- Hardcover: 656 pages
- Publisher: North Holland; Revised edition (1982)
- Language: English
- ISBN-10: 0444860177
- ISBN-13: 978-0444860170
- Product Dimensions: 7 x 1 x 9.5 inches
- Shipping Weight: 2.8 pounds (View shipping rates and policies)
- Average Customer Review: 6 customer reviews
- Amazon Best Sellers Rank: #242,781 in Books (See Top 100 in Books)
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Analysis, Manifolds and Physics, Part 1: Basics Revised Edition
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..This book belongs on the shelf of every mathematically inclined physicist and every mathematician who is interested in physics...
The high quality of French mathematics, combined in this volume with the wide professional expertise of the authors in mathematical physics, has resulted in a work of great value. ... I can wholeheartedly recommend it to anyone who aspires to participate in the exciting developments in modern elementary particle physics and relativity.
... The scope of the coverage is unusually wide and the material treated with more rigour than is customary in a mathematical physics text, because only then can the results be used correctly and fruitfully...
Top customer reviews
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One needs to have had a first course in Analysis as prerequisite: Chapters One and Two offer a Review of such.
Even allowing for such prerequisite, there are a few things to keep in mind:
(1) If terms such as Convex Set, Equivalence Relation, Accumulation Point, Cauchy Sequence remain unfamiliar,
then the first thirty pages will be somewhat of a blur. Thus, familiarity with introductory Analysis is recommended.
(2) Integration and Measure, reviewed Pages 31-56, are summarized in lucid fashion. Follow along to the examples,
Pp. 43-44, excellent examples which bring the substance of Lebesgue Dominated Convergence Theorem to the fore.
(3) Clifford Algebras will be introduced as Problem One (Pages 63-67).
(4) Chapter Two: "Happiness is a Banach Space," Implicit Function and Inverse Function Theorem given due account.
The pinnacle,here, being Problems Two (Page 100) and Three (Page 105) : These conclude the Two Chapter summary
of Analysis--careful study of the Review will enable the student to peruse, with but little difficulty, the entire Textbook.
And, what, then, awaits the eager student ?
(1) Differentiable Manifolds : "...generalize the idea of a parametric representation of a surface" (Page 111).
The concept of Pull-Backs is presented most clearly. You will meet the definition of "Germs" (Page 118).
Highlighting: Fibre Bundles (introduced Page 124), on the way to Lie Brackets and Lie Derivatives.
The "...covariant vector at x can also be defined as an equivalence class of triples..." is especially interesting.
Lie Groups get detailed discussion and culminate with Problem One (Page 162) entailing such concepts as
Lagrangians and Hamiltonians. We learn: "The Euler-Lagrange equation is invariant under a change of natural
fiber coordinates, it is not invariant under a general change of fiber coordinates." (Page 164).
We delight in the relationship (derived Page 167) between Trace and Determinant.
Problem Four (Page 171) prods us to use Euler Angles to show that SO(3)--the group of rotations in R^3--is a Lie Group.
We learn: "The natural metric on SU(2) and SO(3) has a very simple expression in the coordinate system defined by the Euler Angles.
We learn: "The 2-sphere is not a group manifold. Never the less the 2-sphere has something to do with the three-dimensional
(2) Fourth Chapter: Integration on Manifolds.
"...The true nature of the integrand has been left obscure..." (Page 187). Here we meet Differential Forms.
All proceeds apace, with another pass at Pull-Backs: "We shall study their properties more explicitly than before."
Orientation defined on Page 201: "...We shall limit our study on integration to orientable manifolds."
Stoke's Theorem is given a proof on Page 211, after discussion of simplex and chains. Very interesting, indeed.
(Note the typo in the Equation on next-to-bottom Page 225, the Equation serves as prelude to the final Equation).
Immersion, embedding, submersion defined. An amusing exercise is to compare the Example at bottom Page 235
to the same example as expounded in Goldstein (Classical Mechanics,1980, Second Edition, Page 15).
The exposition of Integral Invariants of Classical Dynamics (Pages 253-258) is a delightful excursion of fundamentals.
Highlight: Problem Two, "Show that an electromagnetic field defined on an arbitrary manifold can not necessarily
be derived from a potential," and "Let M be a spacetime manifold with wormholes, show that the wormholes will
appear to be electric charges."
(3) Fifth Chapter, Riemannian Structures. An example displays a manifold which is orientable, but not time-orientable.
Connections are introduced. Covariant Derivatives are introduced. Curvature and Ricci Tensors are introduced.
Highlight: Maxwell's Equations and Gravitational Radiation (Pages 318-323).
"We shall construct various quantities analogous to the quantities used in the case of the electromagnetic field
to define a pure radiation field. The Riemann Tensor can be separated into two sets analogous to the electric and
(4) Sixth Chapter: Distributions. Reading : "We shall devise a concept more general than a norm."
Regularization is elucidated ( Page 352). Examples of Distributions are elucidated (Pages 367-371).
The examples detailed in this chapter offer much needed experience in utilization of distributions.
Sobolev Spaces introduced. "We restrict the space of distributions by the use of Hilbert and Banach space techniques."
Problem Four asks "to derive solutions of the Schrodinger Equation, given solutions of the diffusion equation, using the
Lebesgue Dominated Convergence Theorem." (Answer follows on Page 449 ! ).
The final Chapter is an elaboration of the Infinite-Dimensional Case.
As one quickly ascertains, this is a rather advanced presentation of fundamentals.
The style does take getting used to--especially if one's primary training is oriented in Physics.
However, if one possesses the mathematical maturity and stamina to persevere, all will come into focus.
This is an exceptionally interesting Textbook of advanced mathematical methods applied to Physical problems.
Unless you are in a place where all this material you can attend from lectures, this is the book that if you are (or want to be) a mathematical physicist must try to read 'a little every day', hoping that eventually things will start focusing and you will catch up.
It should be considered in a sense as THE modern analogue of Synge & Schild's Tensor Calculus - it has the same selection of topics but now all on manifolds: Analysis on Manifolds, Riemannian geometry, Integration, Connections, plus distributions and aplications to PDEs and selected topics of infinite-dim geometry.
So you have here a source-book that will not only allow you to formulate, in a modern way, physical laws (differential geometry) but also help you to study them (PDEs).
It is a profitable reading for someone who is somewhat versatile with elementary abstract mathematics, say at the level of Geroch's Mathematical Physics (algebra, topology, measure theory, functional analysis), but once you get going, you never stop!
Start off from chapter 3 and get back (or look up Geroch) if you need an explanation of a word you don't understand or have forgotten.
After you have a basic understanding a Riemannian geometry from this book you'll hopefully be able to reach Mme Choquet's new book on GR and the Einstein Equations, it is a continuation and uses the same notation, or volume 2 of the book under review on various topics in mathematical physics.
Most importantly, keep your cool and don't get intimidated!