The exterior derivative of a function is the one-form
(1)
|
written in a coordinate chart . Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k-forms using the formula
(2)
|
when is a -form and where is the wedge product.
The exterior derivative of a -form is a -form. For example, for a differential k-form
(3)
|
the exterior derivative is
(4)
|
Similarly, consider
(5)
|
Then
(6)
| |||
(7)
|
Denote the exterior derivative by
(8)
|
Then for a 0-form ,
(9)
|
for a 1-form ,
(10)
|
and for a 2-form ,
(11)
|
where is the permutation tensor.
It is always the case that . When , then is called a closed form. A top-dimensional form is always a closed form. When then is called an exact form, so any exact form is also closed. An example of a closed form which is not exact is on the circle. Since is a function defined up to a constant multiple of , is a well-defined one-form, but there is no function for which it is the exterior derivative.
The exterior derivative is linear and commutes with the pullback of differential k-forms . That is,
(12)
|
Hence the pullback of a closed form is closed and the pullback of an exact form is exact. Moreover, a de Rham Cohomology class has a well-defined pullback map .