Suppose we define the Lie derivative on a tensor [itex]T[/itex] at a point p in a manifold by(adsbygoogle = window.adsbygoogle || []).push({});

[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

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# Lie derivative of a function

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