Answer
a) $-\dfrac{y}{1+y^2z^2}i-\dfrac{z}{1+z^2x^2}j-\dfrac{x}{1+x^2y^2}k$
b) $\dfrac{y}{1+x^2y^2}+\dfrac{z}{1+y^2z^2}+\dfrac{x}{1+z^2x^2}$
Work Step by Step
a) Consider $F=A i+B j+C k$
Then $curl F=\begin{vmatrix}i&j&k\\\dfrac{\partial}{\partial x}&\dfrac{\partial }{\partial y}&\dfrac{\partial }{\partial z}\\A&B&C\end{vmatrix}$
$curl F=[C_y-B_z]i+[A_z-C_z]j+[B_x-A_y]k$
$curl F=(0-\dfrac{y}{1+(yz)^2})i+(\dfrac{z}{1+(xz)^2}-0)j+(0-\dfrac{x}{1+(xy)^2})k=-\dfrac{y}{1+y^2z^2}i-\dfrac{z}{1+z^2x^2}j-\dfrac{x}{1+x^2y^2}k$
b) $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
$div F=\dfrac{\partial [\arctan (xy) ]}{\partial x}+\dfrac{\partial [\arctan (yz) ]}{\partial y}+\dfrac{\partial [\arctan (zx)]}{\partial z}=\dfrac{y}{1+x^2y^2}+\dfrac{z}{1+y^2z^2}+\dfrac{x}{1+z^2x^2}$
