Exterior Derivative

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 as a linear operator 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)

For the following component formulas, write a -form as , with antisymmetric coefficients and Einstein summation over all indices. Denote the exterior derivative by

(8)

Then for a 0-form ,

(9)

for a 1-form ,

(10)

and for a 2-form in three dimensions,

(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 a linear operator and commutes with the pullback map induced by a smooth map . 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 .