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= Welcome to MWiki = | = Welcome to MWiki = | ||
− | == | + | == Theorem of the month == |
− | === | + | === Green's theorem === |
− | + | Given neighbourhood relations <math>B \subseteq {D}^{2}</math> for some <math>h</math>-domain <math>D \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 D, {D}^{-} := \{(x, y) \in D : (x + h, y + h) \in D\}</math>, and a simply closed path <math>\gamma: [a, b[\rightarrow \partial D</math> followed anticlockwise, choosing <math>\curvearrowright B \gamma(t) = \gamma(\curvearrowright A t)</math> for <math>t \in [a, b[, A \subseteq {[a, b]}^{2}</math>, the following equation holds for sufficiently <math>\alpha</math>-continuous functions <math>u, v: D \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>:<div style="text-align:center;"><math>\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></div> | |
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==== Proof: ==== | ==== Proof: ==== | ||
− | + | Wlog the case <math>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>\iota</math>, and every <math>h</math>-domain is a union of such sets. It is simply shown that<div style="text-align:center;"><math>\int\limits_{\gamma }{u\,dBx}=-\int\limits_{(x,y)\in {{D}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}</math></div>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<div style="text-align:center;"><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 {{D}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square</math></div> | |
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== Recommended reading == | == Recommended reading == |
Revision as of 22:23, 30 April 2022
Welcome to MWiki
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]:
Proof:
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
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