- Differentiation of Differential Forms

Just as taking the gradient of a function increases the rank by one, we

would like to define dα to be a p+ 1-form, resulting in a map

d : ∧p −→∧p+1 . (15.6)

The obvious place to add an extra d is again to take f to df , which suggests

that we should set

dα = df ∧ dxI (15.7) and extend by linearity. Before summarizing the properties of this new

operation, called exterior differentiation, we consider some examples.