## THe Differential of a Function is a One Form

Let
$\mathbf{a} =(a_1 , a_2 , a_3) \in \mathbb{R}^3$
be a vector.
\begin{aligned} d_{\mathbf{x}} (\mathbf{a}) &= \frac{\partial}{\partial x_1} (\mathbf{x})a_1 + \frac{\partial}{\partial x_2} (\mathbf{x})a_2 + \frac{\partial}{\partial x_3} (\mathbf{x})a_3 \\ &= \frac{\partial}{\partial x_1} (\mathbf{x})dx_1 (\mathbf{a}) + \frac{\partial}{\partial x_2} (\mathbf{x})dx_2 (\mathbf{a}) + \frac{\partial}{\partial x_3} (\mathbf{x})dx_3 (\mathbf{a}) \end{aligned}

The coefficient functions are the
$\frac{\partial}{\partial x_2} (\mathbf{x})$
.
This shows that the differential of a function is a one form.