Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int \int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$

I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a given surface. I also understand that the integral is essentially a summation of a quantity.

However, why is $curl \space \vec{F}$ dotted with $\vec{n}$? Setting aside the fact that integrals require scalar functions (i.e. a dot with some vector is necessary), the dot product says that vector $\vec{a} \cdot \vec{b} = 0$ if $\vec{a}$ and $\vec{b}$ are orthogonal. If you visualize some surface $S$ with a curl vector $\vec{a}$, assuming it is measuring the circulation of particles actually on the surface, and a normal vector $\vec{b}$, dotting $\vec{a}$ and $\vec{b}$ would yield $0$, or, in other words, not the circulation.

What am I missing? Thanks.

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