Talk:Aufgaben:Problem 9
From Ferienserie MMP2
My idea is to transform \( g^{ij} \nabla_i(\partial_j f) \) into \(\Delta f \). From there we have to show \( L_g(f) = \Delta(f) \) as in notes_new at pg 82.
The first part should be:
$$ g^{ij} \nabla_i \partial_j + \underbrace{ \nabla_i g_{ij}}_\text{= 0, from a)} = g^{ij} \nabla_i \partial_j + \underbrace{( \nabla_i g_{ij}) \partial_j}_\text{= 0} = g^{ij} \nabla_i \partial_j + \underbrace{( \nabla_i g^{ij})\partial_j}_\text{= 0} = \nabla_i (g^{ij} \partial_j) = \nabla_i \partial^i = \Delta$$
f is not compactly supported so I don't think you can use this, although maybe they just forgot to write that. It does makes sense that we should use this identity, considering exercise (a) Carl (talk) 15:57, 20 June 2015 (CEST)
