Answer
Divergence $F = 0$
Curl $F =( -12xy^3+40x^2z^4)i+(14y^3z+3y^4)j+(-16xz^5-21y^2z^2)k$
Work Step by Step
Consider the vector field:
$F(x,y,z)=7y^3z^2i-8x^2z^5j-3xy^4k$
Let $F$ be the vector field:
$F(x,y,z)=f(x,y,z)i+g(x,y,z)j+h(x,y,z)k$
$f(x,y,z) = 7y^3z^2$
$g(x,y,z) = -8x^2z^5$
$h(x,y,z) = -3xy^4$
Divergence:
$$div F=\frac{∂f}{∂x}+\frac{∂g}{∂y}+\frac{∂h}{∂z}$$
div $F=\frac{∂(7y^3z^2)}{∂x}+\frac{∂(-8x^2z^5)}{∂y}+\frac{∂(-3xy^4)}{∂z}$
$=0+0+0$
div$F=0$
$$curl F=(\frac{∂h}{∂y}-\frac{∂g}{∂z})i+(\frac{∂f}{∂z}-\frac{∂h}{∂x})j+(\frac{∂g}{∂x}-\frac{∂f}{∂y})k$$
$=(-12xy^3+40x^2z^4)i+(14y^3z+3y^4)j+(-16xz^5-21y^2z^2)k$
$curlF=(-12xy^3+40x^2z^4)i+(14y^3z+3y^4)j+(-16xz^5-21y^2z^2)k$
