Customer Reviews

Vector Calculus

5 star:		(15)
4 star:		(7)
3 star:		(7)
2 star:		(5)
1 star:		(28)

 

 

Having read some of the reviews written here, I feel compelled to write some brief comments on Marsden's text. If you're the kind of math student who has always learned through repetition of techniques presented by a teacher or professor in non-mathematical language (I'd say this is the majority of non-math major undergraduates), this textbook is a bad choice. Marsden's approach is relatively simple, providing ample explanations that are concise and clear for any individual well-adjusted to "reading Math"... by that, I mean a student who is fluent in mathematical notations and in comprehending proofs. For example, test yourself (here's an actual quotation from the text, taken from the discussion on limits and continuity):

Definition, Open Sets: "Let U be a subset of R^n, we call U an open set for every point x in U there exists some r > 0 such that D(x) is contained within U." [in that sense, we might imagine the open set U to be the open disk or ball of radius r and center x denoted by D(x)]

Now, this is a simple concept; but if you find yourself struggling to understand this definition in a superficial way, you might have some problems with Marsden's text. This is the kind of language that is used throughout the text--its rather bourgeois when compared to other multivariable calculus texts (another reviewer makes the comment that Marsden's text is the self-proclaimed 'aristocrat' of Calculus III texts--I find such a comment quite fitting). If you are one of those struggling individuals that nevertheless decides to use this text, here's some advice:

(1) First, read the introductory section on prerequisites and notation--put some time into understanding how to read math and believe me, it will pay off in a big way.

(2) You have to think of this text as all or nothing. If you don't use it, fine; if you do, YOU MUST rely on the text throughout the entire course, i.e. read everything (except perhaps for the historical information)... this is critical because by reading all the chapters and sections, you'll find that you'll begin to develop an ability to read the text efficiently, i.e. your math-reading skills will increase dramatically. If instead, you choose to rely on your class notes and turn to this book when you need clarification, you won't find it simply because you won't understand what you're reading.

(3) This is obvious: work all example problems as you read the text. I've found the examples to be easier than those in the exercises (other reviewers have made this observation as well), but with some help from your TA (or novel thinking on your part), you should never feel lost in attempting to solve these problems (given that you've invested the time to really understand those examples).

(4) If you have experience in Linear Algebra, you're in a much stronger position to succeed with the text. Even though Marsden introduces the topics you'll need, obviously, the experience can only help you.

(5) In difference to the common opinion, Calculus III is actually an easy course--only right up until the end (chapters 7 & 8), does the material become much more difficult. Unfortunately, in my opinion, the most poorly written chapters in the text are chapters 7 and 8. So, a word of advice: if you find yourself coasting throughout the course, be aware that the material gets harder--and be prepared for that dramatic change in difficulty. Though Marsden 'gets the job done' in explaining the material, for all students, I'd recommend thinking about a supplement text for these 2 chapters (or very good class notes; one learns chapters 7 & 8 by working through example problem after example problem).

(6) Buy (or better yet go to your college's library and find) the student solutions manual! Assign yourself problems to practice before tests; as with all quantitative and physical science courses, one learns by solving problems, bottom line.

In summary, Marsden is a good text; its simple for the trained math reader (by that, I don't mean only professors!), but very very difficult to inexperienced students. It's not light reading, and it's going to challenge you; but for those that put in the time, you'll find that the book is very logical and well-organized in its presentation of the material.

While some of my peers deem "Vector Calculus" to be a fine integration of theory and practice, I'd have to COMPLETELY disagree. From a teaching stand point, it is one of the worst texts out there (at least for a first course). At my university, some of the instructors have tried to use it as the text for the second half of a four quarter calculus sequence. This attempt has met with terrible failure, in my opinion. Most of my students (math majors and engineering students) found the book difficult and perplexing with few examples that pertained to the material they were required to learn. Luckily, the professor for my course was very good at conveying the ideas present without alluding to the text; nevertheless, I spent countless hours in discussion helping my students understand material that most standard texts would have clearly elucidated for them. In fact, at numerous points, the text becomes so involved with its own pedagogy that it neglects to delinate between important, must-know theorems and simply interesting facts.

In addition, only the very first exercises in a given section are useful for most students. A number of the later questions become interesting problems in some upper div. class, but have no bearing on the course at hand. Quite a few of them are not difficult but require "tricks" which often discourage the students by giving them the impression that they don't get the material simply because they couldn't come up with the solutions to these extraneous questions.

I would strongly recommend Stewart's text (for those of you on the West Coast) and Salas and Hille's text (for those of you in the Southwest).

Prehaps, Marsden's text would be o.k. for a more advanced course on vector calc. or as a go-between supplement for a more rigorous text.

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.

Many have noted this book as being "difficult to understand" or "not a good introduction". Some looking for a more proof-based approach found it short-coming.

I am barely in my second year as a physics student and this book makes perfect sense. I've often read ahead, did the assigned homework for the class, and returned to the book after lectures were given and found I had very little misunderstandings from the book's point of view... my finished homework needed very little corrections. If I can understand this book and its content anybody can! I am no winner of some math competition, my highest math course in high school was Advanced Trig, and the books little reference to earlier mathematicians and historical anecdotes about famous mathematicians and physicists is actually intriguing! I came from horrible high schools with crappy history departments (and math departments), so the books historical anecdotes are nice detours that makes me want to learn more about these great guys!

With that said I have taken courses in proof-based Advanced Calculus (single-variable) and introductory Real Analysis, those classes were invigorating and practically gave me heart palpitations! But to much sadness this book does not take that route... it is not meant for that. The book takes the "applied mathematics" approach for direct applications to engineering and physics. The proofs are elementary (in-accordance to an Analysis books) and are meant to stimulate the imaginative and theoretical approaches in the minds of soon-to-be physicists and engineers. So, yes its proof-based approach is short-coming for a reason.

I have no idea why some reviewers (gallantly stating their resume of math competitions they've became victorious in) have stated its difficulty as so. I'm a poor kid in college on scholarships (due to my minority status, and not for direct scholastic purposes or merit) and I enjoyed this book... like I said if I can understand it anybody can.

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.

Ch. 1 on the geometry of Euclidean space is unreasonably lengthy (93 pages). Most representations and interpretations are too childish for a student of vector calculus (for example the interpretation of vector addition in terms of flying birds on p. 6). And most examples are numerical substitutions in formulas, which is obviously not the subject of vector calculus.

Mathematical definitions are not precise and notions, such as surface, which deserve some discussion before they appear for the first time, are assumed well known and used in examples without having being previously defined. For example the notion of a surface is first used in example 3 (p. 72) of spherical coordinates but discussed in ch.7.

The discussion of mathematical entities in terms of physical and geometrical objects is too poor. For example in p. 95 the reader is just told that the temperature is a function T(x, y, z) without any further discussion. In the same location the example that the velocity is a function V:R4->R3 is rather confusing and Fig. 2.1.1 depicting the velocity as a non-tangent vector of the streamlines (it is also questionable if they are in fact streamlines, since the velocity depends on time) creates confusion rather than illustrates vector valued functions.

Graphs of functions of several variables and related notions are presented on pages 96-105. This is the first "contact" of the student with surfaces, but the notion of a surface itself is left out of the discussion. Also the discussion does not create physical intuition for functions f:R3->R3, as would, for example, a discussion in terms of deformations, rigid rotations etc.

Limits are according to the authors "not necessary to completely master in order to work problems in differentiation". In this spirit, the 12-line definition of a limit on p. 112 is vague, not understandable, not usable and cannot be easily remembered. As a consequence, the following examples are handled purely descriptively and then the reader is invited to intuitively understand the results of the next two theorems on the uniqueness of limits and some properties of limits. No proof, of course.

After the discussion of limits and continuity the reader is expected to grasp the notion of f:R2->R function differentiability on p. 133 as the limit of a formula extending over the width of a page and involving the norm of a vector on the denominator. In addition, the reader is requested to understand that the expression on the numerator is a "good approximation" - the notion of goodness again being undefined. Following this definition comes a discussion of tangent planes, but this is not properly related to the notion of a differential and differentiation.

After the presentation of the general case of differentiability (p. 134), which additionally involves a norm and a matrix in the numerator, the reader will be so frustrated as to completely give up any effort to get a working knowledge of the subject and continue reading just to pass the exams.

Mathematical proof seems to be something like the "poor relative" in this book. There are some instances of the word throughout, but even these are in fact calculations rather than mathematical proofs. Mathematical reasoning, logic and thinking are totally absent. Is mathematics according to the authors not the right place to teach them to scientists and engineers?

The way multiple integrals are treated in ch. 5 differs from the way antiquity scientists (Archimedes ca 200 B.C.) used to treat them in notation, the complexity of evaluated examples and the rules of integration. But the way the latter are formulated is so unclear, that their understanding and applicability by the student is questionable. For example the theorem of integrability of piecewise continuous functions is called "Theorem 2: Integrability of bounded functions" (p. 330) which is misleading; it gives the erroneous impression that all bounded functions are integrable. In the same theorem the hypothesis, i.e. piecewise continuous functions, is not explicitly stated; instead it is supposed that "the set of points where f is discontinuous lies on a finite union of graphs of continuous functions" - a not so easily grasped expression esp. by the novice. Nothing is said about singularities encountered in many problems of mathematics, physics and engineering. No mention of the necessity for an integral freed from the restrictions of the Riemann integral.

Summarizing, the book is unreasonably sized, not concise, most examples do not make sense, mathematical rigor is absent, discussions do not synchronize well with material ordering, there are points that might create confusion and misunderstandings and as such is not expected to help those wanting to get a working knowledge of calculus and is not useful as a reference. There are many books on calculus and analysis available. I mention the excellent book by Courant and John, "Introduction to Calculus and Analysis" in two volumes (2nd was split in two parts in latest edition) for its perfect balance between mathematical rigor and physical intuition, the readability, the easily understood arguments used by the authors and the carefully selected exercises.
Introduction to Calculus and Analysis, Vol. II/2 (Classics in Mathematics)Introduction to Calculus and Analysis, Vol. II/1 (Classics in Mathematics)

I used this book when I was an undergraduate. At the time, I found the lack of practical problems and motivation frustrating and difficult to read. I did not have an apprieciation for the beauty of the presentation. Now that I have a Ph.D. and am a professor in Mechanical Engineering working in the area of Computational Mechanics, I find myself pulling this book off my shelf often to review fundamental vector concepts, especially in Chapters 6 and 7 on integrals over paths and surfaces and Green's, Stokes', Gauss', theorems from a modern and clean derivation. I would highly recommend this book to graduate students but do not recommend the book for undergraduate students.

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!

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!!!

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.