On Green's Theorem
نویسندگان
چکیده
where the line integral is taken in a positive sense around the curve C. In [l] Bochner investigated under which conditions (1) holds. There it was shown that if A and B have certain regularity properties and if the integrand on the right of (1) behaves well, then (1) does hold. Here we shall prove (1) under what may be regarded as the weakest possible hypotheses. This question was also treated by Shapiro in [3], and though he assumes certain regularity of A and B, namely, the existence of the differential, he allows certain exceptional sets which we cannot allow. The proof of our theorem is modeled after the proof of the Looman-Mensov theorem as contained, for example, in [2]. We will not deal with the topological difficulties involved so that our theorem will only treat the case in which Q is a rectangle, whose sides are parallel to the coordinate axes.
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