微分的本质

微分有两个含义:1.对于与时间有关的函数(称之为动态函数)f而言,微分df表示在无限小的时间dt内函数f的瞬时增加量,即df=f(t dt)?f(t);2.对于与时间无关的函数(称之为静态函数)g而言,微分dg表示g的微小部分,所有dg之和等于g。因为时间总是从过去走向未来,所以时间的微分dt总是恒大于0的正实数。df与dt之比称为函数f的瞬时增加率或导数,而非变化率。变化率包括增加率与减少率两种情况。所有的高阶微分都是无意义的,从来也没有被使用过,应予以彻底抛弃。

The Essence of Differential

There are two meanings of differential: I. When furtction f is dependent on time, which is called a dynamic function, differential df is the instantaneous increment of function f in an infinitesimal time dt, i.e. df=f(t dt) ?f(t). 2. When function g is independent of time, which is called a static function, differential dg is a small part of function g, the sum of all dg is equal to g. The differential of time dt is always a positive real number because time always flows from the past to the future. The ratio of df over dt is called the instantaneous rate of increment of function f, not rate of change. Rate of increment is different from rate of change, which consists of rate of increment and rate of decrement. All differentials of higher order are meaningless and have never been used and should be abandoned.