Consider 2 vector fields $\vec{F}$ and $\vec{G}$. If the closed surface integrals
$$ \oint_S \vec{F} \cdot d\vec{S} = A \tag{1} $$
$$\oint_S \vec{G} \cdot d\vec{S} = B \tag{2} $$
for any surface $S$, can we conclude $\vec{F} \propto \vec{G}$? This would definitely make the above integrals a true statement. But given equations $(1)$ and $(2)$ alone, I don't think the proportionality necessarily follows. It would be the easiest conclusion to make, but it might not be the only conclusion. I'm tempted to say "yes, they are proportional", only because the statement holds true for every closed surface. If the statement were true only for a single closed surface, there's no way you can conclude that they are proportional. Then again, I'm still hesitant to conclude proportionality because I am aware of the following: if the closed line integral of two vector fields are $0$ for every closed loop, this just means that they are conservative. And I know of 2 vector fields (springs and inverse-square) which are conservative but aren't proportional.
For "physics reasons" I would really like $\vec{F}$ and $\vec{G}$ to be proportional. Are $(1)$ and $(2)$ sufficient to make this conclusion? If I also know that $\vec{F}$ and $\vec{G}$ are parallel, do I now have enough to conclude proportionality? This is not true of my problem at hand, but if $\nabla \times \vec{F} = 0$ and $\nabla \times \vec{G} = 0$, is this enough to conclude that the fields are proportional? Thanks in advance. I'm hoping the addition of the parallel requirement is enough to guarantee proportionality
If the flux of $\vec{F}$ over all closed surfaces is constant, then it can be shown that $\vec{\nabla}\cdot \vec{F}=0$. Hence $\vec{F}$ is constant.