Definition of Curl as a Limit
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. \[\mathbf{F} |_{P} \cdot \mathbf{n} = lim_{\Delta S \rightarrow 0} \frac{1}{\Delta S } \oint_C \mathbf{F} \cdot d \mathbf{r}\]
