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Theorem of the month

Green's theorem

Given neighbourhood relations [math]B \subseteq {A}^{2}[/math] for some simply connected [math]h[/math]-set [math]A \subseteq {}^{(\omega)}\mathbb{R}^{2}[/math], infinitesimal [math]h = |dBx|= |dBy| = |\curvearrowright B \gamma(t) - \gamma(t)| = \mathcal{O}({\hat{\omega}}^{m})[/math], sufficiently large [math]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]\gamma: [a, b[\rightarrow \partial A[/math] followed anticlockwise, choosing [math]\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)[/math] for [math]t \in [a, b[, D \subseteq {[a, b]}^{2}[/math], the following equation holds for sufficiently [math]\alpha[/math]-continuous functions [math]u, v: A \rightarrow \mathbb{R}[/math] with not necessarily continuous partial derivatives [math]\partial Bu/\partial Bx, \partial Bu/\partial By, \partial Bv/\partial Bx[/math] and [math]\partial Bv/\partial By[/math]:

[math]\int\limits_{\gamma }{(u\,dBx+v\,dBy)}=\int\limits_{(x,y)\in {{A}^{-}}}{\left( \frac{\partial Bv}{\partial Bx}-\frac{\partial Bu}{\partial By} \right)dB(x,y)}.[/math]

Proof:

Wlog the case [math]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]\iota[/math], and every simply connected [math]h[/math]-set is a union of such sets. It is simply shown that

[math]\int\limits_{\gamma }{u\,dBx}=-\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}[/math]

since the other relation is given analogously. Since the regions of [math]\gamma[/math] where [math]dBx = 0[/math] do not contribute to the integral, for negligibly small [math]t := h(u(s, g(s)) - u(r, g(r)))[/math], it holds that

[math]-\int\limits_{\gamma }{u\,dBx}-t=\int\limits_{r}^{s}{u(x,g(x))dBx}-\int\limits_{r}^{s}{u(x,f(x))dBx}=\int\limits_{r}^{s}{\int\limits_{f(x)}^{g(x)}{\frac{\partial Bu}{\partial By}}dBydBx}=\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square[/math]

Recommended reading

Nonstandard Mathematics