Answer
Conservative and $f(x,y)=e^x \sin y+k$
Work Step by Step
When $F(x,y)=ai+bj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}$
Here, $a$ and $b$ are first-order partial derivatives on the domain $D$.
Then, we have $a_x=e^x \cos y; b_y=e^x \cos y$
Here, $\dfrac{\partial a}{\partial y} = \dfrac{\partial b}{\partial x}$
Thus, the vector field $F$ is conservative.
Need to determine $f$ with $F=\nabla f$
Here, $\int f(x,y) dx=e^x \sin y+g(y)$
In order to find the constant $g(y)$ we will have to integrate $f$ with respect to $x$ with a function of $y$.
Thus, $f_y(x,y)=e^x \cos y+g'(y)=0$
and $\int g(y) dy=k$
Hence, our answer is:
$f(x,y)=e^x \sin y+k$
