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## Theorem of the month

### Green's theorem

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

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

#### Proof:

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

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

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

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