Differential of a smooth function ($d{f}_p(v)=v(f)$)

For a real-valued smooth function on a smooth manifold, its differential of a smooth map is a differential one-form: $d{f}_p$ maps a tangent vector to the directional derivative of the function. In local coordinates, $df={\sum }_i({\partial }_{i}f)d{x}^i$. The exterior derivative extends this operation from functions to higher-degree differential forms.