# Schaum's Outline of Advanced Calculus, Third Edition (Schaum's

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SCHAUMSoutlines

Advanced CalculusThird 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 InstituteHartford Graduate Center

Schaums 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 Spiegels 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 1Sets. 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 25Definition 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. Cauchys Convergence Criterion. Infinite Series.

Chapter 3 FUNCTIONS, LIMITS, AND CONTINUITY 43Functions. 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 71The 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. LHospitals Rules. Applica-tions.

Chapter 5 INTEGRALS 97Introduction 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.

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Chapter 6 PARTIAL DERIVATIVES 125Functions 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. Eulers Theorem on Homogeneous Functions. Implicit Functions. Jacobians. Partial Derivatives Using Jacobi-ans. Theorems on Jacobians. Transformations. Curvilinear Coordinates. Mean Value Theorems.

Chapter 7 VECTORS 161Vectors. 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 195Applications 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 221Double Integrals. Iterate