\[\omega^2 = f_1 dx_2 \wedge dx_3 + f_2 dx_3 \wedge dx_1 +f_3 dx_1 \wedge dx_2\]
is defined on a region \[D\]
of \[\mathbb{R}^3\]
with surface \[S\]
then the statement \[\int_D d \omega^2 = \int_S \omega^2 \]
(1) is equivalent to the Divergence Theorem.To see this not that with
\[\omega^2\]
as above,\[d \omega^2 = (\frac{\partial f_1}{\partial x_1} + \frac{\partial f_2}{\partial x_2} + \frac{\partial f_3}{\partial x_3})dx_1 \wedge dx_2 \wedge dx_3\]
\[\frac{\partial f_1}{\partial x_1} + \frac{\partial f_2}{\partial x_2} + \frac{\partial f_3}{\partial x_3}\]
is the divergence of the vector field \[\mathbf{F} = (f_1,f_2,f_3)^T\]
Substituting for
\[\omega^2 , \: d \omega^2\]
in (12) gives\[\int_D (\frac{\partial f_1}{\partial x_1} + \frac{\partial f_2}{\partial x_2} + \frac{\partial f_3}{\partial x_3})dx_1 \wedge dx_2 \wedge dx_3 = \int_S f_1 dx_2 \wedge dx_3 + f_2 dx_3 \wedge dx_1 +f_3 dx_1 \wedge dx_2 \]
This is equivalent to the Divergence Theorem.