Mathematical-tools

Didier Sornette who is potentialy nobelisable realises with this book a great present for everybody interested to the recent progresses toward the physics of critical phenomena.
Particularly, it shows the way that some seismologists follow toward the ultimate goal to predict the event of large earthquakes. If this task was impossible yesterday,Didier Sornette shows that it is now became realisable.

Sornette's book is quite an achievement both in quantity and
quality. The presentation remains informal and quite readable; it reads like a physics textbook, not a math textbook. The references are very extensive (a total of 832! altogether) and they are a very valuable component of the book. In fact much of the book is about the reference material. You might choose to read the book instead of the 832 references... I think this is
the point...

There's probably nothing wrong with this book besides the fact that it throws it all at you at a high degree of sophistication and in as terse a way as possible, it seems. It's a unique and beautiful achievement but because it is so dense with information and insight, it seems every word counts for ten and you'll want to read several chapters again and again. Also, even though there is a clear unifying theme from chapter to chapter, the book simply ends almost like in the middle of a sentence. After machinegunning out 392 pages of material at research level spanning quite a few scientific fields, there is absolutely no attempt to put it all together. It's up to you to do it and it almost seems like the author is indirectly suggesting you start reading it all over again to "get it"... So, for the second edition, perhaps the author will be bold and add ten pages of wrap up material at the end so that this will read less like an atlas. Apart from that, it's the best!

I would rate this book as a gem! To calibrate that let me say that I think Weinreich's Geometrical Vectors and Foster and Nightingale's General Relativity are gems. Chapter 1 gives a beautiful, clear and concise introduction to Clifford Algebra in flat 3-space using Dirac's anti-commuting gamma matrices. If you have ever wondered about off-hand comments that rotations are double reflections and why half angles enter into this business this is the place to get enlightened. In an amusing series of photographs the author illustrates the 4-pi periodicity of certain objects. The object here is a copy of MTW's Gravitation - one of the more imaginative uses of this tome. As an example of the application of the CA results the chapter ends with a treatment of the spinning top without using Euler's equations for rigid body motion. If you have ever struggled through Goldstein's Classical Mechanics treatment of this problem, from Euler angles to infinitessimal rotations to d-Omega which is not a differential of a vector to dyadics to body diferentials and space differentials to Euler's equations, you will really appreciate Snygg's direct solution using CA. Sure, I know Goldstein's has to be a general treatment of solid body motion and thus more complex so he can treat more general problems, but it is good to find a more direct solution that is cristal clear and only a few pages long. This chapter is real little gem. Chapter 2 takes CA to Minkowsky 4-space rotations. Chapters 3 and 4 take you to flat n-dimensional spaces and curved subspaces embedded in them. Again beautiful explanations are presented of the meaning of tangent spaces, parallel transport and how the covariant derivative arises naturally in curved spaces. I had the silly hope that with Clifford numbers and their products all would be well and done. Unfortunately the exterior product wedges its nose under the tent flap and pretty soon the exterior derivative and its side-kick the co-differential operator soon follow it into the tent. All this is explained in Chapter 7. With Chapter 5 the learning curve steepens with the introduction of Fock-Ivanenko 2-vectors and the curvature 2-vector (or 2-form) and finally the curvature tensor. Chapter 6 solves the field equations for the Schwartzchild metric based on the F-I 2-vector approach. Chapter 8 on the Dirac equation is again an approach different than that found in the usual texts. Chapter 9 derives the Kerr metric, something you won't find in MTW published 8 to 10 years after Kerr's papers. Unfortunately the starting point is some obscure problem from an earlier chapter and Snygg does not provide the delightful physical insight of earlier examples. However, there is discussion at the of the chapter. While you might be able to solve the field equations for the Schartzchild metric on your own, once you know it can be done, I certainly would not be able to do so for the Kerr metric. Snygg takes you through step by step, none of them particularly difficult, but the sequence is certainly not something I would have found by myself. Chapter 10 I only skimmed, the index notation, with underscored and bracketed indices, becomes overloaded for my level of sophistication. Chapter 11 organizes all the matrix stuff together, again a beautiful, straightforward and clear presentation. Here is shown how to construct a matrix representation for the gammas. As you might expect, the book is a veritable beehive of sub- and superscripts over bars and carets Greek and Latin indices and full of gamma gymnastics. Even Pauli's less complementary comment on Dirac algebra comes to mind. The text has a few typos but blessedly few in the Clifford number and gamma indices. By the way, if you expect to find out how to do trace computations on gamma expressions you won't find it here. The explicit form of the gamma matrices is hardly ever mentioned until chapter 11 nor is it needed in the present context.

Snygg's book is a thoroughly delightful introduction to Clifford Algebra and its applications in physics. It is detailed, readable, and at times even humorous... but always clear and educational. Snygg presents Clifford Algebra above all as a practical tool, rather than as the ultimate algebraic representation of spatial geometry. This gives a refreshing alternative to Hestenes' writings which, although quite good, can at times be philosophically pedantic and difficult to connect with standard theory.

From a letter to the author. John, I have to write you to tell you what a wonderful book you wrote. I still can't believe how good it is. Yesterday I was waiting for a television show to begin in ten minutes and I picked your book up while I sat front of the TV set. When I finally looked up 45 minutes later, I had missed the show! In 35 years as an algebraic topologist, I have tried to learn various things about Clifford algebras because of their role in K-theory and in the Atiyah- Singer Index theorem, and more recently because of the Seiberg-Witten equations. With only mediocre intensity, mostly browsing, I have had little success. In the month since I met you and bought your book, I have browsed through it while occupied with several other competing projects. In the process I have internalized the classification of Clifford algebras, learned how physicists use Dirac's equation, what they are doing when they talk about gauge theory, understood Hodge duality much better and so the codifferential operator. And I still have only browsed through a small portion of the text. I think we mathematicians should study your book to learn how to improve our own levels of exposition. Sincerely, Daniel Henry Gottlieb

This book has an enormous number of mathematical functions implemented. Unfortunately, the quality of the code is poor. However, if you buy a copy of A&S, you can use the code as a starting point and it would be faster than writing it from scratch. If you can find the routines you need in the Cephes collection at netlib, I would strongly encourage you to use those instead. On the other hand, there are many routines in this book that I have not seen implementations for elsewhere.

I have used programs from this book for theta functions and for spheroidal wavefunctions, or rather for the corresponding eigenvalues. For the theta functions I could not find equivalent programs elsewhere. A program for the spheroidal wavefunctions from another recipe book returned erroneous values for certain parameter sets. Baker's code, on the other hand, appeared to have been carefully checked, and functioned correctly (as far as I could determine) for a wide range of parameter values. I had no great difficulty with compilation when I used Turbo C, in which apparently the programs were developed; however, Watcom C choked on both. The problems were minor, or should have been, except that the code read as though it were intended for a "most-confusing-C-code" contest.

Baker does not have a table of dependencies in his book, but a DOS executable which is included will give the information, more or less, while engendering homicidal feelings toward the author!

If the code were cleaned up the book would be a "must-have" for physicists and applied mathematicians.

This book is not for the novice! However, if you need accurate values for the less-common transcendental functions, and are comfortable with C-language programming, then you will *certainly* want a copy.

I have tried to use all of the fuctions in this book and found them to be accurate and precise, except for a few problems that occur with derivatives of the Hankel functions.

Todd R. Downey, Oak Ridge National Laboratory, Photometrics Group, Life Sciences Division