上周和boss见面了三次, 来港以后最频繁的一周. 前两次都没有布置太多的任务, 但是周四见面时, boss给了一篇他正在reivew的文章, 说叫我读读看. 发现是关于道路拥挤收费的, 也是我原research proposal的题目. 文章是以continuum network equilibrium作为下层均衡模型的, 但是continuum这个东东小熊以前从没有读过, 于是只能现学现卖.

一读才晓得自己的数学底子薄呀, 多元微积分的知识全没了, 什么Green’s Theorem, Stokes’ Theorem, Divergence Theorem通通忘光. 于是立马从图书馆搞了几本书回来复习. 这还不是重点, 重点是这篇文章还用了 Calculus of Variations, 所谓变分法是也. 以前听说过, 但是不晓得到底是干什么用的, 只是依稀记得自己有本只有五十来页的小书讲这个, Byerly W. Introduction to the Calculus of Variations (1917, pp48).

这一读就不得了, 这本仅有48页的小书, 言简意赅深入浅出, 是我读过最轻松愉快的数学书. 强烈推荐各位兄弟姐妹们来读, 可能现在对你的研究学习没有直接关系, 但是说不定哪天就用得上, 变分法在物理力学这些行当中应用非常普遍. 读了这本小书, 你就会晓得一下经典问题的解答:

1. 为什么平面上两点之间直线最短?
2. 为什么对于给定长度的闭合曲线, 圆覆盖的面积是最大的?
3. 最速降线是啥形状的?

这本书仅需基本的微积分知识和一点点高中物理常识. 特别适合在地铁上, 在马桶上, 在茶余饭后随手捏来看看想想. 而且这本书是1917年Harvard U出版的, 拿在手上亦有怀古思幽的情趣.

Update

当年此书出版时, 发在Bulletin of the American Mathematical Society上的Book Review: Archibald R. C. Bull. Amer. Math. Soc. 24 (1917), 97-98.

Introduction to the Calculus of Variations. By W. E. BYERLY. Cambridge, Harvard University Press, 1917. 8vo. 48 pp. Cloth. Price 75 cents.

THIS little book is the first of a series of ” Mathematical Tracts for Physicists “. It indicates in admirably clear style the solution of a number of examples involving some of the fundamental ideas of the calculus of variations. As the subject owed its origin to the attempt to solve a rather narrow class of problems in maxima and minima, the eight pages of the Introduction are mainly taken up with a discussion of three simple examples: the shortest line, the curve of quickest descent, the minimum surface of revolution.

The integrals of the Lagrange equations arising in connection with the second and third of these examples are given in the second chapter (pages 9-22) which is entitled: ” Variations. Notation and nomenclature. Illustrative problems.” We find here also a second solution of the two-dimensional shortest line problem (polar coordinates), and a solution of the problem of the geodesic line joining two given points on the surface of a sphere. Section 10 is devoted to isoperimetrical problems. The seven examples that are given for solution involve slight developments of the text.

In Chapter II I (pages 23-28), on ” Problems involving several dependent variables,” Hamilton’s principle and its application are considered. Chapter IV (pages 29-33) on ” Multiple integrals ” contains (1) the derivation of the differential equation of minimal surfaces, and, by means of Hamilton’s principle, (2) the derivation of the differential equation for small transverse oscillations of a stretched elastic string.

” Variation of limits ” and the ” Principle of least action ” are the topics of the last chapter. In each of the last three chapters are examples to be solved.

No references to the literature of the subject are to be found in the tract.