Question

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.) F(x, y) = (yex + sin(y))i + (ex + x cos(y))j

Answer 1

If

$\nabla f=(ye^x+\sin y)\,\vec\imath+(e^x+x\cos y)\,\vec\jmath$

then

$\dfrac{\partial f}{\partial x}=ye^x+\sin y\implies f(x,y)=ye^x+x\sin y+g(y)$

Differentiating with respec to $y$ gives

$\dfrac{\partial f}{\partial y}=e^x+x\cos y=e^x+x\cos y+g'(y)$

$\implies g'(y)=0\implies g(y)=C$

So $F$ is indeed conservative, and

$f(x,y)=ye^x+x\sin y+C$