Schaum's Outline of Advanced Calculus, Third Edition (Schaum's Outline Series PDF

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SCHAUM’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 New York Chicago ...

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SCHAUM’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 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 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 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

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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. 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

25

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

43

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 LeftHand 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 Derivatives. Differentiability in an Interval. Piecewise Differentiability. Differentials. The Differentiation of Composite Functions. Implicit Differentiation. Rules for Differentiation. Derivatives of Elementary Functions. HigherOrder Derivatives. Mean Value Theorems. L’Hospital’s Rules. Applications.

Chapter 5

INTEGRALS

97

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.

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Contents

viii Chapter 6

PARTIAL DERIVATIVES

125

Functions of Two or More Variables. Neighborhoods. Regions. Limits. Iterated Limits. Continuity. Uniform Continuity. Partial Derivatives. HigherOrder 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. 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 Vectors. Rectangular (Orthogonal) Unit Vectors. Components of a Vector. Dot, Scalar, or Inner 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 ∇. Vector Interpretation of Jacobians and Orthogonal Curvilinear Coordinates. Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates. 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 Minima. Method of Lagrange Multipliers for Maxima and Minima. Applications 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 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).

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Contents

Chapter 12

ix 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. Application. 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 Function. 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. Residues. Residue Theorem. Evaluation of Definite Integrals.

INDEX

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C HA P T E R 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 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 number system is foundational to the modern scientific and technological world. It is based on the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Thus, it is called a base ten system. (There is the implication that there are other systems. One of these, which is of major importance, is the base two system.) The symbols were introduced by the Hindus, who had developed decimal representation and the arithmetic of positive numbers by 600 A.D. In the eighth century, the House of Wisdom (library) had been established in Baghdad, and it was there that the Hindu arithmetic and much of the mathematics of the Greeks were translated into Arabic. From there, this arithmetic gradually spread to the later-developing western civilization. The flexibility of the Hindu-Arabic number system lies in the multiple uses of the numbers. They may be used to signify: (a) order—the runner finished fifth; (b) quantity—there are six apples in the barrel; (c) construction—2 and 3 may be used to form any of 23, 32, .23 or .32; (d) place—0 is used to establish place, as is illustrated by 607, 0603, and .007. Finally, note that the significance of the base ten terminology is enhanced by the following examples: 357 = 7(100) + 5(101) + 3(102) 9 7 2 .972 = ᎏ + ᎏ2 ++ ᎏ3 10 10 10

The collection of numbers created from the basic set is called the real number system. Significant subsets of them are listed as follows. For the purposes of this text, it is assumed that the reader is familiar with these numbers and the fundamental arithmetic operations. 1.

Natural numbers 1, 2, 3, 4, . . . , also called positive integers, are used in counting members of a set. The symbols varied with the times; e.g., the Romans used I, II, III, IV, . . . . The sum a + b and product a · b or ab of any two natural numbers a and b is also a natural number. This is often expressed by

1

CHAPTER 1 Numbers

2

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.

3.

4.

Negative integers and zero, denoted by –1, –2, –3, . . . , and 0, respectively, arose to permit solutions of equations such as x + b = a, where a and b are any natural numbers. This leads to the operation of subtraction, or inverse of addition, and we write x = a – b. The set of positive and negative integers and zero is called the set of integers. 2 5 Rational numbers or fractions such as , − , . . . arose to permit solutions of equations such as 3 4 bx = a for all integers a and b, where b ⫽ 0. This leads to the operation of division, or inverse of multiplication, and we write x = a/b or a ÷ b, where a is the numerator and b the denominator. The set of integers is a subset of the rational numbers, since integers correspond to rational numbers where b = 1. Irrational numbers such as 2 and π 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 b ⫽ 0. The set of rational and irrational numbers is called the set of real numbers.

Decimal Representation of Real Numbers Any real number can be expressed in decimal form, e.g., 17/10 = 1.7, 9/100 = 0.09, 1/6 = 0.16666. . . . In the case of a rational number, the decimal expansion either terminates or if it does not terminate, one or a group 1 = 0.142857 142857 142. . . . In the of digits in the expansion will ultimately repeat, as, for example, in 7 case of an irrational number such as 2 = 1.41423 . . . or π = 3.14159 . . . 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 sometimes place dots over the repeating cycle of digits, 1 19 e.g., = 0.1˙4˙2˙ 8˙ 5˙ 7˙, and = 3.16˙ . 7 6 It is possible to design number systems with fewer or more digits; e.g., the binary system uses only two digits, 0 and 1 (see Problems 1.32 and 1.33).

Geometric Representation of Real Numbers The geometric representation of real numbers as points on a line, called the real axis, as in Figure 1.1, is also well known to the student. For each real number there corresponds one and only one point on the line, and, conversely, there is a one-to-one (see Figure 1.1) correspondence between the set of real numbers and the set of points on the line. Because of this we often use point and number interchangeably.

Figure 1.1

While this correlation of points and numbers is automatically assumed in the elementary study of mathematics, it is actually an axiom of the subject (the Cantor Dedekind axiom) and, in that sense, has deep meaning. The set of real numbers to the right of 0 is called the set of positive numbers, the set to the left of 0 is the set of negative numbers, while 0 itself is neither positive nor negative.

CHAPTER 1

Numbers

3

(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 dense set.

Operations with Real Numbers If a, b, c belong to the set R of real numbers, then: 1.

a + b and ab belong to R

Closure law

2.

a+b=b+a

Commutative law of addition

3.

a + (b + c) = (a + b) + c

Associative la...