Answer
a) $ze^xi+(xye^z-yze^x)j-xe^zk$
b) $y(e^z+e^x)$
Work Step by Step
a) Consider $F=A i+B j+C k$
Then $curl F=\begin{vmatrix}i&j&k\\\dfrac{\partial}{\partial x}&\dfrac{\partial }{\partial y}&\dfrac{\partial }{\partial z}\\A&B&C\end{vmatrix}$
$curl F=[C_y-B_z]i+[A_z-C_z]j+[B_x-A_y]k$
$curl F=[ze^x-0]i+[xye^z-yze^x]j+[0-xe^z]k=ze^xi+(xye^z-yze^x)j-xe^zk$
b) $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
$div F=\dfrac{\partial (xye^z)}{\partial x}+\dfrac{\partial (0)}{\partial y}+\dfrac{\partial (yze^x)}{\partial z}=y(e^z+e^x)$
