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.