Differential Forms

Let
$x=(x_1 , x_2 ,...,x_n) \in \mathbb{R}^n$
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
$\alpha_1 dx_1+ \alpha_2 dx_2 + ... + \alpha_n dx_n$

$f_1 , f_2 ,... , f_n$
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
$\mathbf{a} = (-1,-2)$

$\omega_{\mathbf{x}}(\mathbf{a}) = 2 \times -1 + 3 \times -2=-8$