Differential of a smooth function
= Differential of a smooth function
{title2=$df_p(v)=v(f)$}
For a real-valued <smooth function> on a <smooth manifold>, its <differential of a smooth map> is a <differential one-form>: $df_p$ maps a <tangent vector> to the directional derivative of the function. In local coordinates, $df=\sum_i(\partial_i f)dx^i$. The <exterior derivative> extends this operation from functions to higher-degree <differential forms>.