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Theorem of the month

Green's theorem

Given neighbourhood relations [math]\displaystyle{ B \subseteq {D}^{2} }[/math] for some [math]\displaystyle{ h }[/math]-domain [math]\displaystyle{ D \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 D, {D}^{-} := \{(x, y) \in D : (x + h, y + h) \in D\} }[/math], and a simply closed path [math]\displaystyle{ \gamma: [a, b[\rightarrow \partial D }[/math] followed anticlockwise, choosing [math]\displaystyle{ \curvearrowright B \gamma(t) = \gamma(\curvearrowright A t) }[/math] for [math]\displaystyle{ t \in [a, b[, A \subseteq {[a, b]}^{2} }[/math], the following equation holds for sufficiently [math]\displaystyle{ \alpha }[/math]-continuous functions [math]\displaystyle{ u, v: D \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]:

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


Wlog the case [math]\displaystyle{ D := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial D \rightarrow {}^{(\omega)}\mathbb{R} }[/math] is proved, since the proof is analogous for each case rotated by [math]\displaystyle{ \iota }[/math], and every [math]\displaystyle{ h }[/math]-domain is a union of such sets. It is simply shown that

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

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

[math]\displaystyle{ -\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 {{D}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square }[/math]

Recommended reading

Nonstandard Mathematics