Question that we have a disagreement about
The graph
My teammates say that the vector field is conservative. But I do not think so, reason being is because if you draw a closed curve on the vector field and add up all the line integrals I believe you get a value greater than zero. For a vector field to be conservative the line integral must equal zero.
So am I correct to say the vector field is not conservative. and So on the question part c it would be none of the above.
On
Looks to me that the $y$ component of the vector field is proportional to $x$, i.e. the vector field looks a bit like $0dx+xdy$.
That one does not satisfy any of (a)-(c) because the necessary condition for a vector field $f (x,y)dx+g (x,y)dy $ to satisfy those conditions is $\frac {\partial f}{\partial y}=\frac {\partial g}{\partial x} $, which is not the case here ($0\ne 1$).
Thus my guess is that the vector field from the problem also does not satisfy (a)-(c).
You are correct. To see that the vector field is not conservative, it will suffice to demonstrate a loop which has a non-zero line integral. You can pretty easily do this with this vector field by starting at the origin, travelling right along the x-axis, travelling north, then returning to the y-axis, and from there back to the origin. All of the components of this path will have integral $0$, except for the second, which has a positive integral.