Vector Calculus

Rating:
Summary: Excellent for self-study
Review: I purchased this book to further my self-study of Physics. Simply put, this is the best written book I have read on any math subject (although Spivak's Calculus is also very good, but is on a different topic).

A wonderful feature of this book is the problems - and complete solutions - provided throughout. The problems are not so extensive that you will skip them (there about 10-20 per chapter, as opposed to many other texts which have fifty or more) nor so difficult that you will give up in frustration. Indeed, for self study, the simple fact that there are answers is key; the fact that they are not of the sort where you are asked to derive new math or otherwise are treated as if you are a "student who believes he is to become the next Einstein" is a refreshing change.

This book covers vectors (of course), various vector integrals (line, surface, and volume integrals), and gradient, divergence and curl (if you ever wondered what that odd upside down triangle symbol is, you will learn), before moving into more advanced topics. It briefly covers "suffix notation" and tensors, as well as transformations into different coordinate systems. In sum, about half the book is wonderful and immediately useful, and the other half will become more useful as I encounter it in the course of learning physics.

That said, I wish it went into more detail in the "suffix notation" section, which is quite confusing (although several readings helps) as well as more on tensors, which extensively employs the suffix notation.

I have been told, without confirmation, that Springer is known for it's clarity; this book certainly is extremely clear, well written, and has allowed me to finally begin to understand many parts of mathematical physics which were previously Greek to me.

Rating:
Summary: Excellent Review/Introduction
Review: I'm currently using this book to review vector calculus for comps. By "review," I really mean "learn" as line integrals escaped me in my multivariable calc class two years ago. I've tried since then using Stewart's "Calculus" (3rd ed.), the book used in the above course, but to no avail. The presentation in Matthews "Vector Calculus" is far superior to Stewart's in that the examples are much clearer and the the steps in solving each type of problem are spelled out explicitly. Matthews tries to tie each subject closely to physical applications, which even as someone who prefers his math pure, I find this approach very helpful and most appropriate to the subject.

Matthews moves quickly from an intro to vector algebra to line, surface, and volume integrals, to divergence and curl and more. Those looking for theoretical depth won't find it here. That's not to say the book is not rigorous or merely "conceptual" In general the rigor is on a level with 90 percent of calculus books. That said, I think this book provides an excellent introduction to vector calculus for physics or engineering students or a very solid review for math students. In fact, the only thing that keeps the reader from mastering the material is the shortage of problems. The problems in the book are good and full solutions can be found in the back; there just aren't enough to get a mastery of the material, but you can go to the library for another calc text with more problems. Again, the strength of the book is the clarity of the explanations so after reading through the new material, you can easily move on to more exercises outside the book.

The book is designed with self-study in mind thus it is self-contained. I've found it extremely useful, and I'm not someone who can easily teach himself mathematics. In short, Matthews has written a book that is accessible to someone studying on their own yet rigorous enough to be useful. After this book, I'm already considering purchasing other books in the same series.

Rating:
Summary: Nice and Succinct Book on the Subject
Review: If there is one thing I adore about Springer books is that they are cheap, to the point, and very accessible. This book is no exception. I used this book for self study after I took a Calculus III course. My understanding of the concepts I learned in class improved two-fold. There is really nothing negative I can say about this book. It is probably the best buy I've ever made yet. I wish all college texts were like this.

Rating:
Summary: not 80% but...
Review: Someone has said in a review that
Its mathmathical rigor is about 80%
but as a student majoring in math I
cannot agree with that. I can't give
it more than 60%.
It's the kind of the book which will
never be met in the math class.
but that does not mean this book is not
good.

it lacks in mathmatical rigor but that's
not problem. it's quite clearly written,
easy to read, and most of all It explains
you the 'meaning' of the equation.
lots of math book you should try youself
very hard to understand its meaning.
they give you the proofs of theorems
definitions of mathmatical objects
but not the meaning nor its context.
that's something you should find out.

but this book gives you that. It explains
you what is the meaning of it.

If you are majoring in math. this can be
a gread secondary text. If you are not.
then this should be your first choice.

Rating:
Summary: nice read, no rigor, too few exercises
Review: This book is quite easy to read, and it gives a good intuitive picture of the subject. Mathematicly it is not of so much value. I think its very good for someone who wants to study electromagnetic fields, or some other fields in physics and want to be able to calculate different integrals (perhaps there are more applications than physics).

Physicists often do non-rigorous arguments, and it is very possible to do so and still be certain what you are doing makes sence. Rigor can take up too much of your time :)

But if you want to go deep into physics I would recommend spending your money on some deeper and more rigourous text, it will be useful in the long run.

There are some incorrect proofs, that I believe are absolute nonsense, but as i said, this book is only good for your feeling of vector calculus and ability to calculate integrals.

I would give the book a 4/5 but you simply cant learn calculation without many problems of varying difficulty. If this is your only book, it is absolutely essential that you get some kind of collection of exercises too. There are only 9 exercises on the chapter on curvilinear coordinates (cylindrical, spherical, etc.)

I think you can get a better book for the money, but its not too bad.

Rating:
Summary: Good but Brief
Review: This book is very clear, but is mostly a collection of work with very little explanation. This is good when you understand the concepts; there is less to lead you astray. However, if you don't grasp an idea from what is provided, there is little recourse. I used this book in conjunction with the (ubiquitous) Stewart multivariate calculus book. It's not very expensive so I reccomend it.

Rating:
Summary: Great book when used in the right context.
Review: This book is very useful as long as you know better than to expect to use it alone. The only way I ever used this book is as an adjunct to my textbook (Stewart's multivariable). Where my big text by Stewart is far, far more rigorous on any given topic, this book by Matthews just punches through to the core concepts. When the big book is making you crazy just relax, take a deep breath and pick up this one. Once you understand the most basic fundamentals of what is happening, go back to your big textbook and get to work figuring out the more subtle mathematical nuances of gradient or curl or whatever. There are also some really useful example problems. Given the low price this book is a definite winner.

Rating:
Summary: Great book when used in the right context.
Review: This book is very useful as long as you know better than to expect to use it alone. The only way I ever used this book is as an adjunct to my textbook (Stewart's multivariable). Where my big text by Stewart is far, far more rigorous on any given topic, this book by Matthews just punches through to the core concepts. When the big book is making you crazy just relax, take a deep breath and pick up this one. Once you understand the most basic fundamentals of what is happening, go back to your big textbook and get to work figuring out the more subtle mathematical nuances of gradient or curl or whatever. There are also some really useful example problems. Given the low price this book is a definite winner.