A vector field is given by \[ \overline {F} = (x^2 + xy^2)i + (y^2 + x^2 y)j. Show that F is irrotational and find its potential. Such that F=∇ϕ.

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written 3.8 years ago by

RakeshBhuse • 3.2k

• modified 3.8 years ago

Solution :

We have,

$$\quad \vec{F}=(x^{2}+xy^2) \hat{i}+(y^2+x^2y)\hat{j} $$

$\begin{aligned} \operatorname{Curl} \vec{F} &=\nabla \times \vec{F}\\ &=\left(\hat{i} \frac{\partial}{\partial x}+\hat{j} \frac{\partial}{\partial y}+\hat{k} \frac{\partial}{\partial z}\right) \times\left(y^{2} \hat{i}+2 x y \hat{j}-z^{2} \hat{k}\right) \end {aligned} $

$ \space\space\space\space\space\space\space\space\space\space =\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^{2}+xy^2 & …

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