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

$\mathbf f(x,y)=(ye^x+\sin y)\,\mathbf i+(e^x+x\cos y)\,\mathbf j$ is continuous, so $\mathbf f(x,y)$ is indeed conservative. So we know there is some scalar function $f(x,y)$ such that $\nabla f(x,y)=\mathbf f(x,y)$.

$\dfrac{\partial f}{\partial x}=ye^x+\sin y$
$\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int(ye^x+\sin y)\,\mathrm dx$
$f(x,y)=ye^x+x\sin y+g(y)$

$\dfrac{\partial f}{\partial y}=\dfrac{\partial(ye^x+x\sin y+g(y))}{\partial y}$
$e^x+x\cos y=e^x+x\cos y+g'(y)$
$0=g'(y)$
$\implies g(y)=C$

So we have

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