送交者: kobe1 于 January 05, 2009 16:21:47:
回答: 读古典 由 kobe1 于 January 01, 2009 17:17:46:
从下面的引文可以知道求解微积分有3种方法:power series,symbolic calculus,和geometric calculus。
而牛顿的《原理》用geometric calculus。
牛顿为什么选了可用直观几何图形表现的微积分法?
物理学家费曼也是应用直观几何图形的高手。
古典永远是我们取自不尽的宝库。
摘自"Visual Complex Analysis, Tristan Needham, Oxford University Press"的前言
It is fairly well known that Newton's original 1665 version of the calculus was different from the one we learn today: its essence was the manipulation of power series, which Newton likened to the manipulation of decimal expansions in arithmetic. The symbolic calculus - the one in every standard textbook, and the one now associated with the name of Leibniz - was also perfectly familiar to Newton, but apparently it was of only incidental interest to him. After all, armed with his power series, Newton could evaluate an integral like ∫e(-x**2)dx just as easily as ∫sinxdx. Let Leibniz try that!
It is less well known that around 1680 Newton became disenchanted with both these approaches, whereupon he proceeded to develop a third version of calculus, based on geometry. This "geometric calculus" is the mathematical engine that propels the brilliant physics of Newton's Principia.
然后举了一个带图的直角三角形+相似小直角三角形导出微分微分的例子.
Only gradully did I come to realize how naturally this mode of thought could be appplied-almost exactly 300 years later! - to the geometry of the complex plane.