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Firstly, the AP Physics C equation sheet here (the page I am talking about is page 5) uses $\oint$ for Gauss's law, Ampere's law, and Faraday's Law. However, the Wikipedia page about Gauss's law uses a symbol I can't seem to get here (Wikipedia says its \oiint), but it's $\iint$ with a circle around it, like $\oint$. The page about Ampere's Law says that $\oint$ is a closed line integral, and that $\iint$ is a 2-D surface integral. Finally, the page on Faraday's Law uses just a single $\int$ for a surface integral, and $\oint$ for a line integral. My teacher uses the $\oint$ symbol for a surface integral.

However, on a Wikipedia page for the integral symbol, it says that \oiint is a closed surface integral, $\oint$ is a contour integral, and that $\iint$ is simply a double integral. However, on the Stoke's Theorem page, they use $\iint$ for a surface integral and $\oint$ for a line integral. The actual page on Wikipedia for line integrals simply uses $\int$, and the page for surface integrals uses $\iint$, which is what my calculus book (here) uses.

Which one is the most common way/the right way (if there is one)?

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The circle is used for emphasis, so it could be there or not (of course, in the case of closed curves or surfaces).

Using one or several integral symbols is also a matter of choice. As one grows in abstract thinking, an integral is an integral, and so $\iint$ is not really conveying more information that $\int$ if the rest of the context is clear. This is not different than going from $\int f(x)\,dx$ to $\int f$.

Finally, $\oint$ for an arbitrary surface integral makes no sense, as the canonical meaning of the circle is that the region of integration is closed (no boundary).

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