Geometrical Methods of Mathematical Physics. By B. F. SCHUTZ. Cambridge University Press, 1980. 250 pp. £20 (hardback), £7.95 (paperback).
Geometrical Methods of Mathematical Physics. By B. F. SCHUTZ. Cambridge University Press, 1980. 250 pp. £20 (hardback), £7.95 (paperback).
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DOI：
10.1017/s0022112082210901
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发表时间：
1982-02
影响因子：
3.7
通讯作者：
J. M. Stewart
中科院分区：
文献类型：
--
作者：
J. M. Stewart
流体力学可能仍然是理论物理中微分几何没有重大影响的一个主要分支；主要问题被认为存在于分析中，几何是次要考虑的。然而，正如卡尔坦最先认识到的那样，几何和分析是紧密交织在一起的，这一事实主导了现代相对论和量子物理学。它为流体动力学提供了什么Z Schutz试图为应用数学家提供一本关于现代微分几何的初级文本。相当明智的是，他专注于三个主要主题：可微流形和张量微积分，李导数和群，以及微分形式。作为引言，这三章都很出色。这需要大量钻研文献才能产生同等的治疗方法。对谎言衍生品的处理特别好。(然而，J F M的读者可能希望提醒，稍微复杂一点的文本有两个令人钦佩的文本。乔奎特-布鲁哈特、德维特-莫雷特和迪拉德-布莱克的《分析、流形和物理》(North-Holland，1977)是一本百科全书式的面向数学的文本，它彻底探索了几何分析界面。Arnol‘d(Springer Verlag，1978)所著的经典力学的数学方法做了同样的事情，但以一种更直观的面向物理的方式。任何理解哈密尔托尼亚力学的人都应该可以毫不费力地从这本书中学到概念。令人费解的是，舒茨的书目中没有阿诺德这本令人钦佩的书。不幸的是，这本书中关于物理应用的潜在最激动人心的一章引用了热力学、哈密顿力学、电磁学、流体动力学和相对论的例子，可能太肤浅了。首先浏览这一章的持怀疑态度的流体动力学家不太可能找到太多东西来促使他阅读这本书的其余部分。正因为如此，本文根据布兰登·卡特的原始观点，对舒茨记法中的环流守恒、涡度守恒和螺旋度守恒作了简要的注解。拉丁字母表示向量，希腊字母表示形式，d表示外导数，A表示外积，0.0表示向量u和p形式w在相邻指数上的收缩，EPV是向量场u的Lie导数。基本结果是：University Press，1980。250页。20美元(精装本)，L7.95美元(平装本)。
Fluid dynamics remains perhaps the one major branch of theoretical physics in which differential geometry has made no significant impact; the major problems are thought to lie in analysis, the geometry being a subsidiary consideration. However, as first realized by Cartan, geometry and analysis are intimately interwoven, and this fact has dominated modern relativity and quantum physics. What does i t offer for fluid dynamics Z Schutz has tried to provide an elementary text on modern differential geometry for the applied mathematician. Rather wisely he has concentrated on three main topics: differentiable manifolds and tensor calculus, Lie derivatives and groups, and differential forms. As an introduction these three chapters are excellent. It would require a great deal of delving in the literature to produce equivalent treatments. The treatment of Lie derivatives is particularly good. (However, readers of J F M may wish to be reminded that a t a slightly higher level of sophistication there are two admirable texts. Analysis, Manifolds and Physics by Choquet-Bruhat, de Witt-Morette & Dillard-Bleick (North-Holland, 1977) is an encyclopaedic mathematically oriented text which explores the geometry-analysis interface thoroughly. Mathematical Methods of Classical Mechanics by Arnol’d (Springer Verlag, 1978) does the same but in a more intuitive physically oriented manner. Anyone who understands Hamil tonian mechanics should have no difficulty in picking up the concepts from this book. Inexplicably Arnol’d’s admirable book is missing from Schutz’s bibliography. Unfortunately the potentially most stimulating chapter in the book on physical applications citing examples from thermodynamics, Hamiltonian mechanics, electromagnetism, fluid dynamics and relativity is perhaps too superficial. The sceptical hydrodynamicist who skims this chapter first is not likely to find much to make him read the rest of the book. Because of this there follow here some brief notes on conservation of circulation, vorticity and helicity in the notation of Schutz, based on an original idea of Brandon Carter. Latin letters denote vectors, Greek letters forms, d the exterior derivative, A the exterior product, 0.0 denotes the contraction of a vector u and p-form w across adjacent indices, ,Epv is the Lie derivative with respect to the vector field u. The fundamental results are: University Press, 1980. 250 pp. $20 (hardback), L7.95 (paperback).