATIVES 65
The concept and definition of a derivative. Right- and left-hand derivatives.
Differentiability in an interval. Piecewise differentiability. Differentials.
The differentiation of composite functions. Implicit
differentiation. Rules for differentiation. Derivatives of elementary functions.
Higher order derivatives. Mean value theorems. L’Hospital’s
rules. Applications.
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CHAPTER 5 INTEGRALS 90
Introduction of the definite integral. Measure zero. Properties of definite
integrals. Mean value theorems for integrals. Connecting integral and
differential calculus. The fundamental theorem of the calculus. Generalization
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 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 partial
derivatives. Differentials. Theorems on differentials. Differentiation
of composite functions. Euler’s theorem on homogeneous functions.
Implicit functions. Jacobians. Partial derivatives using Jacobians. Theorems
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 (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 involving r. Vector
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
Applications to geometry. Directional derivatives. Differentiation under
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.
CHAPTER 10 LINE 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. Conditions
for a line integral to be independent of the path. Surface integrals.
The divergence theorem. Stoke’s theorem.
CHAPTER 11 INFINITE SERIES 265
Definitions of infinite series and their convergence and divergence. Fundamental
facts concerning infinite series. Special series. Tests for convergence
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 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 convergent
integrals. Evaluation of definite integrals. Laplace transforms.
Linearity. Convergence. Application. Improper multiple integrals.
CHAPTER 13 FOURIER 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. Differentiation
and integration of Fourier series. Complex notation for Fourier
series. Boundary-value problems. Orthogonal functions.
CHAPTER 14 FOURIER INTEGRALS 363
The Fourier integral. Equivalent forms of Fourier’s integral theorem.
Fourier transforms.
CHAPTER 15 GAMMA AND BETA FUNCTIONS 375
The gamma function. Table of values and graph of the gamma function.
The beta function. Dirichlet integrals.
CHAPTER 16 FUNCTIONS OF A COMPLEX VARIABLE 392
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. Residues. Residue theorem. Evaluation of definite integrals.
INDEX 425