Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach 1st Edition

I thought it was about time that I start writing reviews of books that I think are special, unique, or beautiful, and I decided to begin with the Hubbards’ Vector Calculus. Actually, a single Theorem in the book is what decided for me: when I saw this Theorem, I knew immediately how genuinely special this book is, and I felt compelled to share.I purchased a used copy of the first edition nine years ago, and although I did wander through it here and there when I first got it, it has mostly been ignored along with other math books that I have kept or bought for various reasons. Only recently have I started going through it more regularly, but still rather lightly, and strictly for recreational purposes.Some quick background for context: I have a degree in Math, so the material is quite well-known to me. It is clear from what I have been reading that this book is a unique labor of love, with more insights than I can begin to count, but, again, it was that one Theorem that closed the circle for me and told me everything I needed to know about the superior quality of this book: Theorem 1.5.10 on page 78. The Theorem actually has a title: Elegance is Not Required.The sense of humor involved in actually expressing this as a "Theorem" made me immediately fall head-over-heels in love with this book. Even if the rest of it fails to impress – it won't, I've already looked; and it can’t, because it possesses too much élan - this "Theorem" alone was worth it. The “Theorem” says that if you ever have to make a quantity less than epsilon, it is perfectly acceptable to make the quantity less than, say 5*epsilon, or radical(epsilon^3) – as long as the expression containing epsilon goes to 0.Anyone who has ever had a pedantic professor or grader tell you that you have to adjust your epsilons along the way in order to end up with JUST epsilon on the right of the inequality will appreciate this “Theorem”.In any case, if you enjoy sophisticated mathematics presented in a unique, friendly, often humorous, and still fully rigorous way, then you will both enjoy, and take away much from, this book. And, by the way, the presentation of Differential Forms is wonderful. That the authors are the type of people who, in the kindest way possible, and for Mathematics’ own good, feel compelled to occasionally pull the rug out from under the feet of mathematical dogma is an extra special treat.The spirit of the book is endearing. The mathematics is beautiful. The style is superb.

Excellent book.The condition of the book was a little lower than I had been led to believe.

I was recommended this book and am grateful I brought it, though I never have had to use it in a course. The price is way too high, so it is worth looking at before hand, I did not do this but I had the money.From a students point of view things that make this book good are the side notes, how the book mixes calculus with linear algebra and the variety of topics.The side notes are the things any good lecturer would tell you in class, but not put in the notes. The kind of things that give you insight into what the ideas mean or how they were developed. These are spread through out the book such that there is space reserved on each page for it.I loved the first half of the book, it is amazing clear in the way it shows how ideas are linked between different topics of mathematics. The pace has a vivid flow, with the proofs being just as vivid. The references to other parts of the book are great, sometimes the author even refers to a main theorem later on in the side notes so that you can keep in mind it is related.The last thing I liked was it was not a typical text book, it was written for anyone who has interest in the ideas and loves to understand mathematics. I had used standard text books for topics like linear algebra and calculus (involving an author called Howard Anton), I never liked them since they did not teach things in a logical order and were amazingly bland, no personality at all. I can say that I have gone over all the start chapters, and even though I new a lot of the maths before hand I felt I learnt just from rereading it all. I was picking up new ways to look at things even though I thought I had a pretty good idea of what the math meant.The only thing to watch out for is that the book could be a bit challenging if you have not seen multi-variable calculus before. I was not in this position, but it goes in depth with wonderful ideas, so it is good to be comfortable with the foundations to get the most out of it.