Answer
Conservative
Work Step by Step
As we know that $\text{curl} F =(\dfrac{\partial R}{\partial y}-\dfrac{\partial Q}{\partial z})i +(\dfrac{\partial P}{\partial z}-\dfrac{\partial R}{\partial x}) k+(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y})k $
A vector field is conservative iff the $\text{curl} F =0$
Given: $F=y \sin z i+x \sin z j+xy \cos z k$
Now, $\text{curl} F =(x \cos z-x \cos z) i+(y \cos z-y \cos z)j +(\sin z-\sin z) k=0$
This shows that the vector field is Conservative.
