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\[\begin{equation} \begin{aligned} \omega_{\mathbf{x}_0} (\mathbf{a}) &=f_1 (\mathbf{x}_0) dx_1 (\mathbf{a})+f_2 (\mathbf{x}_0) dx_2 (\mathbf{a})+...+ f_n (\mathbf{x}_0) dx_n (\mathbf{a}) \\ &=f_1 (\mathbf{x}_0)a_1+f_2 (\mathbf{x}_0) a_2+...+ f_n (\mathbf{x}_0) a_n \end{aligned} \end{equation}\]
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.