## Definition of Curl as a Limit

We can define the curl of a vector at a point as a limit of an integral divided by an area.
Stoke's Theorem states
$\int \int_{\Delta S} (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS = \oint_C \mathbf{F} \cdot d \mathbf{r}$

By The Mean Value Theorem for area integrals
$\int \int_{\Delta S} (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS = \mathbf{F} |_{(x_0 , y_0 , z_0)} \cdot \mathbf{n} \Delta S$
for some
$(x_0 , y_0 , z_0) \in \Delta S$

Hence
$\mathbf{F} |_{(x_0 , y_0 , z_0)} \cdot \mathbf{n} \Delta S = \oint_C \mathbf{F} \cdot d \mathbf{r}$

Divide by
$\Delta S$
to obtain
$\mathbf{F} |_{(x_0 , y_0 , z_0)} \cdot \mathbf{n} = \frac{1}{\Delta S } \oint_C \mathbf{F} \cdot d \mathbf{r}$

Now let
$\Delta S \rightarrow 0$

$(x_0 , y_0 ,z_0)$
tends to a fixed point
$P$
and the right hand side tends to a limit.
$\mathbf{F} |_{P} \cdot \mathbf{n} = lim_{\Delta S \rightarrow 0} \frac{1}{\Delta S } \oint_C \mathbf{F} \cdot d \mathbf{r}$