ISBN0716749920

After a couple of weeks of using this textbook, I am mostly happy with it. It does a good job of explaining the material, however, it is not very concise and takes a long discussion to make a point. It is longer than necessary and contains unnecessary features. Other than that, for the money it is a good deal, and thus the four stars.

First of all, what is this book about? It's just your regular multivariable calculus stuff, what some would have as calculus 2 (others as calculus 3).
That being said, from the standpoint of someone forced to live the horrors of another calculus 2 book, where the explanations are simplified to the point of not making any real sense, this is a *much better* book, because at least it attempts to give more detailed explanations, instead of shoving definitions. However, they don't appear to be exceptional and, in fact, some stuff is, well, condensed. I liked the rigor in the notation - very important to get used to healthy habits.
I do think it falls a little short of the Essence-Which-a-Calculus-Text-Must-Have, which is to relate the stuff to Physics and applications in a strong way. To reach that goal without dumbing down the explanations and theorems, or making the mathematics so detached from the applications that you loose the connections between the abstractions is a balance that falls upon an author to achieve. After all, Calculus was invented because of Physics (on that note, I liked McCallum's et al. Multivariable Calculus, which was taylored precisely with that aspect as one of its goals - but it's less mathematically advanced).
My guess is that there has to be a great calculus book for undergraduates out there, somewhere. I'm not sure this is it (Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, by Hubbard & Hubbard seems to be getting raving reviews).
It's also very nicely illustrated. As I looked more carefully, I came to realize great care was invested in crafting the illustrations - they are a notch higher in quality and really convey imporant information - you know, just the little details, or complexity, that really make a difference (but let's not get all hyped up about it - today, any modern book is - but this is book is very nice).
So, in a nutshell, although I can't vouch for the outstanding quality of the book, my message to those that complained about this being the textbook chosen at their alma mater is: rejoice! You've got a better book than I did!

Note: this review is about the 3rd edition; have only consulted the text (i.e., did not work through the whole book in a class).

This book's target audience is a little unclear. Ostensibly, this is a somewhat more rigorous treatment of multivariable calculus than a typical second-year sequence, but in fact this book is absolutely deficient as an analytical text. There are very few proofs in the book--the proofs of most theorems are relegated to an "internet supplement"--and the ones that are included are at far too low a level and fail to do what the theorems of a good text ought to do: gradually and methodically develop the topic. In some cases, such as the implicit function theorem, the statement of the theorem is just plain convoluted, apparently because the authors attempted to strike some kind of balance between being mathematically correct and working within the comfort zone of students coming out of low-level math courses.

Furthermore, nothing in the book is taught at an appropriate level of generality. For example, many "proofs" involve low-level calculations of dot products when it would be far more elegant, not to mention mathematically preferable, to use the general properties of inner product spaces instead. Many theorems and formulas are stated only for cases in which the domain is in two or three dimensions rather than working in n-dimensional vector spaces, and the complex field is essentially absent from the entire work.

So, since the book is not an analytical treatment, is it useful as a "standard" multivariable text? No. It's extremely difficult to learn the material for the first time from this book because there are numerous unexplained leaps, and examples are scarce. The exercises are useless for developing one's understanding; as other reviewers correctly noted, they frequently involve only a brief calculus setup followed by needlessly contorted algebraic operations, and students are likely to second-guess themselves when they arrive at (correct) answers that are so complicated they look wrong.

Part of the problem is that Marsden and Tromba's text is far shorter than the bulky book makes it appear. The margins, type, and spacing are outrageously generous; many pages are devoted to cute but unnecessary and often irrelevant history essays; and the pictures and figures (whose colors are badly aligned) take up huge amounts of space on the page. There is a vast amount of wasted space that could have been occupied by proofs, examples, motivation for the development of the subject, etc. It's just not worth the price of a textbook to have something with so little useful material.

This latest edition looks beautiful. Alas, it's more for the student who just wants to compete with other students and not for the thoughtful thinker.

Problems:
1. The text has no references! Not a single one!
2. The writing comes across as stiff and scared. As if the writers feel not quite adequate or are making every effort to ensure some smart *ss doesn't see through them (the newly-added, historical essays that weren't part of edition 3, but are now part of edition 5 are retarded and show the authors to be about the same)
3. There is no indication in the text that the authors really understand some of the more beautiful and useful ways of looking at vectors (for instance how to handle a vector translation of an object when u can look at the process from both a multiple vector and a single vector view; or to look at the addition of a group of vectors either in parallel [at the same time] or in series [in steps of time])

This is one of the most useless textbooks I've come across in my four years as a Mathematics undergrad.

This is more of a manual than a textbook. As others mentioned, it is wholly inappropriate for self-study and frustrating even with guided study. Methods are poorly explained and the author uses difficult-to-compute integrals or derivatives to demonstrate new material. The focus of the new material is quickly forgotten while searching integral tables, frustration builds up, and learning goes down.

Problem sets require tools or methods never mentioned or demonstrated in the preceding chapters, so it is terribly easy to get stuck or spend half an hour staring at a simple problem that requires an obscure theorem from a calc course you may have taken 3 years ago.

Chapters and sections do a poor job of explaining material. I constantly found myself needing other sources when working through problem sets. Theorems and definitions may be stated factually, but should only be considered a reference. The text does not explain their usage or meaning.

I would not recommend this book for learning under any circumstance.
Vector Calculus doesn't have to be difficult, but this book makes it so.