- Paperback: 160 pages
- Publisher: Westview Press (January 22, 1971)
- Language: English
- ISBN-10: 0805390219
- ISBN-13: 978-0805390216
- Product Dimensions: 5.5 x 0.4 x 8.2 inches
- Shipping Weight: 9.1 ounces (View shipping rates and policies)
- Average Customer Review: 46 customer reviews
- Amazon Best Sellers Rank: #209,158 in Books (See Top 100 in Books)
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Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
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From the Back Cover
A supplementary text for undergraduate courses in the calculus of variations which provides an introduction to modern techniques in the field based on measure theoretic geometry. Varifold geometry is presented through and appraisal of Plateau's problem.
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Top customer reviews
Another thing I like about this book is that it swiftly builds up the multivariable analysis theory without too many pit stops. One thing I hated about Munkres is that he too way too long to develop the multivariable riemann integral. Munkres takes three steps to developing it (rectangles, Jordan-measurable sets, and then open sets), and on each stage he reproves all of the facts that we know the integral should have. Spivak, on the other hand, develops the integral over rectangles, tells you in a sentence how to generalize it to Jordan-measurable sets (that's all that was needed), and then uses partitions of unity to define the more general integral. Spivak's method is faster, gives us a good look at how partitions of unity can be used, and uses the fact that the reader should be able to prove and predict the properties that the integral should have based on the assumption that we've dealt with the single variable case before. The makes Spivak a much quicker and interesting read than any other book on the subject.
While I do like this book, it is not without flaws. The general opinion is that this book is a little too terse on explanations sometimes. For example, the one example Spivak gives on how to take a derivative, he identifies that derivative of the projection mapping with the i-th standard basis element. That is, he is identifying the dual space of R^n with R^n itself, all the while not telling us. While this is a nice trick that can help us take derivatives faster, this should have been mentioned in the text. Chapter 4 is rough as well. Many times Spivak will say that a theorem is obvious in an easy case, then give you a sentence on how to generalize it to the more general case. There have been many times where I have had to write my own proofs because his were lacking in detail. My margins are full of notes and missing steps because of this. For me, this wasn't too bad because I learned the material really well by being so involved, but I can easily imagine many readers being left in the dust. However, this is a good book, and these flaws only detract one star in my opinion.
The next thing we need to ask is what do you need to read this. You will need a very solid understanding of single variable analysis. If you haven't read Rudin's Principles of Mathematical Analysis, Third Edition yet, now is a great time to get yourself a copy. Also, you will need a strong linear algebra background. My personal favorite is Axler's Linear Algebra Done Right, but many people like the book by Hoffman and Kunze. One you have this background, this is the best place to go if you're looking for a quick lesson in multivariable analysis and your first introduction to manifolds.
After you're done with this book, you're going to have to buy a more serious book on manifolds. This book if good for getting your feet wet, but there are so many essential things left out. I prefer the books by John Lee over Spivak's Vol. I, and so I recommend you look into those.
EDIT: I've since purchased Spivak's "A Comprehensive Introduction to Differential Geometry, vol. 1". Chapters 4 and 5 of "Calculus on Manifolds" are essentially contained in this book as well. However, they are explained in much fuller detail, and the book contains much more information regarding the subject. I would greatly suggest it to anyone reading this who felt unfulfilled or yearning for more from chapters 4 and 5.
I recently reread this book and was happy to recall the magic of this great introduction to the real mathematics.