Question Number 219571 by SdC355 last updated on 29/Apr/25
$$\mathrm{pls}\:\mathrm{Help}…..! \\ $$$$\mathrm{prove} \\ $$$$\int\int_{\:\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{g}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=\mathrm{0}\:\leftrightarrows\:\mathrm{div}\:\overset{\rightarrow} {\boldsymbol{\mathrm{g}}}=\mathrm{0} \\ $$
Answered by aleks041103 last updated on 29/Apr/25
$${Let}\:{S}\:{be}\:{a}\:{closed}\:{surface}\:{enclosing}\:{a} \\ $$$${volume}\:{of}\:{space}\:{V},\:{i}.{e}.\:{S}=\partial{V} \\ $$$${By}\:{Gauss}'\:{theorem}: \\ $$$$\oint_{\partial{V}} \overset{\rightarrow} {{g}}.\overset{\rightarrow} {{dS}}\:=\:\int_{{V}} \left(\:{div}\:\overset{\rightarrow} {{g}}\:\right)\:{dV}\:\:=\:\mathrm{0} \\ $$$${If}\:{the}\:{above}\:{is}\:{true}\:{for}\:{any}\:{volume}, \\ $$$${then}\:{there}\:{are}\:{several}\:{ways}\:{to}\:{prove}\:{that} \\ $$$${div}\left({g}\right)=\mathrm{0}. \\ $$
Commented by SdC355 last updated on 29/Apr/25
$$\heartsuit \\ $$