We would like to extend this notion of “gradient” to differential forms of higher rank. Any such form can be written as (a linear combination of terms like) \begin{equation} \alpha = f \,dx^I = f \,dx^{i_1} \wedge … \wedge dx^{i_p} \end{equation} Just as taking the gradient of a function increases the rank by one, we would like to define $d\alpha$ to be a $p+1$-form, resulting in a map \begin{equation} d: \bigwedge\nolimits^p \longmapsto \bigwedge\nolimits^{p+1} \end{equation} The obvious place to add an extra $d$ is again to take $f$ to $df$, which suggests that we should set \begin{equation} d\alpha = df \wedge dx^I \end{equation} and extend by linearity. Before summarizing the properties of this new operation, called exterior differentiation, we consider some examples.