We now have two algebraic maps on differential forms, namely: \begin{align} \wedge:& \bigwedge\nolimits^p \times \bigwedge\nolimits^q \longmapsto \bigwedge\nolimits^{p+q} \\ *:& \bigwedge\nolimits^p \longmapsto \bigwedge\nolimits^{n-p} \end{align} In this chapter, we introduce a third such map, involving differentiation.

Recall that $\bigwedge^0(M)$ is the set of functions on $M$, and that the differential $df$ of any function $f$ is a 1-form. Taking the differential, or “zapping a function with $d$”, is therefore a map \begin{equation} d: \bigwedge\nolimits^0 \longmapsto \bigwedge\nolimits^1 \end{equation} What is this map? We have \begin{equation} df = \Partial{f}{x^i}\,dx^i \end{equation} a 1-form whose components are just the partial derivatives of $f$ — just like the gradient. We therefore identify $df$ with the gradient of $f$.