The general theorem from which Gauss’s law stems, divergence theorem, states that
the volume integral of the divergence $\nabla\cdot\bf F$ of $\bf F$ over $V$ and the surface integral of $\bf F$ over the boundary $\partial V$ of $V$ are related by $$\int_V \left( \nabla\cdot\bf F\right) \, dV = \int_{\partial V} \mathbf{F} \cdot d \bf{a}$$ Wolfram MathWorld
In class, we would write the righthand side of this equation as $$\iint\limits_{\mathrm{closed}} \vec F \cdot \hat n \, dA$$ where (as you might have guessed) $\hat n$ was a unit vector that was normal to the surface at each point. (Actually, we would draw an oval overtop of the double integral sign to show it was closed, but MathJax doesn’t support that.)
Of course, there exist closed surfaces for which $\hat n$ cannot be systematically defined, called nonorientable surfaces.
My question is: How could divergence theorem be applied to nonorientable surfaces? Of course, an obvious solution is to not choose a nonorientable surface over which to evaluate the integral, but that’s not what I’m aiming for.
(If this question is more appropriate for physics SE, please mention so in the comments.)