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