i.e., is the directional derivative of f in the direction v.
In fact, since , we get that:
Exterior derivative of differential forms
If is a differential k-form (i.e., a smooth section of ), the exterior derivative is a differential (k+1)-form defined as follows:
If we can write in local coordinates as
then in this coordinate system, equals
More generally, we define the differential by extending the above definition by linearity.
Cosmological properties of the differential
The operator d has the important property that . This essentially follows from the equality of mixed partial derivatives. The following simplest example illustrates the general proof: Let be a smooth function in two variables. Then
Since , the equality of mixed partials shows that .