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
\\&=d\phi(p)\wedge\alpha(p)+\phi(p)d\alpha(p)\\&=0+0=0\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Dd%5Calpha%28p%29%26%3D%5Cleft%5Bd%28%5Cphi%5Calpha%29%5Cright%5D%28p%29%5C%5C%26%3Dd%5Cphi%28p%29%5Cwedge%5Calpha%28p%29%2B%5Cphi%28p%29d%5Calpha%28p%29%5C%5C%26%3D0%2B0%3D0%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}d\alpha(p)&=\left[d(\phi\alpha)\right](p)\\&=d\phi(p)\wedge\alpha(p)+\phi(p)d\alpha(p)\\&=0+0=0\end{aligned}")
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.
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July 19, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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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