Chapter 4: Differentation

### 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) $$\alpha = f \,dx^I = f \,dx^{i_1} \wedge … \wedge dx^{i_p}$$ 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 $$d: \bigwedge\nolimits^p \longmapsto \bigwedge\nolimits^{p+1}$$ The obvious place to add an extra $d$ is again to take $f$ to $df$, which suggests that we should set $$d\alpha = df \wedge dx^I$$ and extend by linearity. Before summarizing the properties of this new operation, called exterior differentiation, we consider some examples.