Customer Reviews

Vector Calculus (2nd Edition)

5 star:		(9)
4 star:		(1)
3 star:		(2)
2 star:		(8)
1 star:		(8)

 

 

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.

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.

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.

This undergraduate text treats its readers as mathematicians. The organization is terrific, the examples are great, and the treatment of material includes explanation and proof, unlike this text's counterparts. It is fortunate to have a clear and insightful treatment of multivariate calculus at the undergraduate level, to inspire more to seriously consider a math career. I think this book has promise of growing the number of math majors around the country.

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.

I love this book. Unlike most math majors who get through life by memorizing formulas in a relatively brainless activity, this book actually explains why/how things work. If you take the time to go through this book step by step, examples and all, you will get a beautiful understanding of multivariable calculus. I enjoyed both geometry and single variable calculus a lot, and multivariable calculus is a nice combo of those two.

If you are in charge of designing curricula for your Place of Higher Learning, please don't force your students to use this book as anything more than a doorstop. It's full of text, 70% of which is unnecessary filler. One of the author's favorite pedantic methods is using the phrase "convince yourself that [some mathematical idea is true]" instead of actually proving it. This might be acceptable if the author had balanced out the massive seas of empty text with massive seas of examples. But keeping with the theme of the book ("useless"), there are few examples, which are what usually makes a textbook valuable.

The one bright spot is reading the uncomfortably forced praise on the back cover.

I've read Courant (Vol II), Apostol, Kaplan, and less so, Widder, Marsden & Trombda, and come under the impression that any of these books will better cover the material than Colley's Vector Calculus.

This book was used for the final of my college's calculus sequence, and I only used it for the exercises.

What I didn't like most was the way it was organized. Let's see, I flip to a random page - parametrized curves and Kepler's Laws. It goes, wham, definition, definition, example, theorem, proposition, theorem, definition, proposition, step 1, step 2, step 3, theorem... Very disorganized. No proofs. No mathematical motivations. It feels like an outline.

Only worse, because no indication or motivation is given to lead in to important concepts. The index looks nearly the same as any advanced calculus textbook - yet when you flip to any entry, you'll realize it's just in boldface, lying on the tangent to some example or exercise question.

The content is not very extensive. In Kaplan, Apostol, a chapter is dedicated to linear algebra, whereas Colley only discusses vector algebra - this comes back say in the extrema of functions, in the discussion of quadratic forms. The content starts with multivariable calculus, differential operators, multiple integrals, and ends off at Stokes' theorem etc. before a discussion on series.

I don't understand the purpose of this book. It has an applied slant - you can go through all of the exercises just plugging into the 'propositions/definitions' - but is heavy on notation (not rigor), because you can finish these exercises without actually appreciating any one of the theorems/definitions/propositions in a way that will teach you how to formulate a proof.

This book tries to do a great thing, but fails at it.

I can list several examples where this book really goes into detail, but unless you have another source, you wont understand this.

Simply put, this book is like a list of directions for me. Not every book I have seen (I have a few from older classmates) covers everything thats in this book.

I see that I have to learn, for example, the Hessian Criterion for Constrained Extrema, but I will go to another source to do it.

If I didnt have this book, I wouldn't know about the Hessian Criterion (unless someone told me or I found it), but I wont really learn from the book.

This textbook tells you what you should learn, and you go and learn it.

The reason for the poor instruction is the lack of explanations, and sometimes repeated usage of the same variables, which, as one might guess, in multi-variable calculus, is EXTREMELY confusing and shouldn't be allowed.

If your required to get this book, expect that you'll see what Calculus 3 is about, but dont expect to learn it.

Professor Colley's book excels in all the areas one would look for including abundant examples, fine graphics, excellent graded problems, clear writing, good organization and so on. It stands out particularly for the author's sensitive presentation which not only presents the material in a clear, logical form but in such a way as to anticipate the questions of the reader. The use of geometric intuition is especially effective. Not being a great talent at mathematics, I found that this book clarified many ideas that I had not understood before. How the negative critics came up with their ideas is a mystery.