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