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) = (xy cos(xy) + sin(xy))i + (x2 cos(xy))j

Answer 1

If

$\nabla f=(xy\cos xy+\sin xy)\,\vec\imath+x^2\cos xy\,\vec\jmath$

then we should have

$\dfrac{\partial f}{\partial y}=x^2\cos xy\implies f(x,y)=x\sin xy+g(x)$

Differentiating with respect to $x$ gives

$\dfrac{\partial f}{\partial x}=xy\cos xy+\sin xy=\sin xy+xy\cos xy+g'(x)$

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

So $F$ is indeed conservative, and

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