Talk:Coordinate Representation of Divergence

Latest changes

Latest change involving $g$ is not an error, but just a change of notation. The determinant of metric tensor is often denoted by $g$. In my source, however, $g$ denotes the metric tensor without explicitly presenting the local basis, that is, $g = g_{\mu\nu} \d x^\mu \otimes \d x^\nu$. Then the determinant of $g$ is written as $\det g = \map \det {g_{\mu \nu}}$. The rest of proof seems fine, but one still needs to link it properly. Also, I believe that I used a different symbol for the inner product sign. I have to check that, but only after I return from holidays.--Julius (talk) 22:37, 26 December 2021 (UTC)

I would very strongly prefer that the notation remain the same as the source work presents it. If the change from $\det g$ to $g$ is indeed just a notational difference, then we need to set up an "Also presented as" section and explain the meaning and reasons for the differences.

In all cases, as you say, the underlying objects need to be carefully and fully defined, in order that, for a start, the notation $\struct {M, g}$ is completely understood. As it stands, the page Definition:Riemannian Manifold may be improved by including the notation and its meaning.

Perhaps in the proof itself we implement a "recall" clause for the $\struct {M, g}$, in order to clarify the thinking of the reader and not to demand a prior knowledge of what all this stuff means. --prime mover (talk) 23:19, 26 December 2021 (UTC)

I think we simply need to add the determinant of a metric tensor as the third option to Definition:Determinant, next to the determinant of a matrix and a linear operator. And, indeed, there is no excuse to exclude $\struct {M, g}$ from the page that defines this notion. All these standard abreviations should reside in definition pages.--Julius (talk) 18:37, 27 December 2021 (UTC)