Deriving Stoke's Theorem From Green's Theorem

Suppose a curve  
  with interior  
  defined on  
  has a smooth parametrization  
\[\mathbf{p}(s) = (p_1 (s) , p_2 (s)) \]
  is the distance along the curve.
The unit tangent to  
\[\mathbf{T} = ( \frac{p'_1(s)}{\sqrt{p'^2_1(s)+p'^2_2(s)}} , \frac{p'_2(s)}{\sqrt{p'^2_1(s)+p'^2_2(s)}} )\]

The unit normal to  
  is given by  
\[\mathbf{n(s)} =\mathbf{T'} = ( \frac{p''_1(s)}{\sqrt{p''^2_1(s)+p''^2_2(s)}} , \frac{p''_2(s)}{\sqrt{p''^2_1(s)+p''^2_2(s)}} )\]

The circulation of a force  
  is defined as  
\[\oint_C F_1 \: dx + F_2 \: dy = \oint_C \mathbf{F} \cdot d \mathbf{r} ds \]

\[curl \mathbf{F} = (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}) \mathbf{k} \]

Hence Green's Theorem  
\[\oint_C F_1 \: dx + F_2 \: dy = \int_B (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y})dx \: dy\]

can be written
\[\oint_C \mathbf{F} \cdot \mathbf{T} ds = \int \int_B curl \mathbf{F} \; dx \: dy\]

This is called Stokes Theorem.

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