Multivariable Calculus with Matrices / Edition 6

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This is the most extensively visual book in the market—highlighted by hundreds of Mathematica and MATLAB generated figures throughout. It now contains a full chapter of material on matrices and eigenvalues up front. All of Multivariable Calculus has been rewritten with matrix notation. Chapter topics include infinite series, vectors and matrices, curves and surfaces in space, partial differentiation, multiple integrals, and vector calculus.
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Editorial Reviews

From The Critics
This textbook for a one-semester course consists of chapters 10-15 from , and covers infinite series, vectors and matrices, curves and surfaces in space, partial differentiation, multiple integrals, and vector calculus. The sixth edition integrates matrix methods with the more traditional approaches. The CD-ROM contains interactive examples and true- false questions. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780130648181
  • Publisher: Pearson Education
  • Publication date: 2/28/2002
  • Edition description: 6TH BK&CDR
  • Edition number: 6
  • Pages: 544
  • Product dimensions: 8.10 (w) x 10.80 (h) x 0.70 (d)

Table of Contents

10. Infinite Series.
Introduction. Infinite Sequences. Infinite Series and Convergence. Taylor Series and Taylor Polynomials. The Integral Test. Comparison Tests for Positive-Term Series. Alternating Series and Absolute Convergence. Power Series. Power Series Computations. Series Solutions of Differential Equations.

11. Vectors and Matrices.
Vectors in the Plane. Three-Dimensional Vectors. The Cross Product of Vectors. Lines and Planes in Space. Linear Systems and Matrices. Matrix Operations. Eigenvalues and Rotated Conics.

12. Curves and Surfaces in Space.
Curves and Motion in Space. Curvature and Acceleration. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates.

13. Partial Differentiation.
Introduction. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Multivariable Optimization Problems. Linear Approximation and Matrix Derivatives. The Multivariable Chain Rule. Directional Derivatives and Gradient Vectors. Lagrange Multipliers and Constrained Optimization. Critical Points of Multivariable Functions.

14. Multiple Integrals.
Double Integrals. Double Integrals over More General Regions. Area and Volume by Double Integration. Double Integrals in Polar Coordinates. Applications of Double Integrals. Triple Integrals. Integration in Cylindrical and Spherical Coordinates. Surface Area. Change of Variables in Multiple Integrals.

15. Vector Calculus.
Vector Fields. Line Integrals. The Fundamental Theorem and Independence of Path. Green's Theorem. Surface Integrals. The Divergence Theorem. Stokes' Theorem.

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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 entire chapters of the fifth edition have been replaced in the table of contents by two new ones; most of the remaining chapters have been extensively rewritten. Nearly 160 of the book's over 800 worked examples are new for this edition and the 1850 figures in the text include 250 new computer-generated graphics. Almost 800 of its 7250 problems are new, and these are augmented by over 330 new conceptual discussion questions that now precede the problem sets. Moreover, almost 1100 new true/false questions are included in the Study Guides on the new CD-ROM that accompanies this edition. In summary, almost 2200 of these 8650-plus problems and questions are new, and the text discussion and explanations have undergone corresponding alteration and improvement.


The current revision of the text features

  • Early transcendentals fully integrated in Semester I.
  • Differential equations and applications in Semester II.
  • Linear systems and matrices in Semester III.

Complete coverage of the calculus of transcendental functions is now fully integrated in Chapters 1 through 6—with the result that the Chapter 7 and 8 titles in the 5th edition table of contents do pot appear in this 6th edition.

A new chapter on differential equations (Chapter 8) nowappears immediately after Chapter 7 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 7) and interesting applications of both first- and second-order equations. Chapter 10 (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.

Linear systems and matrices, ending with an elementary treatment of eigenvalues and eigenvectors, are now introduced in Chapter 11. The subsequent coverage of multivariable calculus now integrates matrix methods and terminology with the traditional notation and approach—including (for instance) introduction and extensive application of the chain rule in matrix-product form.


Conceptual Discussion Questions. The set of problems that concludes each section is now preceded by a brief Concepts: Questions and Discussion set 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 Guides that 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 Examples feature 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 videos demonstrate how students can interact with these live examples.
  • Homework Starters for 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 Resources support most of the over 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, and graphing calculator investigations are provided. Computer algebra system commands can be copied and pasted for interactive execution.
  • Hyperlinked Maple Worksheets contributed by Harald Pleym of Telemark University College (Norway) constitute an interactive version of essentially the whole textbook. Students and faculty using Maple can change input data and conditions in most of the text examples to investigate the resulting changes in step-by-step solutions and accompanying graphs and figures.
  • PowerPoint Presentations provide classroom projection versions of about 350 of the figures in the text that would be least convenient to reproduce on a blackboard.
  • Website. The contents of the CD-ROM together with additional learning and teaching resources are maintained and updated at the textbook website, which includes a Comments and Suggestions center where we invite response from both students and instructors.

Computerized Homework Grading System. About 2000 of the textbook problems are incorporated in an automated grading system that is now available. Each problem solution in the system is structured algorithmically so that students can work in a computer lab setting to submit homework assignments for automatic grading. (There is a small annual fee per participating student.)

New Solutions Manuals. The entirely new 1900-page Instructor's Solutions Manual (available in two volumes) includes a detailed solution for every problem in the book. These solutions were written exclusively by the authors and have been checked independently by others.

The entirely new 950-page Student Solutions Manual (available in two volumes) includes a detailed solution for every odd-numbered problem in the text. The answers (alone) to most of these odd-numbered problems are included in the answers section at the back of this book.

New Technology Manuals. Each of the following manuals is available shrink-wrapped with any version of the text for half the normal price of the manual (all of which are inexpensive):

  • Jensen, Using MATLAB in Calculus (0-13-027268-X)
  • Freese/Stegenga, Calculus Concepts Using Derive (0-13-085152-3)
  • Gresser, TI Graphing Calculator Approach, 2e (0-13-092017-7)
  • Gresser, A Mathematica Approach, 2e (0-13-092015-0)
  • Gresser, A Maple Approach, 2e (0-13-092014-2)


In preparing this edition, we have taken advantage of many valuable comments and suggestions from users of the first five editions. This revision was so pervasive that the individual changes are too numerous to be detailed in a preface, but the following paragraphs summarize those that may be of widest interest.

  • New Problems. Most of the almost 800 new problems lie in the intermediate range of difficulty, neither highly theoretical nor computationally routine. Many of them have a new technology flavor, suggesting (if not requiring) the use of technology ranging from a graphing calculator to a computer algebra system.

  • Discussion Questions and Study Guides. We hope the 330 conceptual discussion questions and 1080 true/false study-guide questions constitute a useful addition to the traditional fare of student exercises and problems. The True/False Study Guide for each section provides a focus on the key ideas of the section, with the single goal of motivating guided student reading of the section.

  • Examples and Explanations. About 20% of the book's worked examples are either new or significantly revised, together with a similar percentage of the text discussion and explanations. Additional computational detail has been inserted in worked examples where students have experienced difficulty, together with additional sentences and paragraphs in similar spots in text discussions.

  • Project Material. Many of the text's almost 40 projects are new for this edition. These appear following the problem sets at the ends of key sections throughout the text. Most (but not all) of these projects employ some aspect of modern computational technology to illustrate the principal ideas of the preceding section, and many contain additional problems intended for solution with the use of a graphing calculator or computer algebra system. Where appropriate, project discussions are significantly expanded in the CD-ROM versions of the projects.

  • Historical Material. Historical and biographical chapter openings offer students a sense of the development of our subject by real human beings. Indeed, our exposition of calculus frequently reflects the historical development of the subject—from ancient times to the ages of Newton and Leibniz and Euler to our own era of new computational power and technology.


  • Introductory Chapters. Instead of a routine review of precalculus topics, Chapter 1 concentrates specifically on functions and graphs for use in mathematical modeling. It includes a section cataloging informally the elementary transcendental functions of calculus, as background to their more formal treatment using calculus itself. Chapter 1 concludes with a section addressing the question "What is calculus?" Chapter 2 on limits begins with a section on tangent lines to motivate the official introduction of limits in Section 2.2. Trigonometric limits are treated throughout Chapter 2 in order to encourage a richer and more visual introduction to the limit concept.

  • Differentiation Chapters. The sequence of topics in Chapters 3 and 4 differs a bit from the most traditional order. We attempt to build student confidence by introducing topics more nearly in order of increasing difficulty. The chain rule appears quite early (in Section 3.3) and we cover the basic techniques for differentiating algebraic functions before discussing maxima and minima in Sections 3.5 and 3.6. Section 3.7 treats the derivatives of all six trigonometric functions, and Section 3.8 (much strengthened for this edition) introduces the exponential and logarithmic functions. Implicit differentiation and related rates are combined in a single section (Section 3.9). The authors' fondness for Newton's method (Section 3.10) will be apparent.

    The mean value theorem and its applications are deferred to Chapter 4. In addition, a dominant theme of Chapter 4 is the use of calculus both to construct graphs of functions and to explain and interpret graphs that have been constructed by a calculator or computer. This theme is developed in Sections 4.4 on the first derivative test and 4.6 on higher derivatives and concavity. But it may also be apparent in Sections 4.8 and 4.9 on l'Hopital's rule, which now appears squarely in the context of differential calculus and is applied here to round out the calculus of exponential and logarithmic functions.

  • Integration Chapters. Chapter 5 begins with a section on antiderivatives—which could logically be included in the preceding chapter, but benefits from the use of integral notation. When the definite integral is introduced in Sections 5.3 and 5.4, we emphasize endpoint and midpoint sums rather than upper and lower and more general Riemann sums. This concrete emphasis carries through the chapter to its final section on numerical integration.

    Chapter 6 begins with a largely new section on Riemann sum approximations, with new examples centering on fluid flow and medical applications. Section 6.6 is a new treatment of centroids of plane regions and curves. Section 6.7 gives the integral approach to logarithms, and Sections 6.8 and 6.9 cover both the differential and the integral calculus of inverse trigonometric functions and of hyperbolic functions.

    Chapter 7 (Techniques of Integration) is organized to accommodate those instructors who feel that methods of formal integration now require less emphasis, in view of modern techniques for both numerical and symbolic integration. Integration by parts (Section 7.3) precedes trigonometric integrals (Section 7.4). The method of partial fractions appears in Section 7.5, and trigonometric substitutions and integrals involving quadratic polynomials follow in Sections 7.6 and 7.7. Improper integrals appear in Section 7.8, with new and substantial subsections on special functions and probability and random sampling. This rearrangement of Chapter 7 makes it more convenient to stop wherever the instructor desires.

  • Differential Equations. This entirely new chapter begins with the most elementary differential equations and applications (Section 8.1) and then proceeds to introduce both graphical (slope field) and numerical (Eider) methods in Section 8.2. Subsequent sections of the chapter treat separable and linear first-order differential equations and (in more depth than usual in a calculus course) applications such as population growth (including logistic and predator-prey populations) and motion with resistance. The final two sections of Chapter 8 treat second-order linear equations and applications to mechanical vibrations. Instructors desiring still more coverage of differential equations can arrange with the publisher to bundle and use appropriate sections of Edwards and Penney, Differential Equations: Computing and Modeling 2/e (Prentice Hall, 2000).

  • Parametric Curves and Polar Coordinates. The principal change in Chapter 9 is the replacement of three separate sections in the 5th edition on parabolas, ellipses, and hyperbolas with a single Section 9.6 that provides a unified treatment of all the conic sections.

  • Infinite Series. After the usual introduction to convergence of infinite sequences and series in Sections 10.2 and 10.3, a combined treatment of Taylor polynomials and Taylor series appears in Section 10.4. This makes it possible for the instructor to experiment with a briefer treatment of infinite series, but still offer exposure to the Taylor series that are so important for applications. The principal change in Chapter 10 is the addition of a new final section on power series methods and their use to introduce new transcendental functions, thereby concluding the middle third of the book with a return to differential equations.

  • Vectors and Matrices. After covering vectors in its first four sections, Chapter 11 continues with three sections on solution of linear systems (through elementary Gaussian elimination), matrices (through calculation of inverse matrices), and eigenvalues and eigenvectors and their use in classification of rotated conks. This introduction of linear systems and matrices provides the preparation required for the matrix-oriented multivariable calculus of Chapter 13. The intervening Chapter 12 (Curves and Surfaces in Space) includes discussion of Kepler-Newton motion of planets and satellites. The chapter includes also a brief application of eigenvadues to the discussion of rotated quadric surfaces in space.

  • Multivariable Calculus. Appropriately enough, the introduction and initial application of partial derivatives is traditional. But, beginning with the introduction of the multivariable chain rule in matrix-product form, matrix notation and terminology is used consistently in the remainder of Chapter 13. This approach affords a more clear-cut treatment of differentiability and linear approximation of multivariable functions, as well as of directional derivatives and Lagrange multipliers. We conclude Chapter 13 with a classification of critical points based on eigenvalues of the (Hessian) matrix of second derivatives (thereby generalizing directly the second derivative test of single-variable calculus). As a final illustration of the utility of matrix methods, this approach unifies the standard discriminant-based classification of two-variable critical points with the analogous classification of critical points for functions of three or more variables. Matrix methods (naturally) are needed less frequently in Chapters 14 (Multiple Integrals) and 15 (Vector Calculus), but appear whenever a change of variables in a multiple integral is involved.


The present version of the text is accompanied by a more traditional version that treats transcendental functions later in single variable calculus and omits matrices in multivariable calculus. Both versions of the complete text are also available in two-volume split editions. By appropriate selection of first and second volumes, the instructor can therefore construct a complete text for a calculus sequence with

  • Early transcendentals in single variable calculus and matrices in multivariable calculus;
  • Early transcendentals in single variable calculus but traditional coverage of multivariable calculus;
  • Transcendental functions delayed until after the integral in single variable calculus, but matrices used in multivariable calculus;
  • Neither early transcendentals in single variable calculus nor matrices in, multivariable calculus.
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