The Lie derivative looks sort of familiar as a derivative, but we have another sort of derivative on the algebra of differential forms: the “exterior derivative”. But this one doesn’t really look like a derivative at first, since we’ll define it with some algebraic manipulations.
If 
is a 
-form then 
is a 
-form, defined by
![\displaystyle\begin{aligned}d\omega(X_0,\dots,X_k)=&\sum\limits_{i=0}^k(-1)^iX_i\left(\omega(X_0,\dots,\hat{X_i},\dots,X_k)\right)\\&+\sum\limits_{0\leq i<j\leq k}(-1)^{i+j}\omega\left([X_i,X_j],X_0,\dots,\hat{X_i},\dots,\hat{X_j},\dots,X_k\right)\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Dd%5Comega%28X_0%2C%5Cdots%2CX_k%29%3D%26%5Csum%5Climits_%7Bi%3D0%7D%5Ek%28-1%29%5EiX_i%5Cleft%28%5Comega%28X_0%2C%5Cdots%2C%5Chat%7BX_i%7D%2C%5Cdots%2CX_k%29%5Cright%29%5C%5C%26%2B%5Csum%5Climits_%7B0%5Cleq+i%3Cj%5Cleq+k%7D%28-1%29%5E%7Bi%2Bj%7D%5Comega%5Cleft%28%5BX_i%2CX_j%5D%2CX_0%2C%5Cdots%2C%5Chat%7BX_i%7D%2C%5Cdots%2C%5Chat%7BX_j%7D%2C%5Cdots%2CX_k%5Cright%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}d\omega(X_0,\dots,X_k)=&\sum\limits_{i=0}^k(-1)^iX_i\left(\omega(X_0,\dots,\hat{X_i},\dots,X_k)\right)\\&+\sum\limits_{0\leq i<j\leq k}(-1)^{i+j}\omega\left([X_i,X_j],X_0,\dots,\hat{X_i},\dots,\hat{X_j},\dots,X_k\right)\end{aligned}")
where a hat over a vector field means we leave it out of the list. There’s a lot going on here: first we take each vector field 
out of the list, evaluate on the remaining vector fields, and apply to the resulting function. Moving to the front entails moving it past 
other vector fields in the list, which gives us a factor of 
, so we include that before adding the results all up. Then, for each pair of vector fields and 
, we remove both from the list, take their bracket, and stick that at the head of the list before applying . This time we apply a factor of 
before adding the results all up, and add this sum to the previous sum.
Wow, that’s really sort of odd, and there’s not much reason to believe that this has anything to do with differentiation! Well, the one hint is that we’re applying to a function, which is a sort of differential operator. In fact, let’s look at what happens for a 
-form — a function 
:
That is, 
is the 
-form that takes a vector field 
and evaluates it on the function . And this is just like the differential of a multivariable function: a new function that takes a point and a vector at that point and gives a number out measuring the derivative of the function in that direction through that point.
As a more detailed example, what if is a -form?
![\displaystyle d\omega(X,Y)=X\left(\omega(Y)\right)-Y\left(\omega(X)\right)-\omega\left([X,Y]\right)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+d%5Comega%28X%2CY%29%3DX%5Cleft%28%5Comega%28Y%29%5Cright%29-Y%5Cleft%28%5Comega%28X%29%5Cright%29-%5Comega%5Cleft%28%5BX%2CY%5D%5Cright%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle d\omega(X,Y)=X\left(\omega(Y)\right)-Y\left(\omega(X)\right)-\omega\left([X,Y]\right)")
We’ve got two terms that look like we’re taking some sort of derivative, and one extra term that we can’t quite explain yet. But it will become clear how useful this is soon enough.
July 15, 2011
Posted by John Armstrong |
Differential Topology, Topology |
9 Comments
