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

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

then

`(\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

`(\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`