## 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{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}

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.