Vector Calculus

Rating:
Summary: The Perfect Book for Vector Calculus
Review: After you've learned calculus from a book like Stewart's, this is the definite next step. It includes lots of proofs for most of the theorems. I love the way the textbook is organized. And all the explanations are incredibly clear. The choice of notation makes all the formulas a lot clearer than in other books. Although I am not as big a fan of math as I am of physics, I must say that I really enjoyed reading this book a lot, and believe you will too. This book seriously makes advanced calculus incredibly fun!

Rating:
Summary: Relatively weak as a standard textbook on vector calculus
Review: I am well aware of the usefulness of these reviews in determining the applicability of a book for self-study; so let me address this quickly. This has got to be the worst vector calculus book available if you're looking to study the subject on your own!!! This book is frustrating and dry; please consider other self-study options!

Unfortunately, most people who use this text are required to for a class, and for whatever reason, this book has become somewhat of a standard at many universities. I used this book a while back in a Vector Calculus class at UT Austin, and I was largely disappointed by its contents.

First of all, the author of the book is dry and completely uninspiring. That's not to say that people read calculus books like novels, but the author presents the material from a strictly technical and theoretical perspective. Further adding to its blandness, the author (or the publisher) has opted for the cost-effective choice of using no color in the book. The graphs and figures are confused and lacking - often difficult to understand.

Now, the obvious rebuttal to my accusations will come from purists (hardcore math majors). I am, myself, a math (and physics) major, and though I am not saying that this text is completely inaccessible, I have to say that the author wrote this book wholly without imagination or sincerity. There is no emphasis on vector calculus' usefulness to applied mathematical sciences or other areas of math (if I do recall, though, a bit is addressed in association with integral theorems).

The only reason I give this book two stars is that the later parts of the book offer a peak at more advanced topics in geometry.

Last, and perhaps most inexcusable, the book requires an errata as a full supplement (I'm not exaggerating). This book is littered with errors, and not just grammatical typos! I suffered a couple of times on assignments due to incorrect formulas in the book. For example, the edition of the book I used gave the incorrect formula for the second derivative test! Now come on, they're actually charging people for this!!!

Rating:
Summary: Difficult to learn multivariable/vector calc from this book
Review: I have had several other math courses prior to and concurrent with the one in which I used Marsden and Tromba's Vector Calculus, including some of the toughest 400-level undergrad math courses at Cornell. I have done well in all of them and have understood the textbooks just fine. In my single-variable calculus courses, I got rare perfect scores on some of the exams. In high school, I ranked #8 in Maryland in the statewide math competition. Furthermore, reading comprehension is one of my greatest strengths. On standardized tests such as the GRE and SAT, I always get a perfect or near-perfect score on the reading comprehension questions. When I used Marsden and Tromba's Vector Calculus in my multivariable calculus class, I read the chapters both before and after the corresponding lectures. I spent many hours over each one, trying to understand it and working through those examples that were given. In spite of all of this, I found most of Marsden and Tromba's Vector Calculus extremely difficult to understand. (Chapter 1 was the biggest exception--it was easy.) I consider this especially problematic in a multivariable calculus course because I think it is very difficult to learn the material by lecture.

Essentially, for most of the material in Marsden and Tromba's Vector Calculus, I did not understand it until after I had learned the material by doggedly slogging through problems without the benefit of prior understanding. (By the way, many of the problems from Marsden and Tromba's Vector Calculus, at least the problems we were assigned by our professors, were far too difficult. A lot of these problems required tricks or unnecessarily difficult steps, rather than just having us practice the material we were supposed to be learning. And yet I don't think the professors were just assigning us the harder questions from the book.)

I can understand why faculty members like this book. They understand the material already. They look at this book and they see the material presented succinctly and in a way that resembles, more than the ways in most textbooks, the way that academic mathematicians do math. The problem with this way is that it is is extremely difficult for a person to understand when learning the material for the first time. Understanding the material is necessary for becoming proficient in math. Without that, a high-level presentation style is of little use. With this book, the self-described "aristocrat of multivariable calculus textbooks," I believe that a student sees a high-level presentation style, but has a hard time building understanding.

For one section, late in the course, I picked up another text instead (an old edition of Adams, which was the only multivariable calc book left at a used booksale I went to). Even though the notation in Adams was different from what I'd been seeing so far in the semester, I understood the material quickly and learned it better.

If you are a faculty member, I urge you to select, or push for the selection of, another textbook. If you are a student assigned this book, I suggest that you might consider the following:
- Use another multivariable calculus textbook in conjunction with it. Perhaps there is some multivariable calc book that is designed to be an auxiliary text, as the Schaum's Guides are.
- Print out this review and/or others of the same book from Amazon and show them to your professor, either to ask for advice on avoiding an experience like mine or to raise their awareness about how this book may be for students.

Rating:
Summary: mathematical beauty at its best
Review: I love this textbook and highly recommend it to any math major. The book is fairly technical, but the explanations are brilliant. And the authors really do try to give the student an intuitive understanding of what the underlying concepts mean (e.g. the explanation of the curl operator). Like any technical text, don't expect to read it like a novel. Applications are of periphery importance in the text (if it doesn't have "Applications" in the title, don't expect applications!)

Rating:
Summary: I recommend the study guide (to the fourth edition)
Review: The study guide to the fourth edition contains summaries of what one should learn from each chapter. I found these helpful in clarifying what the textbook was trying to present, and wished I had started using the study guide earlier in the semester than I did. I do not see a study guide to the fifth edition, so I suppose that one may not exist, but the summaries in the study guide to the fourth edition are still quite usable in conjunction with the fifth edition because the fifth edition of Vector Calculus is quite similar to the fourth edition.

Rating:
Summary: APESTA!!!
Review: This book .... big time, the examples are too naive and the explanations are too incomplete, if you have never seen a number before this book will confuse you, and if you know some math this book will confuse you even more. DON'T BUY THIS BOOK, UNDER ANY CIRCUNSTANCES!

Rating:
Summary: Best Single Volume on Vector Calculus
Review: This book in its fourth edition benefits from revisions to earlier editions. For a short quick reference the book by Schey is delightful. Marsden and Tromba give a more thorough and complete work. It is well laid out and has good illustrations. One could look at this book as Calc III with applications. For most people that part of the calculus sequence was far too quick and terse. This book is the antidote. It is definitely worth considering for use in the junior-senior level course on vector calculus.

Rating:
Summary: Worst Calculus book!!!
Review: This is a pretty good book, but it is somewhat more difficult to use for self-study. The 'easier' subjects are understood fast enough, but when it gets more difficult, you'll probably need some help to master the topics well. However, I liked working with it. It really amazes me that some reviewers think it's no good at all. Because it is.

Rating:
Summary: This book is terrible!!!
Review: This is the worst math book I've ever used, the explanations are cryptic, the proofs are incomplete and naive, the examples are dumb and unexplaining. If you want to understand anything about vector calculus, don't even touch this book.

Rating:
Summary: Poorly written introduction to vector calculus
Review: This was the required textbook for my calculus 3 course, and I found it very difficult to use. The example problems are neither useful nor enlightening; they are usually the simplest, most intuitive cases of the type of problem at hand and do not help students who are seeing this material for the first time learn how to think about more complex problems and concepts. There are errors in the answer key. The illustrations and graphs are sparse and done entirely in orange, black, and grey (in contrast to the Stewart text, with rich, useful graphics that really help students learn to visualize the material). The text is poorly written and often difficult to understand, not in terms of mathematical concepts but simply in terms of figuring out what the author is trying to communicate. Many of the exercises are poorly worded, confusing, and far too many require stupid "tricks" to solve - i.e., all of the "calculus" content is contained in a simple one-line setup of the problem, but solving the problem requires another page and a half of algebraic gymnastics involving unintuitive substitutions and so on.

There are two separate calculus 3 courses at my university. The other course used the Stewart text this semester, and their average exam scores were about twenty points higher than ours. If you must use this book, try to get a copy of Stewart; it helped me get through several problems and concepts that I would never have understood using only this book. Every mathematician and math major I have shown this book to has agreed with me that it is very poorly written, especially for an introductory text.