First of all, don’t be fooled by the “prerequisite” in the preface. You should have completed one term of linear algebra, analysis and multi-variable calculus. As I have already self learnt these topic, I took this book as another self learning challenge.
This book is really well written. I really like its compactness and conciseness, though this means not much explanation is given usually. I have read through chapter 1 to 4,
these chapter give complete building block for doing calculus on higher dimensions. By the time I finally get to the stoke theorem, I was really excited. This definition about forms, chains, d and ' are quite complicated. But after all, the stoke theorem and the problems can explain everything about stuff like curl, div, grad, and other “stoke type theorems” you learnt from multi-variable calculus course. So the effort have paid off.
I and have done all the exercises. There are few reasons why you must do the exercise:
1. This book lacks example, it is filled with definition, theorems, and proofs of theorems. You can understand nothing if you omit the exercises.
2. Most of the exercises are quite doable. I can do many of them by myself (despite the fact that have I no teacher to seek for help). Though sometimes I am stuck, I usually find the answer from the internet.
3. Some exercises are targeted at the given theorem. They explain why such condition in the theorem is necessary, and what counter-example will appear if they don’t. Sometimes the text even rely on results proven in previous exercises.
By the way, you must get involved in the context. Sometimes when you feel the context is not enough, search from the internet, and you learn more.
As for the downside, this books contain quite a few flaws. Sometimes the assumption of exercise problems are not clearly stated. Now I have finished chapter 4, the reason I choose to pause here is because I found that in chapter 5, the definition of manifold is over simplified. I don’t feel it is suitable to learn such a important concept from this thin final chapter. I will probably seek for another book. So after all, I learnt nothing about manifold from this “calculus on manifolds”.
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