## Proof That a 1 - form is a Real Valued Linear Function for a Fixed Point

Theorem
If
$\omega_{\mathbf{x}}$
is a 1 - form on a region
$D \subseteq \mathbb{R}*n$
and
$x$
is a fixed point in
$D$
then
$\omega_{\mathbf{x}}$
is a real valued function on
$\mathbb{R}^n$
.
We can write
$\omega_{\mathbf{x}} =f_1 (\mathbf{x}) dx_1 +f_2 (\mathbf{x}) dx_2 +...+ f_n (\mathbf{x}) dx_n$

For a =fixed point
$x_0 \in D$
,
$\omega_{\mathbf{x}_0} =f_1 (\mathbf{x}_0) dx_1 +f_2 (\mathbf{x}_0) dx_2 +...+ f_n (\mathbf{x}_0) dx_n$

Let
$\mathbf{a} , \: \mathbf{b} \in \mathbb{R}^n , \: \alpha in \mathbb{R}$
then
\begin{aligned} \omega_{\mathbf{x}_0} (\mathbf{a}) &=f_1 (\mathbf{x}_0) dx_1 (\mathbf{a})+f_2 (\mathbf{x}_0) dx_2 (\mathbf{a})+...+ f_n (\mathbf{x}_0) dx_n (\mathbf{a}) \\ &=f_1 (\mathbf{x}_0)a_1+f_2 (\mathbf{x}_0) a_2+...+ f_n (\mathbf{x}_0) a_n \end{aligned}

Since the
$f_i (\mathbf{x}_0) , \: a_i$
are real numbers
$\omega_{\mathbf{x}_0}$
is a real valued function on
$\mathbb{R}^n$

$\omega_{\mathbf{x}_0}$
is linear by this result.