13 of 17 people found the following review helpful
mixed bag: many virtues but many weaknesses,
By A Customer
This review is from: Manifolds, Tensor Analysis, and Applications (Applied Mathematical Sciences) (v. 75) (Hardcover)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.
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Initial post: Jul 24, 2012 1:16:29 AM PDT
The only alternative for infinite-dimensional manifolds is Lang's book, but Abraham-Marsden-Ratiu is so much better. There is no volume on this subject in the otherwise excelent multi-volume book on analysis by Dieudonne. I think this book deserves all five stars. Try last available edition, it has many improvements and corrections.
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