一类含未知函数重积分竞赛题的研究与推广

Generalization of a class of competition problemson multiple integral containing an unknown function

考虑历届全国大学生数学竞赛题中含有未知函数重积分的试题,给出这类题型一般性的推广;利用格林公式、高斯公式,将这类题目中未知函数进行替换求解,给出了两个比较自然的解法.此类型题的推广不仅提高了学生的计算能力,还加深了对坐标变换、格林公式与高斯公式的理解与运用.

The problems on multiple integral containing an unknown function in the national college mathematics competitions are put under examination,and generalizations of this class of problems are given.Using Green’s and Gauss’s formula,the unknown functions in such problems are replaced,and two natural solutions are presented.Our work not only improves students’computing ability,but also deepens the understanding and application of coordinate transformation,Green’s and Gauss’s formula.