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Question Number 219423 by SdC355 last updated on 24/Apr/25

Answered by SdC355 last updated on 24/Apr/25

S^→ (u,v)= { (((5/4)(1−(v/(2π)))cos(2v)(1+cos(u))+cos(2v))),(((5/4)(1−(v/(2π)))sin(2v)(1+cos(u))+sin(2v))),((((10v)/(2π))+(5/4)(1−(v/(2π)))sin(u))) :} 0≤u≤2π , −π≤v≤2π F^→ (x,y,z)=−xe_1 ^→ −ye_2 ^→ −ze_3 ^→ ∫∫_S F^→ ∙dS^→ =??

$$\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}\left({u},{v}\right)=\begin{cases}{\frac{\mathrm{5}}{\mathrm{4}}\left(\mathrm{1}−\frac{{v}}{\mathrm{2}\pi}\right)\mathrm{cos}\left(\mathrm{2}{v}\right)\left(\mathrm{1}+\mathrm{cos}\left({u}\right)\right)+\mathrm{cos}\left(\mathrm{2}{v}\right)}\\{\frac{\mathrm{5}}{\mathrm{4}}\left(\mathrm{1}−\frac{{v}}{\mathrm{2}\pi}\right)\mathrm{sin}\left(\mathrm{2}{v}\right)\left(\mathrm{1}+\mathrm{cos}\left({u}\right)\right)+\mathrm{sin}\left(\mathrm{2}{v}\right)}\\{\frac{\mathrm{10}{v}}{\mathrm{2}\pi}+\frac{\mathrm{5}}{\mathrm{4}}\left(\mathrm{1}−\frac{{v}}{\mathrm{2}\pi}\right)\mathrm{sin}\left({u}\right)}\end{cases} \\ $$$$\mathrm{0}\leq{u}\leq\mathrm{2}\pi\:,\:−\pi\leq{v}\leq\mathrm{2}\pi \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\left({x},{y},{z}\right)=−{x}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} −{y}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −{z}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\int\int_{\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=?? \\ $$

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