Why is $\oint_{C_{r_{0}}} \mathbf{v} \cdot \mathbf{r}' \mathrm ds$ called the circulation of the flow around $C_{r_{0}}$? I was reading the physical interpretation of curl from the textbook' Advanced Engineering Mathematics, Ervin Kreyzig. I have a small doubt in the explanation.

 



Doubt:- $C_{r_{o}}$ is a circle of radius $r_o$.Why does $\oint_{C_{r_{o}}}\textbf{v}\textbf{.}\textbf{r'} ds$ is called the
  circulation of the flow around $C_{r_{o}}$? How does it measure the
  extend to which the corresponding fluid motion is a rotation around
  the circle $C_{r_{o}}$? Please help me. Except this statement, I
  understood the rest of the statement.

My attempt:-
I come across the Calculus, M.J strauss, G.L Bradly, K.J Smith. I got some ideas, I can write, What I understood from there. Still I have little bit doubt. $\textbf{r'}(s)$ denote the unit tangent vectors to the given curve $C_{r_o}$ right? $\textbf{v}\textbf{.}\textbf{r'}(s)$ denote the tangential component of velocity. Right? If we plot in the this tangential component as a function of points in the space. I can think about the geometry. It is the curved surface area. Still, I don't know, How it measures the tendancy of the fluid to rotate. Please help me.
 A: This comes from understanding the geometric interpretation of the dot product and thinking about what would happen in the event of a purely rotational vector field.
The dot product $\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta$ is a scaled version of $\mathbf{a}$'s projection onto $\mathbf{b}$ and vice-versa. In the event that $\mathbf{b}$ is a unit vector, it is exactly just the orthogonal projection of $\mathbf{a}$ onto $\mathbf{b}$. That is, if we decompose $\mathbf{a}=\mathbf{a}_{\|}+\mathbf{a}_{\perp}$ into components parallel and perpendicular to $\mathbf{b}$ respectively, then the dot product is $\mathbf{a}\cdot\mathbf{b}=(\mathbf{a}_{\|}+\mathbf{a}_{\perp})\cdot\mathbf{b}=\mathbf{a}_{\|}\cdot\mathbf{b}$ which will be the signed length of $\mathbf{a}$'s projection onto the oriented $\mathbf{b}$-axis.
Say you have a constant vector field $\mathbf{F}$ in the plane which represents liquid flowing in a vaguely northernly direction (i.e. the $y$-coordinate of $\mathbf{F}$ is positive). If $\mathbf{F}$ points straight up, then the amount of liquid that passes through a unit length horizontal line segment will be $\|\mathbf{F}\|$, however if the direction of the flow is angled at all it is intuitively clear that less liquid will pass through the line segment per unit time. More precisely, the rate of flow through the horizontal line segment will be the dot product of $\mathbf{F}$ with the unit normal pointing straight north - this represents the vertical component of the flow. (If $\mathbf{F}$ were southernly-directed, this rate would be negative.)
Now imagine there is a mystical horizontal line with a ball on it. The line magically negates all vertical force acting on the ball, but allows horizontal force to act on it. So when the liquid is moving, the ball doesn't move up or down at all, it only moves left and right. The rate of change of how much it moves to the right (east) is the dot product of $\mathbf{F}$ with the unit vector pointing east.
The moral here is that the dot product $\mathbf{F}\cdot\mathbf{s}$ will measure the component of the field $\mathbf{F}$ along the direction of $\mathbf{s}$. If you want to measure the flow across a surface, for instance, one must take the dot product $\mathbf{F}\cdot\mathbf{n}$ of the vector field $\mathbf{F}$ with the surface's unit normal vector $\mathbf{n}$, yielding the Divergence Theorem. If instead you want to measure how much $\mathbf{F}$ flows around a circle, say, you need to completely discard the components of $\mathbf{F}$ that are moving across the boundary (the components parallel to the unit normal) and settle on the components of $\mathbf{F}$ that are moving along te boundary (the components parallel to the unit tangent vectors). This yields the definition of circulation.
