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

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.

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