Divergence
In the context of vector fields in
with its usual meaning. In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field.
It can be generalized to a vector field
Equivalently,
And therefore, by one of the formulas for Lie derivative, exterior derivatives, bracket, interior product (the number 2)
since
(I don't know yet how to fill the gap
A function
Change of the volume form
Suppose we have in
But since
Change of variables
(Named as Jacobi's lemma in Muriel_2014)
Lemma. Suppose the coordinate change in
Suppose, also, that we have a vector field
where
Proof
Denote
Now, observe that
(we are simply computing the same thing in other coordinates).
So therefore
and according to
which is the desired formula
In particular, if
Computation from a covariant derivative
See Schuller GR-connections. Given any covariant derivative operator
But I think that this notion only coincides with the divergence above when we have a Riemannian metric inducing both the connection and the volume form.