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Theorem of the month
Green's theorem
Given neighbourhood relations [math]\displaystyle{ B \subseteq {A}^{2} }[/math] for some simply connected [math]\displaystyle{ h }[/math]-set [math]\displaystyle{ A \subseteq {}^{(\omega)}\mathbb{R}^{2} }[/math], infinitesimal [math]\displaystyle{ h = |dBx|= |dBy| = |\curvearrowright B \gamma(t) - \gamma(t)| = \mathcal{O}({\hat{\omega}}^{m}) }[/math], sufficiently large [math]\displaystyle{ m \in \mathbb{N}^{*}, (x, y) \in A, {A}^{-} := \{(x, y) \in A : (x + h, y + h) \in A\} }[/math], and a simply closed path [math]\displaystyle{ \gamma: [a, b[\rightarrow \partial A }[/math] followed anticlockwise, choosing [math]\displaystyle{ \curvearrowright B \gamma(t) = \gamma(\curvearrowright D t) }[/math] for [math]\displaystyle{ t \in [a, b[, D \subseteq {[a, b]}^{2} }[/math], the following equation holds for sufficiently [math]\displaystyle{ \alpha }[/math]-continuous functions [math]\displaystyle{ u, v: A \rightarrow \mathbb{R} }[/math] with not necessarily continuous partial derivatives [math]\displaystyle{ \partial Bu/\partial Bx, \partial Bu/\partial By, \partial Bv/\partial Bx }[/math] and [math]\displaystyle{ \partial Bv/\partial By }[/math]:
Proof:
Wlog the case [math]\displaystyle{ A := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial A \rightarrow {}^{(\omega)}\mathbb{R} }[/math] is proved, since the proof is analogous for each case rotated by [math]\displaystyle{ \iota }[/math], and every simply connected [math]\displaystyle{ h }[/math]-set is a union of such sets. It is simply shown that
since the other relation is given analogously. Since the regions of [math]\displaystyle{ \gamma }[/math] where [math]\displaystyle{ dBx = 0 }[/math] do not contribute to the integral, for negligibly small [math]\displaystyle{ t := h(u(s, g(s)) - u(r, g(r))) }[/math], it holds that