So far for a smooth function , gradient of
is a vector field defined as follows
.
Since the gradient of is nothing but a vector field then it is reasonable to talk about the divergence of a vector field
. To be exact, we define
where is a vector field. All above was discussed in this topic.
In order to go further, I need to spend some time talking about . Precisely, if
is a smooth function, then can we write down explicitly
in local coordinates?
Having the definition of divergence operator yields
where presents a pairing between
and its dual space
. If, in local coordinates,
, we then have
.
Thus,
.
As a consequence,
.
We now look at the Christoffel symbols. Clearly, by definition
where the main theorem posted here has been used. Thus,
As a consequence,
.
The above acting on a smooth function
is called the Laplace-Beltrami operator.