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