So in class we are currently going over the (classical) Kelvin Stokes theorem, and a variety of equivalences. Namely, one theorem we have is as follows:
Let $F$ be a $C_1$ vector field mapping $\mathbb{R}^3$ to $\mathbb{R}^3$ The following statements are equivalent
- There exists a vector field $G$ so that
$$ F = \nabla \times G $$
- The following integral depends only on the boundary of the surface $S$:
$$ \int_SF \cdot n \ \mathrm{d}\sigma $$
- The following integral is $0$ over any closed surface: $$ \int_{\partial S} F\cdot T \ \mathrm{d}s $$ It is easy to show that $1$ implies $2$ and $3$. I am having issues showing that $2$ and $3$ imply $1$. I saw a proof of an analogous statement in a vector calculus book (it was a proof of similar equivalences using green's theorem) that constructed a triangular path and applied Green's theorem in a similar fashion. I am thinking of applying the same methodology, but I am not sure what surface I should construct. I was hoping someone could provide me with a hint/some inspiration.