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Question Number 19341 by tawa tawa last updated on 09/Aug/17

Commented by tawa tawa last updated on 09/Aug/17

$$\left(1\right)\mathrm{Using}{\mathrm{Stoke}}^{\prime }\mathrm{s}\mathrm{theorem},\mathrm{find}\oint {\mathrm{y}}^2\mathrm{dx}+{\mathrm{z}}^2\mathrm{dy}+{\mathrm{x}}^2\mathrm{dz},\mathrm{where}\mathrm{\Gamma }\mathrm{is}\mathrm{the}\mathrm{closed}$$$$\mathrm{curve}\mathrm{A}\to \mathrm{B}\to \mathrm{C}\to \mathrm{A}\mathrm{and}\mathrm{A}=(1,0,0),\mathrm{B}=(0,0,1),\mathrm{C}(0,1,0).$$

Commented by tawa tawa last updated on 09/Aug/17

$$2)\mathrm{Consider}\overrightarrow{\mathrm{F}}=(\mathrm{x}+\mathrm{y}-4)\mathrm{i}+\left(3\mathrm{xy}\right)\mathrm{j}+(2\mathrm{zz}+{\mathrm{z}}^2)\mathrm{k},$$$$\mathrm{s}=(\mathrm{x},\mathrm{y},\mathrm{z})\in {\mathbb{R}}^3\mid \mathrm{z}=4-{\mathrm{x}}^2-{\mathrm{y}}^2,\mathrm{z}⩾0)$$$$\mathrm{Find}\int \int (\overrightarrow{▽}\cdot \overrightarrow{\mathrm{F}})\cdot \overrightarrow{\mathrm{d}s}$$

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