Can-you-help-me-pls-Evaluate-S-F-dS-Parametric-Surface-S-u-v-2-sin-u-cos-v-e-1-2-sin-u-sin-v-e-2-cos-v-u-e-3-Vector-Field-F-x-y-xe-1-ye-2-ze

Question Number 199686 by MathedUp last updated on 07/Nov/23

$${Can}\:{you}\:{help}\:{me}..???\:{pls}…. \\ $$$$\: \\ $$$$\: \\ $$$$\mathrm{Evaluate}\:\int\int_{\:\boldsymbol{\mathcal{S}}} \:\hat {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\hat {\boldsymbol{\mathrm{S}}} \\ $$$$\mathrm{Parametric}\:\mathrm{Surface} \\ $$$$\hat {\boldsymbol{\mathrm{S}}}\left({u},{v}\right)=\left(\mathrm{2}+\mathrm{sin}\left({u}\right)\right)\mathrm{cos}\left({v}\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +\left(\mathrm{2}+\mathrm{sin}\left({u}\right)\right)\mathrm{sin}\left({v}\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +\left(\mathrm{cos}\left({v}\right)+{u}\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\mathrm{Vector}\:\mathrm{Field}\:\hat {\boldsymbol{\mathrm{F}}}\left({x},{y}\right)={x}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{y}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +{z}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$

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