Answer
The function $f$ is a conservative vector field.
and
$\phi=x \cos y+y \sin x+K$
Work Step by Step
Here, we have $f(x,y)=\cos y +y \cos x$ and $g(x,y)=\sin x- x \sin y$
In order to find the function as a conservative vector field, we must have $\dfrac{\partial f}{\partial y}=\dfrac{\partial g}{\partial x}$
Thus, $\dfrac{\partial f}{\partial y}=-\sin y+\cos x$ and $\dfrac{\partial g}{\partial x}=-\sin y+\cos x$
This means that the the function $f$ is a conservative vector field.
Next, we will find the potential function.
Here, $\phi=x \cos y+y \sin x+k(y)$ and $\dfrac{\partial \phi}{\partial y}=\sin x-x \sin y \implies k'(y)=0$
Therefore, $\phi=x \cos y+y \sin x+K$
