## Proof That 1 - Forms are Semi - Distributive

Theorem
The 1 - form is semi distributive that is, if
$\omega_1 , \: \omega_2 , \: \omega_3$
are 1 - forms ans
$f,\: g$
are functions, then
$(f \omega_1 + g \omega_2) \wedge \omega_3 = f \omega_1 \wedge \omega_3 + g \omega_2 \wedge \omega_3$
Proof
Let
$f, \: g$
be real valued functions on a domain
$D \subseteq \mathbb{R}^n$
and let
$\omega_1 ,\: \omega_2 ,\: \omega_3$
be 1 - forms defined on
$D$
&. Then
$\omega_{1 \mathbf{x}} =f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n$

$\omega_{2 \mathbf{x}} =g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n$

$\omega_{3 \mathbf{x}} =h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n$

Expanding the left hand side gives
\begin{aligned} & ( \omega_{1 \mathbf{x}} + \omega_{2 \mathbf{x}}) \wedge \omega_3 \\ &=(f(f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n)+ g(g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n)) \wedge (h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n ) \\ &= \sum_{i=1}^n \sum_{j=1}^n ( h_i (ff_i +gg_i) -h_i (ff_j +gg_j)) dx_i \wedge dx_j \end{aligned}

Expanding the right hand side gives
\begin{aligned} & f \omega_1 \wedge \omega_3 + g \omega_2 \wedge \omega_3 \\ &= f(f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n) \wedge (h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n)\\ &+ g(g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n) \wedge (h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n) \\ &=\sum_{i=1}^n \sum_{j=1}^n (h_jff_i -h_i ff_j) dx_i \wedge dx_j +\sum_{i=1}^n \sum_{j=1}^n (h_jgg_i -h_igg_j)dx_i \wedge dx_j \\ &= \sum_{i=1}^n \sum_{j=1}^n ( h_i (ff_i +gg_i) -h_i (ff_j +gg_j)) dx_i \wedge dx_j \end{aligned}