The Geometry of Jet Bundles (London Mathematical Society Lecture Note Series Book 142) by D. J. Saunders (PDF)

Description

The purpose of this book is to provide an introduction to the theory of jet bundles for mathematicians and physicists who wish to study differential equations, particularly those associated with the calculus of variations, in a modern geometric way. One of the themes of the book is that first-order jets may be considered as the natural generalisation of vector fields for studying variational problems in field theory, and so many of the constructions are introduced in the context of first- or second-order jets, before being described in their full generality. The book includes a proof of the local exactness of the variational bicomplex. A knowledge of differential geometry is assumed by the author, although introductory chapters include the necessary background of fibred manifolds, and on vector and affine bundles. Coordinate-free techniques are used throughout, although coordinate representations are often used in proofs and when considering applications.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is a classic on the theory of jet manifolds. Whenever I read a paper that uses the theory of jets, I am almost certain that the references section is going to include this book and, surely enough, when I take a look, it does. This is the only book out there that accessibly and rigorously treats the theory of jet bundles. When I say accessibly, I mean that a student or a professor who took a decent course in differential geometry can read and understand the book. The examples and exercises also include the application of jet bundles to Lagrangian and Hamiltonian theory of motion (field theory).Contrary to what one of the reviewers say about the book, the book almost never abuses the notation. Whenever it introduces an abbreviation, either what is meant is clear from the context or the book clearly states that it will perform this abbreviation. I suppose if you start reading the book as a reference; that is, start reading a chapter or a section, you might come across such an issue. However, if you start reading the book from atop, this is never a problem.

⭐then ok buy it, but don’t expect to do a self-study with this one. The notation is so sketchy that you have to rewrite the book in case you need to learn the material and not just skim through the book. Abusing notation is a commonplace in the practice of math and at the same time confounding for the newcomers to any topic. Of course there are case where the notation would be unmanageable without simplifications, however having to *guess* that a map is not actually the one you see written but its pullback, doesn’t seem to improve clarity. The book falls in the typical category where exposition aims at anything but communication with the reader. I didn’t give it 1 star because if you learn the material elsewhere it can serve as a “road-map” of how several results connect to each other or of areas where they can be applied (e.g. jet manifolds, calculus of variations). Sometimes a useful big picture is given.

⭐This is a very good book on jets. Jets can be thought of as the systematic generalization, to any order, of concepts such as vector fields, flows, sprays and partial differential equations, carried out through the language of fiber bundles. For example jets can be used to represent a Taylor series expansion of a map. The treatment of the subject is comprehensive and developed both intrinsically and by using coordinates. There is a nice part devoted to the theory of vector-valued forms, graded derivations and the Frolicher- Nijenhuis bracket. I just did not give 5 stars for two reasons: i) on one side the book seems too ambitious, the presented material is so wide that there is the risk for the reader to get lost in the formulas, especially in self-study, ii) on the other side I would have expected to see a little more application of the theory to the calculus of variations. Even so, a very good book on the subject.

⭐I had to study on this book and I must say it scares you at first, but it is very complete and probably the only one focused on bundles and jets. I found useful to read elsewhere some resume to have “the big picture” which is difficult to grasp at the beginning (see my review of Giachetta and others’ New Lagrangian and Hamiltonian Methods in Field Theory). I think it is a complex book because the topic is in itself complex enough, but there are many examples, each proposition is demonstrated, each definition justified, and there are remarkable suggestions to further readings and links to other geometrical topics.

⭐Ottimo libro

⭐

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