Answer
a) $(4y^3z^3 - 2x^3yz)i + (0)j + (3x^2yz^2)k$
b) $x^3z^2 + 3y^4z^2$
Work Step by Step
a)
$P=0$
$Q=x^3yz^2$
$R=y^4z^3$
$curl F = \left(\frac{dR}{dy} - \frac{dQ}{dz}\right)i + \left(\frac{dP}{dz} -\frac{dR}{dx}\right)j + \left(\frac{dQ}{dx} - \frac{dP}{dy}\right)k$
$ =(4y^3z^3 - 2x^3yz)i + (0)j + (3x^2yz^2)k$
b)
$div F = \frac{dP}{dx} + \frac{dQ}{dy} + \frac{dR}{dz}$
$ = \frac{d}{dx}(0) + \frac{d}{dy}(x^3yz^2) + \frac{d}{dz}(y^4z^3)$
$ = 0 + x^3z^2 + 3y^4z^2$
$ = x^3z^2 + 3y^4z^2$
