I am studying for exam and I come across a dilemma.
Assume that:
$$ \vec{F}=\vec{\nabla}\times\vec{A} $$
This implies that we are in a solenoidal or divergence-less vector field.
It is given that $\oint{\vec{F} \cdot d\vec{s}}=0$. Prove that $\int{\vec{F} \cdot d\vec{s}}$ is independent of surface, for any given boundary line.
My attempt to solve this is to cut a closed surface $S$ into any two open surfaces, $S_1$ and $S_2$, as long as they are different from each other but shares a common boundary.
$$ \oint_S{\vec{F} \cdot d\vec{s}}=0 $$
$$ \int_{S_1}{\vec{F} \cdot d\vec{s}} + \int_{S_2}{\vec{F} \cdot d\vec{s}}=0 $$
$$ \int_{S_1}{\vec{F} \cdot d\vec{s}} = -\int_{S_2}{\vec{F} \cdot d\vec{s}} $$
Here comes the dilemma, the two above must be equal to prove that $\int{\vec{F} \cdot d\vec{s}}$ is independent of surface, for any given boundary line. For some reason, the two has opposite sign.
The only reason I can think to solve this is for the $d\vec{s}$ of two integrals to have opposite signs. This will make the integrals equal. But this does not really make sense to me. At least, I can't imagine why this is the case.
Thank you very much!