The Uniqueness of the Exterior Derivative
It turns out that our exterior derivative is uniquely characterized by some of its properties; it is the only derivation of the algebra of degree
whose square is zero and which gives the differential on functions. That is, once we specify that
, that
, that
if
is a
-form, that
, and that
for functions
, then there is no other choice but the exterior derivative we already defined.
First, we want to show that these properties imply another one that’s sort of analytic in character: if in a neighborhood of
then
. Equivalently (given linearity), if
in a neighborhood
of
then
. But then we can pick a bump function
which is
on a neighborhood of
and
outside of
. Then we have
and
And so we may as well throw this property onto the pile. Notice, though, how this condition is different from the way we said that tensor fields live locally. In this case we need to know that vanishes in a whole neighborhood, not just at
itself.
Next, we show that these conditions are sufficient for determining a value of for any
-form
. It will helps us to pick a local coordinate patch
around a point
, and then we’ll show that the result doesn’t actually depend on this choice. Picking a coordinate patch gives us a canonical basis of the space
of
-forms over
, indexed by “multisets”
. Any
-form
over
can be written as
and so we can calculate
where we use the fact that .
Now if is a different coordinate patch around
then we get a different decomposition
but both decompositions agree on the intersection , which is a neighborhood of
, and thus when we apply
to them we get the same value at
, by the “analytic” property we showed above. Thus the value only depends on
itself (and the point
), and not on the choice of coordinates we used to help with the evaluation. And so the exterior derivative
is uniquely determined by the four given properties.
Are you going to cover integration of differential forms or de Rham cohomology?
Well, having introduced the exterior derivative and shown that its square is zero, I’m halfway to de Rham cohomology already, so that wouldn’t be a bad guess.
[…] The really important thing about the exterior derivative is that it makes the algebra of differential forms into a “differential graded algebra”. We had the structure of a graded algebra before, but now we have a degree-one derivation whose square is zero. And as long as we want it to agree with the differential on functions, there’s only one way to do it. […]
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