Question Number 219374 by SdC355 last updated on 23/Apr/25
Commented by SdC355 last updated on 23/Apr/25
![S^→ (u,v)= { (((2+v∙sin(u))sin(2πv))),((v∙cos(u))),(((2+v∙sin(u))cos(2πv)+(2v−2))) :} u∈[−π,π] , v∈[0,(π/2)] and vector field F^→ (x,y,z)=−xe_1 ^→ −ye_2 ^→ −ze_3 ^→ evaluate ∫∫_( S) F^→ ∙dS^→](https://www.tinkutara.com/question/Q219375.png)
$$\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}\left({u},{v}\right)=\begin{cases}{\left(\mathrm{2}+{v}\centerdot\mathrm{sin}\left({u}\right)\right)\mathrm{sin}\left(\mathrm{2}\pi{v}\right)}\\{{v}\centerdot\mathrm{cos}\left({u}\right)}\\{\left(\mathrm{2}+{v}\centerdot\mathrm{sin}\left({u}\right)\right)\mathrm{cos}\left(\mathrm{2}\pi{v}\right)+\left(\mathrm{2}{v}−\mathrm{2}\right)}\end{cases} \\ $$$${u}\in\left[−\pi,\pi\right]\:,\:{v}\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right] \\ $$$$\mathrm{and}\:\mathrm{vector}\:\mathrm{field}\:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\left({x},{y},{z}\right)=−{x}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} −{y}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −{z}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\mathrm{evaluate}\:\int\int_{\:\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}} \\ $$
Commented by Nicholas666 last updated on 23/Apr/25
$$−\mathrm{3}\int\int\int_{\:\:{V}} {dV} \\ $$