Vector Calculus

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.

This book is excellent in that it does exactly what it is meant to do: provide a thorough introduction to vector calculus. The book is not meant to be read only at the level of something like Stewart, but to probe a little deeper. Explanations are thorough, almost to the point of redundancy sometimes, but they hammer the concepts home. There are a lot of good problems in this book as well, mainly concentrated in a few fields of physics (electrodynamics, thermal, gravitation, etc.). That said, this book is by no means an application book. There is about one section each chapter devoted to thorough proofs of some of the more finer points of the theorems presented. For the uninterested reader, this may be skipped, but even these explanations are presented in a way to give the reader a good introduction to proofs and analysis. (Examples in these sections are proofs). All in all, I found this book to be the descriptive, through treatment of vector calculus I was looking for. For something more application and less theory oriented, look to Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, Fourth Edition.

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.

I taught myself vector calculus from this book. It is clear, concise, and fabulously well written. Though it doubtlessly makes an valuable reference, you will enjoy just reading it from cover to cover. Really a treat, worth the price.

The way this book explains calculus relating to multiple variables is somewhat different from old others. After reading from cover to cover, I come to settle many why and why in multiple calculus. If I have to read only two books in college, I will not hesitate to read this first.