If $G$ is a vector field, and $G = ∇g$ for some function $g$, what would line integral $G · ds$ have to be?

Question

If $G$ is a vector field, and $G = ∇g$ for some function $g$, what would line integral $G · ds$ have to be? (Hint: Think of c as a curve whose ending point is the same as its starting point).

Since $c$ has same starting and end point it could be a circle $c(t) = (\cos(t), \sin(t))$, $t ∈ [0,2\pi]$. So by fundamental theorem of line integrals integral of $$G · ds = g(c(2\pi)) - g(c(0)) = g(1,0)-g(1,0) = 0$$ This doesn't seem right. Please correct me where I went wrong.

Answer 1

This is correct. It is a consequence of the gradient theorem.