The operation mathematically means $$(\nabla \times \vec A)\cdot\hat n = \lim_{\Delta S\to\ 0} \frac{\oint\vec A\cdot\ d\vec l }{\left | \Delta S \right |}$$ and the proof of this is quite logical.
In textbooks, I have found that they relate curl to rotation.
But how does the above mathematical treatment show this relation?
Any mathematical help is welcome.
Best Answer
Let $\bf{C}$ be any vector-field. Let there be a closed-curve $\Gamma$. Take the line-integral of $\bf C$ around this complete loop that is, take the tangential-component of the field at any point on the curve & take dot-product of $\mathbf{C}_\text{tangential}\cdot d\mathbf{s}$ & integrate over the whole loop. This is circulation : $$\text{Circulation}_{\Gamma}=\int_{\Gamma} \mathbf{ C} \cdot d\mathbf{s}.$$
Now make partitions on $C$ with an intermediary-bridge, say $B$; this then will make two loops viz; $\Gamma_1 \;\&\; \Gamma_2.$ Now circulation over the whole loop is just line-integral around each sub-loop that is, $$\text{Circulation}_{\Gamma} \\= \text{Circulation}_{\Gamma_1} + \text{Circulation}_{\Gamma_2}\\= (\text{Circulation}_{\Gamma_a} + \text{Circulation}_{\Gamma_{ab}}) + (\text{Circulation}_{\Gamma_b} -\text{Circulation}_{\Gamma_{ab}})\\ =\text{Circulation}_{\Gamma_{a}} + \text{Circulation}_{\Gamma_{b}}.$$
Break into still further smaller loops as $\Gamma_1,\Gamma_2,\Gamma_3 \ldots \Gamma_N.$
Circulation over the whole-loop $$\int_{\Gamma} \mathbf{C}\cdot d\mathbf{s}= \sum_{i=1}^N\int_{\Gamma_i} \mathbf{C}\cdot d\mathbf{s}_i.$$
Now we want to get the infinitesimal-circulation characteristic to a certain coordinate. We then make $N\to \infty;$ however each integral that is $\int_{\Gamma_i} \mathbf{C}\cdot d\mathbf{s}_i \to 0$ as $N\to \infty.$ So, in order to get the finite characteristic which is associated with circulation locally to a point, we divide each integral by the area enclosed by each sub-loop. That is, take the ratio of circulation to loop area$$\frac{\text{Circulation}_{\Gamma_i}}{a_i}.$$ This is our local-property that is the infinitesimal circulation of the vector-field at a certain point (around it) is given by $$\lim_{a_i\to 0} \frac{\int_{\Gamma_i} \mathbf{C}\cdot d\mathbf{s}_i}{a_i}.$$