#### Fredrik

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This is supposed to be an explanation of what the author did on the page before. He had just described how to construct a (complex) Hilbert space from a (real) smooth manifold with a smooth nowhere vanishing volume element, and then moved on to construct operators on that Hilbert space.It is well-known that the divergence of a contravariant vector field can be defined on a manifold with a volume-element [itex]\epsilon_{a_1\cdots a_n}[/itex].

...

For instance, using Lie derivatives we define "[itex]\nabla_av^a[/itex]" by:

[tex]\mathcal L_{v^m}\epsilon_{a_1\cdots a_n}=(\nabla_a v^a)\epsilon_{a_1\cdots a_n}[/tex]

(Note that, since the left side is totally skew, it must besomemultiple of [itex]\epsilon_{a_1\cdots a_n}[/itex].)

I would appreciate if someone who understands what he's talking about could explain it to me, and maybe translate it to a coordinate independent notation.We now introduce some operators. Let [itex]v^a[/itex] be any smooth (complex) contravariant vector field, and [itex]v^a[/itex] any smooth (complex) scalar field on M. Then with each smooth, complex-valued function f on M we may associate the function

[tex]Vf=v^a\nabla_af+vf[/tex]

where [itex]\nabla_a[/itex] denotes the gradient on M.