A traditional and accessible calculus book with a strong conceptual and geometric slant that assumes a background in single-variable calculus. It uses the language and notation of vectors and matrices to clarify issues in multivariable calculus, and combines a clear and expansive writing style with an interesting selection of material. Chapter topics cover vectors, differentiation in several variables, vector-valued functions, maxima and minima in several variables, multiple integration, line integrals, surface integrals and vector analysis, and vector analysis in higher dimensions. For individuals interested in math and calculus.
Table of Contents
(NOTE: Each chapter concludes with True/False Exercises and Miscellaneous Exercises.)
Vectors in Two and Three Dimensions. More About Vectors. The Dot Product. The Cross Product. Equations for Planes; Distance Problems. Some n-Dimensional Geometry. New Coordinate Systems.
2. Differentiation in Several Variables.
Functions of Several Variables; Graphing Surfaces. Limits. The Derivative. Properties; Higher-Order Partial Derivatives; Newton’s Method. The Chain Rule. Directional Derivatives and the Gradient.
3. Vector-Valued Functions.
Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator.
4. Maxima and Minima in Several Variables.
Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema.
Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration.
6. Line Integrals.
Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields.
7. Surface Integrals and Vector Analysis.
Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations.
8. Vector Analysis in Higher Dimensions.
An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem.
Suggestions for Further Reading.
Answers to Selected Exercises.
Physical and natural phenomena depend on a complex array of factors. The sociologist or psychologist who studies group behavior, the economist who endeavors to understand the vagaries of a nation's employment cycles, the physicist who observes the trajectory of a particle or planet, or indeed anyone who seeks to understand geometry in two, three, or more dimensions recognizes the need to analyze changing quantities that depend on more than a single variable. Vector calculus is the essential mathematical tool for such analysis. Moreover, it is an exciting and beautiful subject in its own right, a true adventure in many dimensions.
The only technical prerequisite for this text, which is intended for a sophomore-level course in multivariable calculus, is a standard course in the calculus of functions of one variable. In particular, the necessary matrix arithmetic and algebra (not linear algebra) are developed as needed. Although the mathematical background assumed is not exceptional, the reader will still need to "think hard" in places.
My own objectives in writing the book are simple ones: to develop in students a sound conceptual grasp of vector calculus and to help them begin the transition from first-year calculus to more advanced technical mathematics. I maintain that the first goal can be met, at least in part, through the use of vector and matrix notation, so that many results, especially those of differential calculus, can be stated with reasonable levels of clarity and generality. Properly described, results in the calculus of several variables can look quite similar to those of the calculus of one variable. Reasoning by analogy will thus be animportant pedagogical tool. I also believe that a conceptual understanding of mathematics can be obtained through the development of a good geometric intuition. Although I state many results in the case of n variables (where n is arbitrary), I recognize that the most important and motivational examples usually arise for functions of two and three variables, so these concrete and visual situations are emphasized to explicate the general theory. Vector calculus is in many ways an ideal subject for students to begin exploration of the interrelations among analysis, geometry, and matrix algebra.
Multivariable calculus, for many students, represents the beginning of significant mathematical maturation. Consequently, I have written a rather expansive text so that they can see that there is a "story" behind the results, techniques, and examplesthat the subject coheres and that this coherence is important for problem solving. To indicate some of the power of the methods introduced, a number of topics, not always discussed very fully in a first multivariable calculus course, are treated here in some detail:
- an early introduction of cylindrical and spherical coordinates
- the use of vector techniques to derive Kepler's laws of planetary motion
- the elementary differential geometry of curves in R3, including discussion of curvature, torsion, and the Frenet-Serret formulas for the moving frame
- Taylor's formula for functions of several variables
- the use of the Hessian matrix to determine the nature (as local extrema) of critical points of functions of n variables
- an extended discussion of the change of variables formula in double and triple integrals
- applications of vector analysis to physics
- an introduction to differential forms and the generalized Stokes's theorem
Included are a number of proofs of important results. The more technical proofs are collected as addenda at the end of the appropriate sections so as not to disrupt the main conceptual flow and to allow for greater flexibility of use by the instructor and student. Nonetheless, some proofs (or sketches of proofs) embody such central ideas that they are included in the main body of the text.
New in the Second Edition
I have retained the overall structure and tone of the first edition. New features include the following:
- 220 new exercises of varying levels of difficulty;
- 70 computer-based exercises;
- a new chapter (Chapter 8) on differential forms, parametrized manifolds, and the generalized Stokes's theorem that significantly expands on the first edition;
- an expanded discussion of the implicit function and inverse function theorems
- an expanded discussion of quadratic forms and their role in determining extrema of functions
- various refinements throughout the text, including new examples and explanations.
How To Use This Book
There is more material in this book than can be covered comfortably during a single semester. Hence, the instructor will wish to eliminate some topics or subtopicsor to abbreviate the rather leisurely presentations of limits and differentiability. Since I frequently find myself without the time to treat surface integrals in detail, I have separated all material concerning parametrized surfaces, surface integrals, and Stokes's and Gauss's theorems (Chapter 7), from that concerning line integrals and Green's theorem (Chapter 6). In particular, in a one-semester course for students with little or no experience with vectors or matrices, instructors can probably expect to cover most of the material in Chapters 1-6, although no doubt it will be necessary to omit some of the optional subsections and to downplay many of the proofs of results. A rough outline for such a course, allowing for some instructor discretion, could be the following:Chapter 1: 8-9 lectures
Chapter 2: 9 lectures
Chapter 3: 4-5 lectures
Chapter 4: 5-6 lectures
Chapter 5: 8 lectures
Chapter 6: 4 lectures
Total: 38-41 lectures
If students have a richer background (so that much of the material in Chapter 1 can be left largely to them to read on their own), then it should be possible to treat a good portion of Chapter 7 as well. For a two-quarter or two-semester course, it should be possible to work through the entire book with reasonable care and rigor, although coverage of Chapter 8 should depend on students' exposure to introductory linear algebra, as somewhat more sophistication is assumed there.
The exercises vary from relatively routine computations to more challenging and provocative problems, generally (but not invariably) increasing in difficulty for each section. Each chapter concludes with a set of miscellaneous exercises that review and extend the ideas introduced in the chapter, and occasionally present new applications.
A word about the use of technology. The text was written without reference to any particular computer software or graphing calculator. Most of the exercises can be solved by hand, although there is no reason not to turn over some of the more tedious calculations to a computer. Those exercises that require a computer for computational or graphical purposes are marked with a computer symbol, and should be amenable to software such as Mathematica®, Maple®, or MATLAB.