1. The Question Circulation Raises: How Much Does a Field Spin at a Point?
Recall: Circulation
In the previous topic you learned that the circulation of a vector field $\mathbf{F}$ around a closed curve $C$ is the line integral
$$\oint_C \mathbf{F}\cdot d\mathbf{r}.$$Circulation tells you the total "push" the field gives a particle travelling once around the loop. But that is a global number — it depends on the size and shape of the loop. The natural next question is:
"How much does the field spin at a single point?"
1.1 The Limit Definition — Circulation per Unit Area
To isolate the local spin at a point $\mathbf{a}$, we shrink the loop down to that point and divide by the enclosed area. Pick a unit normal direction $\hat{\mathbf{n}}$ and a tiny oriented loop $C$ enclosing area $A(C)$ in the plane perpendicular to $\hat{\mathbf{n}}$. Then the component of curl $\mathbf{F}$ in the $\hat{\mathbf{n}}$ direction is defined by the limit:
$$\bigl(\nabla\times\mathbf{F}(\mathbf{a})\bigr)\cdot\hat{\mathbf{n}} \;=\; \lim_{A(C)\to 0}\frac{1}{A(C)}\oint_C \mathbf{F}\cdot d\mathbf{r}.$$This is the circulation density — the circulation per unit area as the loop shrinks to a point. The full curl vector is obtained by evaluating this limit for three independent normal directions ($\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$).