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Proof That a Linear Combination of 1 - Forms is a 1 - Form

Theorem
A linear combination of 1 - forms is also a 1 form.
Proof
Let  
\[f, \: g\]
  be real valued functions on a domain  
\[D \subseteq \mathbb{R}^n\]
  and let  
\[\omega_1 ,\: \omega_2\]
  be 1 - forms defined on  
\[D \]
&. Then
\[\omega_{1 \mathbf{x}} =f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n \]

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

\[\begin{equation} \begin{aligned} f \omega_{1 \mathbf{x}} +g \omega_{2 \mathbf{x}} &=f(f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n )+ g(g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n) \\ &=(ff_1 +gg_1)(\mathbf{x}) dx_1 +...+ (ff_n +gg_n)(\mathbf{x}) dx_n \end{aligned} \end{equation}\]

Which is of the same form as  
\[\omega_1 ,\: \omega_2\]
.

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