Is it notationally incorrect to use the contour integration symbol $\oint$ for electric flux $\oint \vec{E}\cdot\,d\vec{A}$, an integral over an area? I've been told that the symbol $\oint$ means that we are taking an integral over a line or contour. However, in physics, I've seen electric flux defined as $\oint \vec{E} \cdot \, d\vec{A}$, which doesn't make sense because we are integrating over an area, not a contour.
Therefore, is it notationally incorrect to define electric flux as $\oint \vec{E} \cdot \, d\vec{A}$? Or am I not understand what $\oint$ means?
 A: The symbol $\oint$ means you're integrating over a closed object (note 'closed' here doesn't mean in the topological sense). For example, a closed loop (a circle $S^1$) or a sphere $S^2$ or higher dimensional analogues. So, the use of the symbol $\oint$ is fine, but it's just redundant, because typically one indicates the domain of integration, $M$, in the subscript position as $\int_M$, and from here we can tell what kind of space we're integrating over (a closed loop, or a closed surface, or a non-closed one).
For the case of electric flux specifically, one may use the symbol $\unicode{x222F}$, because the integration domain is a 2-dimensional closed surface. However, you'll realize as you take more math classes that it is often better to use just one integral symbol, and be explicit about the domain of integration, as in $\int_M$, because that makes everything clearer.
A: It's completely reasonable to be confused by this notation for exactly the reasons you mentioned, so I would personally avoid it. However, some people (more common in Physics than in math, but I've seen it in math) use the circle to denote that the surface where you are integrating over is a closed surface (without boundary). This is important in this flux integral because Gauss's Theorem (Divergence theorem) only applies when you integrate over a closed surface. I think a decent compromise is to use the double integral with loop symbol \oiint from the esint package.
