Cartan’s Formula
It turns out that there is a fantastic relationship between the interior product, the exterior derivative, and the Lie derivative.
It starts with the observation that for a function 
Next we consider the differential 
&=\lim\limits_{t\to0}\frac{1}{t}\left(\left[(\Phi_t)^*(df)\right](Y)-df(Y)\right)\\&=\lim\limits_{t\to0}\frac{1}{t}\left(df((\Phi_t)_*Y)-df(Y)\right)\\&=\frac{\partial}{\partial t}\left[(\Phi_t)_*Y\right](f)\bigg\vert_{t=0}\\&=\frac{\partial}{\partial t}Y(f\circ\Phi_t)\bigg\vert_{t=0}\\&=YXf\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5BL_X%28df%29%5Cright%5D%28Y%29%26%3D%5Clim%5Climits_%7Bt%5Cto0%7D%5Cfrac%7B1%7D%7Bt%7D%5Cleft%28%5Cleft%5B%28%5CPhi_t%29%5E%2A%28df%29%5Cright%5D%28Y%29-df%28Y%29%5Cright%29%5C%5C%26%3D%5Clim%5Climits_%7Bt%5Cto0%7D%5Cfrac%7B1%7D%7Bt%7D%5Cleft%28df%28%28%5CPhi_t%29_%2AY%29-df%28Y%29%5Cright%29%5C%5C%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7D%5Cleft%5B%28%5CPhi_t%29_%2AY%5Cright%5D%28f%29%5Cbigg%5Cvert_%7Bt%3D0%7D%5C%5C%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7DY%28f%5Ccirc%5CPhi_t%29%5Cbigg%5Cvert_%7Bt%3D0%7D%5C%5C%26%3DYXf%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[L_X(df)\right](Y)&=\lim\limits_{t\to0}\frac{1}{t}\left(\left[(\Phi_t)^*(df)\right](Y)-df(Y)\right)\\&=\lim\limits_{t\to0}\frac{1}{t}\left(df((\Phi_t)_*Y)-df(Y)\right)\\&=\frac{\partial}{\partial t}\left[(\Phi_t)_*Y\right](f)\bigg\vert_{t=0}\\&=\frac{\partial}{\partial t}Y(f\circ\Phi_t)\bigg\vert_{t=0}\\&=YXf\end{aligned}")
just as if we took 
So on exact 
and in fact this holds for all differential forms, which follows from these two base cases by a straightforward induction. This is Cartan’s formula, and it’s the natural extension to all differential forms of the basic identity 
1 Comment »
Leave a Reply
RSS Feeds
[…] Cartan’s formula in hand we can show that the Lie derivative is a chain map . That is, it commutes with the exterior […]
Pingback by The Lie Derivative on Cohomology « The Unapologetic Mathematician | July 28, 2011 |