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Stokes's 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.

enter image description here

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Personally, I imagine that dot product roughly as follows...

...disclaimer: I am not going to get rigorous. You should interpret this answer only as a reference point which can help you see things one way (not necessarily the correct one).

As we know, the curl of a vector field measure the "rotational tendency", or just rotation, for each point of the vector field. If we calculate the curl at a given point, we would get a vector. What does this vector measures? How do we interpret it?

If we want to measure the rotational tendency at a point, we would have to find two things: the magnitude (speed) of this rotation and the plane on which it takes place. Alternatively, this rotation is going to have an axis, which (surprisingly...) is the normal vector to the plane. Let's call "curl" exactly this: the vector whose magnitude shows the speed of rotation and whose direction tells us the axis of the rotation.

The sketch below is intended to illustrate the point. Imagine that the surface $S$ is some surface in the space. Let's also imagine that there is some vector field $\vec{F}$ working in the space (not drawn). Let the region $R$ be the $curl$ at the point $(x,y)$ produced by that field. Of course, this is an approximation, but it will get exact when we pass to the limit by making $R$ smaller and smaller. So, that tiny little region $R$ will rotate around an axis with a given (by the $curl\vec{F}$ at that point) speed.

<span class=$curl\vec{F}$ on a surface" />

So far so good, but let's get to the point. We want to measure the curl on a surface and not around a surface or through a surface or whatever. I.e., what do we do if we have something like this?

<span class=$curl\vec{F}$ passing through a surface" />

Our region $R$ is not quite on the surface $S$, right... and this doesn't work for us. We have a rotational tendency at our $(x,y)$ point, but it is not on the surface (or which is the same, its axis of rotation is not in the same direction as the normal to $S$ at $(x,y)$). But we are stubborn and we want our $curl$ and that is. Can't we just "turn" that evil region $R$ a bit so that it is actually on $S$, just like on the previous picture?

Turns out we can. Since our curl at that point has magnitude (speed) and a direction (and is, by and large, a vector), we can just take the component of it in some other direction - the normal to $S$ in our case. This will solve our problem: it will not only "turn" the $curl$ so that it fits nicely on $S$, but it will also "rescale" it so as to reflect its proper effect on $S$.

The obvious way to get the component of the $curl$ on $S$ is to just dot the curl axis with the normal vector, $curl\vec{F}\cdot\vec{n}$.

Cool!

And lastly, what if the axis of rotation of the $curl$ is on $S$? In other words, if the axis of rotation is perpendicular to the normal of the surface? In this case the dot product would be $0$ or in other words, $curl\vec{F}$ at that point would not produce any rotation on $S$.

If we want to do this not just for a point, but for the whole surface, we just add up the $curl$-s of all points on that surface, dilligently dotted with the normals at each point. Since we are working in continuous space, that sum would be represented as a surface integral, as you have shown.

Since you have included a picture of a sphere, I'm attaching a quick sketch involving one. The explanation is pretty much the same: the left region shows a curl is on the sphere and thus is perpendicular to the normal to the sphere at the point. We don't need to find components here, we just take what we have. But the region on the right is kind of "wrong" - it is not on the surface of the sphere and so we take only its component that is "curling" things on the sphere surface.

<span class=$curl\vec{F}$ on a surface of a sphere" />

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