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Find (a) the curl and (b) the divergence of the vector field.
$\mathbf{F}(x, y, z)=x y^{2} z^{2} \mathbf{i}+x^{2} y z^{2} \mathbf{j}+x^{2} y^{2} z \mathbf{k}$

$F(x, y, z)=x y^{2} z^{2} \hat{i}+x^{2} y z^{2} \hat{j}+x^{2} y^{2} z \hat{k}$

$\left|\begin{array}{ccc}{\hat{i}} & {\hat {j}} & {\hat{k}} \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ {p} & {Q} & {R}\end{array}\right|$

vector $=p \hat{i}+Q \hat{j}+R \hat{k}$

$\nabla{x} F=\left[\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right] \hat{i}+\left[\frac{\partial R}{\partial x}- \frac{\partial p}{\partial z} \right] \hat{j}+\left[\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right] \hat{k} \rightarrow$ vector

$\nabla x F=\left [2 y x^{2} z-2 z x^{2} y \right] \hat{i}+\left[2 x y^{2} z-2 z y^{2} x\right] \hat{j}+\left[2 x y z^{2}-2 y x^{2} z^{2}\right] \hat{k}$

$\nabla x F=0 \hat{i}+0 \hat{j}+0 \hat{k}=0$

$\nabla \cdot F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$ Scalar

$\nabla \cdot F=y^{2} z^{2}+x^{2} z^{2}+x^{2} y^{2}$