Analysis, Manifolds and Physics, Part II – Revised and Enlarged Edition 1st Edition by Y. Choquet-Bruhat (PDF)

Description

Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.

User’s Reviews

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

⭐This is a great book for anyone wishing to go beyond elementary mathematical physics. Cecile DeWitt was one of the truly great mathematical physicist of modern time. She will be sorely missed.

⭐Everything OK !

⭐This, a review of the First Edition (published 1977). One needs to have had a first course in Analysis as prerequisite: Chapters One and Two offer a summary, a review of such. Even allowing for such prerequisite, there are a few things to keep in mind:(A) If terms such as convex set, equivalence relation, accumulation point, Cauchy sequence are unfamiliar, then these initial thirty pages will be somewhat of a blur. Thus, familiarity with introductory Analysis is recommended.(B) Integration and Measure, reviewed (pages 31-56) A summary presented in lucid fashion. Follow along to the examples (pages 43-44), these excellent examples bring the substance of Lebesgue dominated convergence theorem to the fore.(C) Clifford Algebras will be introduced as exercise-problem one (Pages 63-67). Chapter Two: “Happiness is a Banach Space,” the implicit function and inverse function theorem given proper account. The pinnacle being Problems Two (page 100) and Three (page 105). These conclude the dual-chapter summary of Analysis–careful study of this review will enable the student to peruse the entire textbook with profit.What, next, awaits the eager student ?(1) Differentiable Manifolds : “…generalize the idea of a parametric representation of a surface” (page 111). Pull-Backs presented 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 exercise-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-dimensionalrotation group.”(2) Fourth Chapter: Integration on Manifolds. Read: “…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 (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 is a delightful excursion of fundamentals (pages 253-258). A 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. A highlight: Maxwell’s equations and gravitational radiation (pages 318-323). Read: “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 andmagnetic components…”(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.(5) In conclusion: 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 possessive of mathematical maturity and stamina to persevere, all will come to fruition.This is an exceptionally interesting textbook of advanced mathematical methods applied to physical problems.Highly Recommended.

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