Question Number 218421 by SdC355 last updated on 09/Apr/25
$$\mathrm{prove}\: \\ $$$$\int_{\:\partial\boldsymbol{\Sigma}} \:\boldsymbol{\omega}=\int_{\:\boldsymbol{\Sigma}\:} \mathrm{d}\boldsymbol{\omega} \\ $$
Commented by Ghisom last updated on 09/Apr/25
$$\mathrm{is}\:\mathrm{this}\:\mathrm{the}\:\mathrm{newest}\:\mathrm{anti}−\mathrm{syntax}? \\ $$
Commented by SdC355 last updated on 09/Apr/25
$$?\:\mathrm{what}\:\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{i}\:\mathrm{really}\:\mathrm{want}\:\mathrm{know}\:\mathrm{why}\: \\ $$$$\int_{\:\partial\boldsymbol{\Sigma}} \boldsymbol{\omega}=\int_{\:\boldsymbol{\Sigma}} \mathrm{d}\boldsymbol{\omega}\:\:\:\mathrm{aka}\:\mathrm{Stoke}'\mathrm{s}\:\mathrm{Theorem} \\ $$$$\: \\ $$$$\:\mathrm{for}\:\mathrm{example}…\:\mathrm{Magnetic}\:\mathrm{Field}\:\boldsymbol{\mathrm{B}}\:; \\ $$$$\oint_{\:\partial\boldsymbol{\Sigma}} \:\boldsymbol{\mathrm{B}}\centerdot\mathrm{d}\boldsymbol{\mathrm{l}}=\int_{\:\boldsymbol{\Sigma}} \:\bigtriangledown×\boldsymbol{\mathrm{B}}\:\mathrm{d}\boldsymbol{\mathrm{S}} \\ $$
Commented by Ghisom last updated on 09/Apr/25
$$\mathrm{the}\:\mathrm{proof}\:\mathrm{of}\:\mathrm{Stoke}'\mathrm{s}\:\mathrm{Theorem}\:\mathrm{demands} \\ $$$$\mathrm{a}\:\mathrm{huge}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{previous}\:\mathrm{knowledge}. \\ $$$$\left.\mathrm{a}\right)\:\mathrm{you}\:\mathrm{have}\:\mathrm{this}\:\mathrm{knowledge}\:\Rightarrow\:\mathrm{it}'\mathrm{s}\:\mathrm{easy} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{have}\:\mathrm{it}\:\Rightarrow\:\mathrm{study}\:\mathrm{mathematics} \\ $$$$\mathrm{it}\:\mathrm{makes}\:\mathrm{no}\:\mathrm{sense}\:\mathrm{to}\:\mathrm{post}\:\mathrm{the}\:\mathrm{proof}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{forum}\:\mathrm{like}\:\mathrm{this}\:\mathrm{one}. \\ $$
Answered by MrGaster last updated on 10/Apr/25
$$\int_{\:\partial\boldsymbol{\Sigma}} \:\boldsymbol{\omega}\equiv\int_{\:\boldsymbol{\Sigma}\:} \mathrm{d}\boldsymbol{\omega} \\ $$$$\boldsymbol{\omega}\in\Omega^{{k}} \left(\mathbb{R}^{{n}} \right),\boldsymbol{\Sigma}\subseteq\mathbb{R}^{{n}+\mathrm{1}} \left(\mathrm{orientable}\right) \\ $$$${d}\boldsymbol{\omega}:=\underset{{i}_{\mathrm{1}} <\ldots{i}_{{k}} } {\sum}\frac{\partial\omega_{{i}_{\mathrm{1}} } \ldots{i}_{{k}} }{\partial{x}_{{j}} }{dx}_{{j}} \wedge{dx}^{{i}_{\mathrm{1}} } \wedge\ldots\wedge{dx}^{{i}_{{k}} } \\ $$$$\int_{\:\partial\Sigma} \:\omega\overset{\mathrm{Stokes}} {\Rightarrow}\int_{\:\Sigma\:} \mathrm{d}\boldsymbol{\omega} \\ $$