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I am reading about differential geometry, and in particular the Lie derivative and its relation to (relativistic) hydrodynamics. In particular, I was wondering if, given two scalar functions $f(\mathbf{x}),~g(\mathbf{x})$, it is possible to solve the following equation:

$$\mathcal{L}_Z[f]=Z^\mu\partial_\mu f=g,$$

where $Z=Z^\mu\partial_\mu$ is an arbitrary vector field. Since I am thinking of hydrodynamics here, I am considering the manifold associated to $Z$ to be Euclidean, but I would be interested to know how it works for arbitrary manifolds.

Is their a theorem from our mathematician friends that guaranties the existence of a solution (I am okay with existence, I don't necessarily need a way to find the solution)? If there is no such proof, what are the restriction one needs to impose on $Z$? If there is a proof, is it also valid for tensor quantities, ie is there a solution to

$$\mathcal{L}_Z[f^{\mu_1...\mu_n}_{\nu_1...\nu_m}]=g^{\mu_1...\mu_n}_{\nu_1...\nu_m}.$$

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  • $\begingroup$ May more relevant and answered quicker on Math Stackexchange. $\endgroup$ – user7757 Feb 8 '14 at 10:47
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    $\begingroup$ Cross-posted to math.stackexchange.com/q/668327/11127 $\endgroup$ – Qmechanic Feb 8 '14 at 22:23
  • $\begingroup$ I'm voting to close this question as partly because it was crossposted. $\endgroup$ – Qmechanic Dec 17 '18 at 17:42