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Title: Manifolds, Tensor Analysis, and Applications (Applied Mathematical Sciences, Vol 75) by Ralph Abraham, Jerrold E. Marsden, T. Ratiu ISBN: 0-387-96790-7 Publisher: Springer Verlag Pub. Date: August, 1996 Format: Hardcover Volumes: 1 List Price(USD): $84.95

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Average Customer Rating: 3.25 (4 reviews)

Rating: 5
Summary: A complete book by very erudite authors
Comment: I actually read this entire book--it is quite long and dense. Actually I took the course from the author Jerry Marsden at Caltech and Tutor (Jerry's friend and co-author) gave a guest lecture while visiting. We flew through the entire thing and ch 9 on lie groups of his mechanics and symmetry text in a short 10 weeks! My background in math was relatively weak when taking the course so it was a little hard to keep up; i.e. I came from an engineering background. Anyway, it is probably the most complete/diverse text I've come across on the subject. Of course, it's actually more of a monograph than a text. Since I've read the whole thing, I have to admit there are "several" typos. But as it is that most people can't even write a damn email without a typo or two, the book really does a good job considering it is 800 pages of mostly dense mathematical rigor. I imagine that if I wrote 800 pages of mathematical symbols in latex, that I might forget a tilde or put something as subscript that should have been superscript here or there! None of these errors really matter too much-they should not hinder one's understanding. All and all I think that this book is a great ref, although I've never seen the index, if one exists. For the beginner, also check out Boothby's book, which covers a lot of the same material but tones it down a bit.

Rating: 5
Summary: A Unique Reference
Comment: Students of mathematical physics in general, and general relativity in particular, face a formidable challenge in attempting to find coherent, readable references on manifold theory and tensor analysis. I think it fair to say that for every well-written work on the subject, there are ten that do more damage than good. Very few texts can claim to (1) be clear enough to assist the person who is studying alone, (2) offer valuable PHYSICAL insight into the subject, and (3) pass the standards of rigor that mathematicians would impose. Abraham, Marsden, and Ratiu manage to accomplish all three of these goals in this profoundly useful text. I studied from the first edition and I have taught from the second. The two chapters on differential forms, Hodge star duality, integration on manifolds, and the generalized Stokes' Theorem alone are worth the price of the entire book. I am unaware of any other reference which which treats differential forms with the same combination of clarity, physical motivation, and mathematical rigor. The concluding chapter on applications offers one of the clearest introductions to the relativistic form of Maxwell's equations to be found in any text. For students of physics who want to see the mathematics "done right," one would be hard pressed to do better than Abraham, Marsden, and Ratiu.

Rating: 2
Summary: mixed bag: many virtues but many weaknesses
Comment: I took a course taught by the 3rd author (Tudor Ratiu) at UCSC using this book; I found both good and bad in it. Much of the bad for me was overcome by the inspiring and energetic presentation by one of the authors. One may view this book as basically a detailed elaboration of the "preliminary" chapters of the book "Foundations of Mechanics" by the 1st 2 authors. The strengths of this book are (a) the treatment which is general enough to include infinite-dimensional manifolds and not just the finite-dimensional case (most books just talk about the finite-dim'l case) and (b) the attempt to cover all theorems "full strength" (in the greatest generality obtaining the strongest conclusions from the weakest hypotheses). Neither of these (not counting the many typos) recommends this as a first or even second text for students, but it's hard to find any other books that treat the material at the same level of generality and precision, which is a must if attempting "hard" global analysis in areas such as fluid mechanics (from a geometric point of view). Correction of the many typos could make this an indispensable reference book for those requiring the techniques discussed. More discussion of finite-dimensional examples before jumping to infinite-dimensional ones (e.g. discussing finite-dimensional Grassmannians before jumping to the infinite-dimensional Banach manifold version) could make this into a tolerable text.

As it is, it's problematic, aggravating, and not for the faint of heart, but not without its virtues.

Possible alternatives for the infinite-dimensional point of view are Lang's manifolds book or some volume of the expensive multi-volume treatise on analysis by Dieudonne.