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Vector Calculus (2nd Edition), by Susan Jane Colley
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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.
- Sales Rank: #2055013 in Books
- Brand: Brand: Prentice Hall
- Published on: 2001-06-08
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x 1.41" w x 8.26" l,
- Binding: Hardcover
- 576 pages
- Used Book in Good Condition
From the Back Cover
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.
About the Author
Susan Coney is currently the Andrew and Pauline Delaney Professor of Mathematics at Oberlin College, having previously served as Chair of the Department.
She received S.B. and Ph.D. degrees in mathematics from the Massachusetts Institute of Technology prior to joining the faculty at Oberlin in 1983.
Her research focuses on enumerative problems in algebraic geometry, particularly concerning multiple-point singularities and higher-order contact of plane curves.
Professor Coney has published papers on algebraic geometry as well as articles on other mathematical subjects. She has lectured internationally on her research and has taught a wide range of subjects in undergraduate mathematics.
Professor Coney is a member of several professional and honorary societies, including the American Mathematical Society, the Mathematical Association of America, Phi Beta Kappa, and Sigma Xi.
Excerpt. © Reprinted by permission. All rights reserved.
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 an important 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 examples—that 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 EditionI 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.
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 subtopics—or 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 lecturesChapter 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.
Most helpful customer reviews
25 of 30 people found the following review helpful.
A solid, thorough treatment of multivariable calculus.
By A Customer
I used Susan Colley's Vector Calculus when I took multivariable calculus in the spring of '99. The book is very well written and I would definitely recommend it to anyone, but most especially to those who have a strong interest in the subject and aren't just fulfilling a requirement. Here is why--
When the reader is presented with an mathematical idea, it is nice to know where that idea comes from, and to be given whatever explanations or proofs are needed. An example of where Colley does this is in the chapter on the chain rule in several variables. This is a difficult chapter and Colley does an excellent job of explaining the underlying concepts (with lots of visual aids) where a less thorough author might have simply offered formulas and methods to solve a few specific types of problems.
Also, Colley introduces vector notation which, although at first unfamiliar, ultimately leads to a better understanding of the relationships between functions of different numbers of variables. For example, instead of the notation f(x,y,z,w,...) we have f(x->) (the arrow indicates that x is a vector). This notation, as well as the extensive use of matrices is very helpful and eliminates much confusion.
The visuals are simple and easy to understand, and the problems are appropriately designed, with plenty of very simple exercises for dealing with basic calculations, as well as very challenging and thought-provoking problems which require plenty of thought and help develop good mathematical intuition and visualization.
Overall this is a very good book, and it appears to me that the other reviews on this page come from neither a good knowledge of the book nor multivariable calculus.
13 of 16 people found the following review helpful.
a very profound and majestic treatment
By A Customer
Of all the math texts I have ever read, this is the first one which really seems infused with great enthusiasm for the subject as well as with humor. It is the textbook that one would use if one didn't want to just memorize techniques and formulas with little understanding, but wanted to have as deep and as beautiful (not to mention fun) appreciation of the subject as possible without being dragged down in minutiae. The people who criticized it were probably frustrated by the book because it really tries to bring the reader into the almost magical world of multivariable calculus so she or he may marvel at it. But to do so takes a great deal of effort, so people who just wanted to know how and not why would certainly prefer a different text. Being a Oberlin student myself, as the critics were, I understand that in the midst of all their other classes and being confronted for the first time with real math (multivariable is definitely a step up in difficulty from ordinary calculus) they could be frustrated by such an approach. But, I'm not an even a math minor and I was so happy to be able to use this text and not your standard blah-blah, humorless, lifeless,and arcane math text. Bottom line: if you want to understand come here; if you want to just do seek another text.
33 of 37 people found the following review helpful.
The best introduction to Vector Calculus ever written
By R. Prabhaharan
The author has written a carefully thought out introduction to the subject whose only assumptions are that you know the most rudimentary coordinate geometry and single variable calculus. From this all the classical subjects in vector calculus are built up using geometric ideas to motivate the definitions of the concepts. Typically the first course in vector calculus tries to get to Stokes Theorem and so on as quickly as possible without explaining what motivated these ideas. Much of the technical apparatus in vector calculus was used in modelling fluid dynamic flows in the nineteenth century, this is where the idea of "vector field" came from. As far as I know, this is the first vector calculus book I've read that defines a vector field, and next to it shows a picture of water flowing out of an upturned cup, with velocity vectors pointing in all directions. Just one picture captures the essence of the definition and immediately renders concrete something very abstract. There are many other examples in the book where a picture is shown of an abstract concept, making the definitions and theorems intuitive.
However, this book is not just pretty pictures, the calculus is built up in a rigorous manner (as far as a first introduction to the subject goes) and by the end of the book you are well placed to read your first book on manifolds and differential geometry. The book is not cheap, but if you think about it in terms of if you wanted to replicate this book you'd need at least 3 other standard textbooks, then its reasonable. Even advanced mathematicians would be surprised how much they could learn by looking at some of the pictures ! This book would be ideal as an appetiser before a main course of graduate differential geometry.
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