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: $F=(e^x \cos y) i+(e^x \sin y)j+xy k$
Now, curl F$=(0-0) i+(0-0)j +(0-0) k=0$
This shows that the vector field is Conservative.
