Vector Calculus (3rd Edition) [Hardcover]

Susan J. Colley (Author)

Book Description

ISBN-10: 0131858742 | ISBN-13: 978-0131858749 | Publication Date: March 26, 2005 | Edition: 3

This text uses the language and notation of vectors and matrices to clarify issues in multivariable calculus. Accessible to anyone with a good background in single-variable calculus, it presents more linear algebra than usually found in a multivariable calculus book. Colley balances this with very clear and expansive exposition, many figures, and numerous, wide-ranging exercises. Instructors will appreciate Colley’s writing style, mathematical precision, level of rigor, and full selection of topics treated.

 

Vectors: 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. 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. Vector-Valued Functions: Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator. Maxima and Minima in Several Variables: Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema. Multiple Integration: Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration. Line Integrals: Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields. Surface Integrals and Vector Analysis: Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations. Vector Analysis in Higher Dimensions: An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem.

 

For all readers interested in multivariable calculus.

Editorial Reviews

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. --This text refers to an out of print or unavailable edition of this title.

About the Author

Susan Colley is the Andrew and Pauline Delaney Professor of Mathematics at Oberlin College and currently Chair of the Department, having also previously served as Chair. 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 Colley has published papers on algebraic geometry and commutative algebra, 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 Colley 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.

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Product Details

• Hardcover: 576 pages
• Publisher: Prentice Hall; 3 edition (March 26, 2005)
• Language: English
• ISBN-10: 0131858742
• ISBN-13: 978-0131858749
• Product Dimensions: 10.1 x 8.2 x 1.2 inches
• Shipping Weight: 2.8 pounds (View shipping rates and policies)
• Average Customer Review: 2.8 out of 5 stars  See all reviews (28 customer reviews)
• Amazon Best Sellers Rank: #30,316 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

29 of 32 people found the following review helpful:

5.0 out of 5 stars The best introduction to Vector Calculus ever written, February 28, 2002

By 

R. Prabhaharan (Singapore / Malaysia) - See all my reviews
(REAL NAME)   

This review is from: Vector Calculus (Hardcover)

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.

26 of 30 people found the following review helpful:

5.0 out of 5 stars Perfect text for year-long course in vector calulus, October 7, 1999

By 

David P. Lang, Ph.D.(Math), Ph.D.(Phil) (Worcester, MA) - See all my reviews

This review is from: Vector Calculus (Hardcover)

I have examined a number of textbooks in multi-variable calculus, and I conclude that this is the best of the recent breed. The theory is well-motivated by discussion and illustrative examples, yet presented as rigorously as possible at this level. The applications (especially to physics) are outstanding, the text examples illuminating, and the exercises both doable and beneficial for enhancing comprehension through sufficient practice. The negative comments that this highly competent and well-structured book has received are entirely unfair, failing to render a realistic assessment of its real value. I wonder whether they are referring to the same obviously proficent author and the same publication (which I read nearly cover-to-cover). The only drawback is that the great amount of material cannot (by the author's own admission) be covered in one semester; it would require an entire academic year to do adequate justice to all the essential topics.

22 of 27 people found the following review helpful:

5.0 out of 5 stars A solid, thorough treatment of multivariable calculus., September 10, 1999

By A Customer

This review is from: Vector Calculus (Hardcover)

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.