- #1
jdstokes
- 523
- 1
Suppose we define the Lie derivative on a tensor [itex]T[/itex] at a point p in a manifold by
[itex]\mathcal{L}_V (T) = \lim_{\epsilon \to 0}\frac{\varphi_{-\epsilon \ast}T(\varphi_\epsilon(p))- T(p)}{\epsilon}[/itex]
where V is the vector field which generates the family of diffeomorphisms [itex]\varphi_t[/itex].
If T is just an ordinary function [itex]f:M \to \mathbb{R}[/itex] then it seems like the numerator of the above expression is [itex]f(p) - f(p) = 0[/itex] which is unusual since I thought the lie derivative of a function was the ordinary derivative [itex]\mathcal{L}_V f = V^\mu \partial_\mu f[/itex]. Can anyone reconcile this?
Thanks
[itex]\mathcal{L}_V (T) = \lim_{\epsilon \to 0}\frac{\varphi_{-\epsilon \ast}T(\varphi_\epsilon(p))- T(p)}{\epsilon}[/itex]
where V is the vector field which generates the family of diffeomorphisms [itex]\varphi_t[/itex].
If T is just an ordinary function [itex]f:M \to \mathbb{R}[/itex] then it seems like the numerator of the above expression is [itex]f(p) - f(p) = 0[/itex] which is unusual since I thought the lie derivative of a function was the ordinary derivative [itex]\mathcal{L}_V f = V^\mu \partial_\mu f[/itex]. Can anyone reconcile this?
Thanks