曲面积分中关于面积定义的探讨

Some notes on the definition of area in surface integral

高等数学与数学分析教材中第一型曲面积分的定义中要求该曲面是可求面积的,但出于篇幅限制,大部分教材没有对曲面的面积的定义进行探讨。曲面面积的定义不能简单地从曲线的弧长的定义推广而来(见正文Schwarz反例)。本文对曲面面积的定义做了一些探讨,并给出了该部分内容教学中的一些建议。

The definition of the surface integral in the textbooks of advanced mathematics and mathematical analysis requires that the surface has a obtainable area,but most textbooks do not discuss the definition of the area of the surface.The definition of the area of the surface cannot be generalized from the definition of the arc length of the curve(see the Schwarz counterexample in the main text).This article discusses the definition of the surface area and gives some teaching suggestions.