A vector field [tex]\mathbf f(x,y)[/tex] is conservative *if and only if* we can find a scalar function [tex]f(x,y)[/tex] for which [tex]\nabla f=\mathbf f[/tex], which means we're looking for a function [tex]f(x,y)[/tex] such that
[tex]\dfrac{\partial f}{\partial x}=e^x\cos y[/tex]
[tex]\dfrac{\partial f}{\partial y}=e^x\sin y[/tex]
Integrating both sides of the first PDE with respect to [tex]x[/tex], we get
[tex]f(x,y)=e^x\cos y+g(y)[/tex]
We're assuming the "constant" of integration is some expression independent of [tex]x[/tex]. However, upon differentiating with respect to [tex]y[/tex], we have
[tex]\dfrac{\partial f}{\partial y}=-e^{\sin y}+\dfrac{\mathrm dg}{\mathrm dy}=e^x\sin y[/tex]
[tex]\implies\dfrac{\mathrm dg}{\mathrm dy}=2e^x\sin y[/tex]
which means [tex]g[/tex] is *not* a function of [tex]y[/tex] alone, contradicting the assumption of the contrary. So our desired [tex]f(x,y)[/tex] does not exist, and therefore [tex]\mathbf f(x,y)[/tex] is *not* conservative.
