A Vector field F is given by \[\bar{F}=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}\] is irroational and hence find scalar point function ? such that F = ? ?

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A Vector field F is given by

$\bar{F}=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}$ is irroational and hence find scalar point function ? such that F = ? ?

written 3.9 years ago by

teamques10 ★ 69k

• modified 3.9 years ago

engineering physics

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written 3.9 years ago by

teamques10 ★ 69k

$Part\ 1: \\[2ex]F=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}\ \\[2ex] \therefore{}r=xi+yj+zk \\[2ex] \displaystyle Curl\ F=\left\vert{}\begin{array}{ ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial{}}{\partial{}x} & \frac{\partial{}}{\partial{}y} & \frac{\partial{}}{\partial{}z} \\ x^2-yz & y^2-zx & z^2-xy \end{array}\right\vert{} \\[2ex]$

$ \displaystyle =0-0+0=0 \\[2ex] \therefore{}F\ is\ a\ conservative\ field. \\[2ex] \therefore{}there\ exists\ a\ scalar\ potential\ of\ F\ such\ that\ \ …

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