Definition talk:Divergence Operator

Is there a way of salvaging this page by putting suitable constraints on the conditions, or would it in fact be best to scrap it completely and leave just the Real Cartesian Space version up?

This is what happens when one is mostly self-taught. --prime mover (talk) 18:01, 15 April 2020 (EDT)

$\ds \operatorname {div} \mathbf f = \sum_{k \mathop = 1}^n \frac {\partial f_k} {\partial x_k}$ holds only for flat spaces, so it should not be present in the general definition. The problem comes from the definition of standard ordered basis, because this is Cartesian basis in disguise. Relaxing this condition would help the general case. To deal with this properly we need differential geometry and manifold analysis. Or we can hide the details behind a red link. --Julius (talk) 19:08, 15 April 2020 (EDT)

Hide whatever details behind a redlink -- because this is all stuff which we will need to cover in due course. Are you able to put this in some sort of order? It would be appreciated. Does a similar consideration apply to gradient and curl? --prime mover (talk) 01:04, 16 April 2020 (EDT)

I will do it. As for gradient and curl, they also suffer from the same problems. Check https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates for comparison. --Julius (talk) 06:27, 16 April 2020 (EDT)

Fixed general definitions for divergence and gradient. Terminology is a bit advanced but general enough to include advanced topics while still reducible to classical results. I did not edit curl, because: 1) it contains only Euclidean case in a correct way, 2) due to its properties, definition through antisymmetric products only works in 3d, and more abstract notions still have this constraint, so not worth trying. Sometimes it is "generalized" by defining it as a commutator, but that is something else. --Julius (talk) 18:04, 16 April 2020 (EDT)

Yes, that does the job. We can develop the terminology in due course. As long as we have the basic Cartesian version that undergraduates encounter in their introductory electromagnetism classes (which is the usual applied-mathematics motivation to get introduced to these things), then we're okay. --prime mover (talk) 18:17, 16 April 2020 (EDT)

As for curl, as long as there's nothing broken, we can leave it as is for the moment, till we actually get round to working on that area of mathematics. The object of this immediate exercise was to correct my mistakes. --prime mover (talk) 18:19, 16 April 2020 (EDT)

The Hodge star operator relies on the Riemannian metric and also on the orientation of an orthonormal basis. There's a co-ordinate-free definition in https://en.wikipedia.org/wiki/Divergence#Definition, i.e. The divergence of a vector field $F(x)$ at a point $x_0$ is defined as the limit of the ratio of the surface integral of $F$ out of the surface of a closed volume $V$ enclosing $x_0$ to the volume of $V$, as $V$ shrinks to zero. It also has an advantage that it gives a more intuitive feel for what it describes, e.g. the extent to which the vector field flux behaves like a source (positive) or a sink (negative) at a given point.-----John Coupe (talk) 19:21, 3 May 2020 (EDT)

So what needs to be done to this page to make it rigorous? --prime mover (talk) 07:16, 4 May 2020 (EDT)

Well, to calculate the surface integral of $F$ out of the surface, you still need induced metric on that surface plus the orientation of the surface normal. But I digress.

As far as I am concerned, there are no serious problems with the definition. I took this from https://ncatlab.org/nlab/show/divergence, which is used by serious folks, so I trust it. It is general enough to include most cases covered in graduate level maths.

I understand that this is an appeal to having more elegance. However, on this site we strive to have all definitions, not the best ones. If anyone has time and knowledge to tie in other definitions (and show their equivalence), he or she is welcome to do it.--Julius (talk) 18:14, 4 May 2020 (EDT)