Exterior Differentiation

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.

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