I'm trying to evaluate the following integral:
$ \int_C(y+\sin x) dx +(z^2+\cos y)dy+(x^3)dz$
Where $C$ is the curve: $c(t) = (\sin t, \cos t, \sin 2t) $. Note that $C$ lies on the surface $z= 2xy$.
Question: Could I just use the parameterization of the surface S $\phi(u,v)=(u, v,2uv) $ with $-1\leq u\leq 1 $ and $-1\leq v \leq 1$ and use Stokes' Theorem to evaluate $ \int_S \nabla \times F\cdot dS$ which should be equal to the required integral?
My only trepidation about the above method is that $c(t)$ may not necessarily be on the boundary of $S$, which would be required to use Stokes' Theorem.