Differential Forms
\[x=(x_1 , x_2 ,...,x_n) \in \mathbb{R}^n\]
with each \[x_i\]
a real number, be an n dimensional vector.By
\[dx_i\]
we denote the function that assigns to a vector \[a \in \mathbb{R}^n\]
its ith component i.e. \[dx_i (\mathbf{a}) =a_i\]
If then
\[\mathbf{a} =(2,1,6)\]
then\[dx_1 (\mathbf{a})=2\]
\[dx_2 (\mathbf{a})=1\]
\[dx_3 (\mathbf{a})=6\]
We can form new functions by taking linear combinations of the
\[dx_i \]
such as
\[\alpha_1 dx_1+ \alpha_2 dx_2 + ... + \alpha_n dx_n\]
I
\[f_1 , f_2 ,... , f_n\]
are real valued functions defined on \[\mathbb{R}^n\]
or a region of \[\mathbb{R}^n\]
\[f_k : D \rightarrow \mathbb{R} \: k=1,2,...,n\]
Then for each
\[\mathbf{x}x \in D\]
we can form the linear combination\[\omega_{\mathbf{x}} = f_1 (\mathbf{x}) dx_1 + f_2 (\mathbf{x}) dx_2 +...+ f_n (\mathbf{x}) dx_n \]
\[\omega_{\mathbf{x}} \]
acts on a vector \[\mathbf{a} \in \mathbb{R}^n \]
\[\omega_{\mathbf{x}}(\mathbf{a}) = f_1 (\mathbf{x}) dx_1 (\mathbf{a}) + f_2 (\mathbf{x}) dx_2 (\mathbf{a}) +...+ f_n (\mathbf{x}) dx_n (\mathbf{a})\]
Example:
\[\omega_{\mathbf{x}} = 2dx_1 + 3 dx_2 \]
and \[\mathbf{a} = (-1,-2)\]
\[\omega_{\mathbf{x}}(\mathbf{a}) = 2 \times -1 + 3 \times -2=-8\]
