不可微格林定理
Green's theorem with no differentiability
摘要：
这一结果是建立在边界可求长的Jordan可测区域上的。新平面积分的被积函数F是平行轴矩形的函数，在非重叠矩形上是可加的，因此在“图形”（即平行轴矩形的有限并集）上是明确定义和可加的。定义它在Jordan可测S上的积分为它的值在包含S的子图的图形上的极限，并且包含在包含S的图形中，因为前者/后者的补数直接扩展以填充S/S的补数。当加法F在收敛到零的意义上是“绝对连续的”时，在每个Jordan可测区域上的积分存在，因为它的幅角所包围的面积收敛到零，或者对于F，圆周线积分$\oint P dx + Q dy$对于$P$，$Q$在S的可求长边界处连续，并且沿着平行于轴的线段可积。因此，平等的这一领域积分与线积分周围的边界，要证明，如下的各种积分的分歧提出：Pfeffer，W.F.《黎曼积分法》，剑桥大学出版社，纽约，1993年。
英文摘要：
The result is established for a Jordan measurable region with rectifiable boundary. The integrand F for the new plane integral to be used is a function of axis-parallel rectangles, finitely additive on non-overlapping ones, hence unambiguously defined and additive on "figures" (i.e. finite unions of axis-parallel rectangles). Define its integral over Jordan measurable S as the limit of its value on the figures, which contain a subfigure of S and are contained in a figure containing S, as the former/complements of the latter expand directedly to fill out S/the complement of S. The integral over every Jordan measurable region exists when additive F is "absolutely continuous" in the sense of converging to zero as the area enclosed by its argument does, or with F the circumferential line integral $\oint P dx + Q dy$ for $P$, $Q$ continuous at the rectifiable boundary of S and integrable along axis - parallel line segments. Thus the equality of this area integral with the line integral around the boundary, to be proved, follows for the various integrals of divergence presented in: Pfeffer, W.F. The Riemann Approach to Integration, Cambridge Univ. Press, New York, 1993.
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