# Intuition on Stokes' Theorem

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.