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Derivation of Green's Theorem From Stoke's Theorem in Terms of Forms

Theorem
Let  
\[D\]
  be a  
\[p+1\]
  - dimensional region and let  
\[\partial D\]
  be its  
\[p\]
  dimensional boundary. We can rewrite Stoke's Theorem in terms of the exterior derivative.
\[\int_D d \omega^p =\int_{\partial D} \omega^p\]
  (1)
From this statement we can derive Green's Theorem by letting  
\[\omega^p\]
  be a 1 - form in  
\[\mathbb{R}^2\]
.
Proof>br> Let  
\[\omega^1 = f_1 dx_1 + f_2 dx_2\]

Then
\[\begin{equation} \begin{aligned} d \omega^1 &=(\frac{\partial f_1}{\partial x_1}dx_1 + (\frac{\partial f_1}{\partial x_2}dx_2 ) \wedge dx_1 + (\frac{\partial f_2}{\partial x_1}dx_1 + (\frac{\partial f_2}{\partial x_2} \wedge dx_2 ) \\ &= (\frac{\partial f_2}{\partial x_1} - \frac{\partial f_1}{\partial x_2})dx_1 \wedge dx_2 \end{aligned} \end{equation}\]

Then (1) becomes  
\[\int_D (\frac{\partial f_2}{\partial x_1} - \frac{\partial f_1}{\partial x_2})dx_1 \wedge dx_2 =\int_{\partial D} f_1 dx_1 + f_2 dx_2\]
  (1)
This is equivalent to Green's Theorem.

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