格林公式成立条件的改进

Improvement on Conditions for Validity of Green's Formula

为扩大格林公式的应用范围,将格林公式中被积函数在有界闭区域边界曲线上连续的条件放宽为有界且除有限个点外处处连续,区域上的二重积分拓宽为从内部向外逼近且收敛的广义重积分,并且允许被积函数在区域内部含有零面积集的不可导点。结果表明,适当改进公式成立的条件,格林公式仍成立。

To expand the application of Green's formula,the continuity condition of the integrand function along the boundary curve of a bounded closed region in Green's formula is relaxed to being all bounded and continuous except a finite number of points.The double integral over the region is extended to a generalized multiple integral that is approximated and converged from the inside outward.Furthermore,the integrand function is allowed to have non-differentiable points with zero area sets inside the region.The improvement shows that Green's formula still holds with the appropriate improvement on the condtion.