Multivariable Calculus

by Edwards, C. Henry; Penney, David E.

• ISBN13: 9780130339676
• ISBN10: 0130339679
• Edition: 6th
• Format: Paperback
• Copyright: 2002-05-21
• Publisher: Pearson

Summary

This book combines traditional mainstream calculus with the most flexible approach to new ideas and calculator/computer technology. It contains superb problem sets and a fresh conceptual emphasis flavored by new technological possibilities. Chapter topics cover polar coordinates and parametric curves, infinite series; vectors and matrices, curves and surfaces in space, partial differentiation, multiple integrals, and vector calculus. For individuals interested in the study of calculus.

Author Biography

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

Table of Contents

About the Authors	xi
Preface	xiii
Functions, Graphs, and Models	1	(52)
Functions and Mathematical Modeling	2	(10)
Project: A Square Wading Pool	11	(1)
Graphs of Equations and Functions	12	(12)
Project: A Broken Tree	23	(1)
Polynomials and Algebraic Functions	24	(9)
Project: A Leaning Ladder	33	(1)
Transcendental Functions	33	(12)
Project: A Spherical Asteroid	45	(1)
Preview: What Is Calculus?	45	(8)
Review: Definitions and Concepts	48	(5)
Prelude to Calculus	53	(48)
Tangent Lines and Slope Predictors	54	(9)
Project: Numerical Slope Investigations	63	(1)
The Limit Concept	63	(12)
Project: Limits, Slopes, and Logarithms	74	(1)
More About Limits	75	(13)
Project: Numerical Epsilon-Delta Limit Investigations	87	(1)
The Concept of Continuity	88	(13)
Review: Definitions, Concepts, Results	99	(2)
The Derivative	101	(92)
The Derivative and Rates of Change	102	(13)
Basic Differentiation Rules	115	(11)
The Chain Rule	126	(7)
Derivatives of Algebraic Functions	133	(9)
Maxima and Minima of Functions on Closed Intervals	142	(10)
Project: When is Your Coffee Cup Stablest?	150	(2)
Applied Optimization Problems	152	(13)
Derivatives of Trigonometric Functions	165	(11)
Successive Approximations and Newton's Method	176	(17)
Project: How Deep Does a Floating Ball Sink?	188	(1)
Review: Formulas, Concepts, Definitions	189	(4)
Additional Applications of the Derivative	193	(78)
Implicit Functions and Related Rates	194	(10)
Project: Investigating the Folium of Descartes	203	(1)
Increments, Differentials, and Linear Approximation	204	(8)
Increasing and Decreasing Functions and the Mean Value Theorem	212	(10)
The First Derivative Test and Applications	222	(10)
Project: Making a Candy Box With Lid	232	(1)
Simple Curve Sketching	232	(10)
Higher Derivatives and Concavity	242	(14)
Curve Sketching and Asymptotes	256	(15)
Project: Locating Special Points on Exotic Graphs	267	(1)
Review: Definitions, Concepts, Results	267	(4)
The Integral	271	(94)
Introduction	272	(1)
Antiderivatives and Initial Value Problems	272	(14)
Elementary Area Computations	286	(12)
Riemann Sums and the Integral	298	(10)
Project: Calculator/Computer Riemann Sums	307	(1)
Evaluation of Integrals	308	(10)
The Fundamental Theorem of Calculus	318	(10)
Integration by Substitution	328	(8)
Areas of Plane Regions	336	(10)
Numerical Integration	346	(19)
Project: Trapezoidal and Simpson Approximations	359	(2)
Review: Definitions, Concepts, Results	361	(4)
Applications of the Integral	365	(62)
Riemann Sum Approximations	366	(10)
Volumes by the Method of Cross Sections	376	(11)
Volumes by the Method of Cylindrical Shells	387	(9)
Project: Design Your Own Ring!	395	(1)
Arc Length and Surface Area of Revolution	396	(9)
Force and Work	405	(11)
Centroids of Plane Regions and Curves	416	(11)
Review: Definitions, Concepts, Results	423	(4)
Calculus of Transcendental Functions	427	(62)
Exponential and Logarithmic Functions	428	(13)
Project: Discovering the Number e for Yourself	441	(1)
Indeterminate Forms and L'Hopital's Rule	441	(8)
More Indeterminate Forms	449	(6)
The Natural Logarithm as an Integral	455	(12)
Project: Natural Functional Equations	466	(1)
Inverse Trigonometric Functions	467	(10)
Hyperbolic Functions	477	(12)
Review: Formulas, Concepts, Definitions	485	(4)
Techniques of Integration	489	(56)
Introduction	490	(1)
Integral Tables and Simple Substitutions	490	(4)
Integration by Parts	494	(7)
Trigonometric Integrals	501	(7)
Rational Functions and Partial Fractions	508	(7)
Trigonometric Substitution	515	(6)
Integrals Involving Quadratic Polynomials	521	(5)
Improper Integrals	526	(19)
Summary	539	(6)
Differential Equations	545	(78)
Simple Equations and Models	546	(12)
Slope Fields and Euler's Method	558	(10)
Project: Computer-Assisted Slope Fields and Euler's Method	567	(1)
Separable Equations and Applications	568	(7)
Linear Equations and Applications	575	(12)
Population Models	587	(11)
Project: Predator-Prey Equations and Your Own Game Preserve	597	(1)
Linear Second-Order Equations	598	(9)
Mechanical Vibrations	607	(16)
Review: Definitions, Concepts, Results	618	(5)
Polar Coordinates and Parametric Curves	623	(58)
Analytic Geometry and the Conic Sections	624	(5)
Polar Coordinates	629	(9)
Area Computations in Polar Coordinates	638	(5)
Parametric Curves	643	(10)
Project:: Trochoid Investigations	652	(1)
Integral Computations with Parametric Curves	653	(8)
Project:: Moon Orbits and Race Tracks	660	(1)
Conic Sections and Applications	661	(20)
Review: Concepts and Definitions	679	(2)
Infinite Series	681	(90)
Introduction	682	(1)
Infinite Sequences	682	(9)
Project:: Nested Radicals and Continued Fractions	691	(1)
Infinite Series and Convergence	691	(11)
Project:: Numerical Summation and Geometric Series	701	(1)
Taylor Series and Taylor Polynomials	702	(13)
Project:: Calculating Logarithms on a Deserted Island	715	(1)
The Integral Test	715	(7)
Project:: The Number π, Once and for All	722	(1)
Comparison Tests for Positive-Term Series	722	(6)
Alternating Series and Absolute Convergence	728	(9)
Power Series	737	(13)
Power Series Computations	750	(8)
Project:: Calculating Trigonometric Functions on a Deserted Island	758	(1)
Series Solutions of Differential Equations	758	(13)
Review: Definitions, Concepts, Results	767	(4)
Vectors, Curves, and Surfaces in Space	771	(78)
Vectors in the Plane	772	(6)
Three-Dimensional Vectors	778	(10)
The Cross Product of Vectors	788	(8)
Lines and Planes in Space	796	(7)
Curves and Motion in Space	803	(14)
Project:: Does a Pitched Baseball Really Curve?	816	(1)
Curvature and Acceleration	817	(13)
Cylinders and Quadric Surfaces	830	(8)
Cylindrical and Spherical Coordinates	838	(11)
Review: Definitions, Concepts, Results	845	(4)
Partial Differentiation	849	(90)
Introduction	850	(1)
Functions of Several Variables	850	(10)
Limits and Continuity	860	(8)
Partial Derivatives	868	(10)
Multivariable Optimization Problems	878	(11)
Increments and Linear Approximation	889	(7)
The Multivariable Chain Rule	896	(11)
Directional Derivatives and the Gradient Vector	907	(11)
Lagrange Multipliers and Constrained Optimization	918	(9)
Project:: Numerical Solution of Lagrange Multiplier Systems	927	(1)
Critical Points of Functions of Two Variables	927	(12)
Project:: Critical Point Investigations	935	(1)
Review: Definitions, Concepts, Results	936	(3)
Multiple Integrals	939	(74)
Double Integrals	940	(7)
Project:: Midpoint Sums Approximating Double Integrals	947	(1)
Double Integrals over More General Regions	947	(7)
Area and Volume by Double Integration	954	(7)
Double Integrals in Polar Coordinates	961	(7)
Applications of Double Integrals	968	(11)
Project:: Optimal Design of Downhill Race-Car Wheels	978	(1)
Triple Integrals	979	(9)
Project:: Archimedes' Floating Paraboloid	987	(1)
Integration in Cylindrical and Spherical Coordinates	988	(8)
Surface Area	996	(5)
Change of Variables in Multiple Integrals	1001	(12)
REVIEW: Definitions, Concepts, Results	1009	(4)
Vector Calculus	1013	(1)
Vector Fields	1014	(5)
Line Integrals	1019	(11)
The Fundamental Theorem and Independence of Path	1030	(7)
Green's Theorem	1037	(10)
Surface Integrals	1047	(10)
Project:: Surface Integrals and Rocket Nose Cones	1057	(1)
The Divergence Theorem	1057	(8)
Stokes' Theorem	1065
Review: Definitions, Concepts, Results	1072
Appendices
A: Real Numbers and Inequalities	1	(5)
B: The Coordinate Plane and Straight Lines	6	(7)
C: Review of Trigonometry	13	(6)
D: Proofs of the Limit Laws	19	(4)
E: The Completeness of the Real Number System	23	(5)
F: Existence of the Integral	28	(5)
G: Approximations and Riemann Sums	33	(3)
H: L'Hopital's Rule and Cauchy's Mean Value Theorem	36	(2)
I: Proof of Taylor's Formula	38	(1)
J: Conic Sections as Sections of a Cone	39	(1)
K: Proof of the Linear Approximation Theorem	40	(1)
L: Units of Measurement and Conversion Factors	41	(1)
M: Formulas from Algebra, Geometry, and Trigonometry	42	(2)
N: The Greek Alphabet	44	(1)
Answers to Odd-Numbered Problems	45	(23)
References for Further Study	68
Index	1	(1)
Table of Integrals	1

Excerpts

Contemporary calculus instructors and students face traditional challenges as well as new ones that result from changes in the role and practice of mathematics by scientists and engineers in the world at large. As a consequence, this sixth edition of our calculus textbook is its most extensive revision since the first edition appeared in 1982.Two chapters of the fifth edition have been combined in a single more tightly organized one. An entirely new chapter now appears in the table of contents, and most of the remaining chapters have been extensively rewritten. About 125 of the book's over 750 worked examples are new for this edition and the 1825 figures in the text include 225 new computer-generated graphics. About 600 of its over 7000 problems are new, and these are augmented by 320 new conceptual discussion questions that now precede the problem sets. Moreover, 1050 new true/false questions are included in the Study Guides on the new CD-ROM that accompanies this edition. In summary, almost 2000 of these 8400-plus problems and questions are new, and the text discussion and explanations have undergone corresponding alteration and improvement. PRINCIPAL NEW FEATURESThe current revision of the text features More unified treatment oftranscendental functionsin Chapter 7, and A new Chapter 9 ondifferential equationsand applications.The new chapter on differential equations now appears immediately after Chapter 8 on techniques of integration. It includes both direction fields and Eider's method together with the more elementary symbolic methods (which exploit techniques from Chapter 8) and interesting applications of both first- and second-order equations. Chapter 11 (Infinite Series) now ends with a new section on power series solutions of differential equations, thus bringing full circle a unifying focus of second-semester calculus on elementary differential equations. NEW LEARNING RESOURCESConceptual Discussion Questions:The set of problems that concludes each section is now preceded by a briefConcepts: Questions and Discussionset consisting of several open-ended conceptual questions that can be used for either individual study or classroom discussion.The Text CD-ROM:The content of the new CD-ROM that accompanies this text is fully integrated with the textbook material, and is designed specifically for use hand-in-hand with study of the book itself. This CD-ROM features the following resources to support learning and teaching: Interactive True/False Study Guidesthat reinforce and encourage student reading of the text. Ten author-written questions for each section carefully guide students through the section, and students can request individual hints suggesting where in the section to look for needed information. Live Examplesfeature dynamic multimedia and computer algebra presentations--many accompanied by audio explanations--which enhance student intuition and understanding. These interactive examples expand upon many of the textbook's principal examples; students can change input data and conditions and then observe the resulting changes in step-by-step solutions and accompanying graphs and figures.Walkthrough videosdemonstrate how students can interact with these live examples. Homework Startersfor the principal types of computational problems in each textbook section, featuring both interactive presentations similar to the live examples and (Web-linked) voice-narrated videos of pencil-and-paper investigations illustrating typical initial steps in the solution of selected textbook problems. Computing Project Resourcessupport most of the almost three dozen projects that follow key sections in the text. For each such project marked in the text by a CD-ROM icon, more extended discussions illustrating Maple,Mathematica,MATLAB,