# Talk:Volume element

WikiProject Mathematics (Rated C-class, Mid-importance)
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Field: Geometry

## ?

This subject needs a lot of work for it to be useful to lay persons. An expert needs to develop this subject.

## Incorrect Volume Element for Sphere

I tried correcting it but it was reverted to the incorrect formula. The correct formula must contain ${\displaystyle dr}$. See <url=http://mathworld.wolfram.com/SphericalCoordinates.html>Wolfram</url>, for example. — Preceding unsigned comment added by Czyx (talkcontribs) 21:34, 17 July 2013 (UTC)

You're confusing integration in spherical coordinates with surface integrals on the sphere. If you're on a sphere, then dr is zero since the radius is constant. Also, the volume element on the sphere computes the area of regions on the sphere. So dV has units of area. The two angle differentials are in natural units (so dimensionless), and ${\displaystyle r^{2}}$ has units of area. A further dr would give units of volume. Sławomir Biały (talk) 06:59, 18 July 2013 (UTC)
The formula calls for the VOLUME ELEMENT. The volume element , as per Wolfram has units of ${\displaystyle length^{3}}$. Your use of "volume element on a sphere" is, at least, confusing. The fact that it has units of area is also non-standard. This is not the standard terminology for VOLUME ELEMENT. If you want to make up your own terminology, then I would suggest that you do it elsewhere. — Preceding unsigned comment added by Tach123 (talkcontribs) 18:40, 13 August 2013 (UTC)
The volume element for an n-manifold has dimensions length^n. So the volume element for a surface has dimensions of area. While I'm flattered that you think I made this up, it is in fact very standard terminology. See, for instance, the book Einstein manifolds by Besse, referenced in the article, Helgason's standard text Differential geometry, Lie groups, and symmetric spaces, or Spivak's also standard Differential geometry: A comprehensive course. Sławomir Biały (talk) 19:46, 13 August 2013 (UTC)
But we are talking about the standard connotation of "volume element" not your volume element for a surface. It is because people like you that wiki is a cesspool, luckily, there is Wolfram where people get things correctly.The volume element in spherical coordinates is given as: <url=http://mathworld.wolfram.com/SphericalCoordinates.html>Wolfram</url> . You seem to have a degree in advanced math but your page fails elementary math. I see what you did, you wrote a paragraph entitled "Volume element of a surface", something that no one cares about but you have nothing on the entry that people do care about, the ACTUAL volume element. I corrected that, please don't delete it again. Tach123 (talk) 22:06, 13 August 2013 (UTC)
The purpose of the example is to illustrate the volume element on a surface by using a very simple example of a surface: the sphere. The general formula was given in the previous section. It's also simply wrong that no one besides me is interested in the volume element of a surface. Indeed, many physical applications require computation of integrals over a surface. (Also, more often than not, Wolfram is simply wrong. Be careful in using that as a source: Wikipedia's mathematical content is generally much more reliable in my experience.) Sławomir Biały (talk) 22:49, 13 August 2013 (UTC)
This is preposterous, you have an advanced math degree and you do not recognize the formula for the volume element. To make matters worse, you claim that Wolfram is wrong. The formula is derived in second year college, advanced math. I'll leave your sand box for yourself. Tach123 (talk) 05:20, 14 August 2013 (UTC)
You're talking about the volume element in Euclidean 3-space. This is a different thing than the volume element on the sphere. As I said, the purpose of the example was to show how to use the formula derived in the preceding section in a simple example. I don't know where you got the idea that I don't recognize the volume element in Euclidean space: indeed, I've taught this very material to thousands of students. My warning regarding Wolfram is that you seem to hold it up as a paragon of scientific accuracy (your exact words: "wiki is a cesspool, luckily, there is Wolfram where people get things correctly"). That's simply not the case in my experience. Sławomir Biały (talk) 10:59, 14 August 2013 (UTC)
The wiki entry you put together is called "Volume Element". It is not called "Volume Element on a Sphere". As such, it is missing any component to make it useful, something flagged by other users in the talk section. By contrast, the Wolfram entry on the subject is quite useful. Yours is not, it is more like an exercise in egotism. Tach123 (talk) 13:52, 14 August 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── As the references I have given show, the term "volume element" is typically used in a more general sense in mathematics than the one you are used to, one that includes area elements on surfaces. There is nothing wrong with this, and the article should include both points of view. I have made some edits to the article to include volume elements in Euclidean spaces as the first instance of a volume element (including spherical coordinates, even though in my opinion this is already redundant with information presented in the very first paragraph of the article) followed by the case of linear subspaces, and then Riemannian manifolds, surfaces, and the example of the sphere. Despite what you seem to be saying, surface integrals are in fact useful concepts: engineers and physical scientists use these on a regular basis. Volume integrals over non-Euclidean manifolds appear regularly in mechanics as well. (Also, I should add that if you wish to be taken seriously, and your comments construed as constructive, you might want to use less inflammatory rhetoric and denigrating personal remarks.) Sławomir Biały (talk) 14:10, 14 August 2013 (UTC)

Finally! Now the entry is even useful. As to the comments, I am sorry but I was frustrated with your attitude, you should look at yourself before you criticize others. Tach123 (talk) 15:20, 14 August 2013 (UTC)

## merge

This article should be merged into Volume form. Jackzhp (talk) 14:22, 28 March 2011 (UTC)

No. A manifold can have a volume element but not volume form. I have moved the example of a surface from volume form here. What it more properly describes is a volume element rather than a volume form, which requires a choice of orientation, and singling out orientation-preserving diffeomorphisms rather than all diffeomorphisms, so it requires more care in that case. Sławomir Biały (talk) 14:59, 28 March 2011 (UTC)
OK. good then,

## sphere example

we can like the result to its corresponding main article, what is needed is to show g, and determinant of g. Jackzhp (talk) 15:40, 28 March 2011 (UTC)

g is the metric tensor on the sphere. Sławomir Biały (talk) 15:41, 28 March 2011 (UTC)

## subspace example

In Euclidean space with dimension n, a subspace is spanned by matrix X with rank k, then the volume element in this subspace is ...? ${\displaystyle \varphi =X\beta ,\lambda ={\frac {\partial \varphi }{\partial \beta ^{T}}}=X,\lambda ^{T}\lambda =X^{T}X}$, In the whole space, the volume element is ${\displaystyle \Pi _{i=1}^{n}dy_{i}}$, in the subspace spanned by the matrix X, the volume element is ${\displaystyle {\sqrt {{\text{det}}\left(\lambda ^{T}\lambda \right)}}\Pi _{j=1}^{k}d\beta _{j}}$?? Jackzhp (talk) 19:47, 28 March 2011 (UTC)

If X is injective, then the volume form is ${\displaystyle {\sqrt {{\text{det}}\left(X^{T}X\right)}}\Pi _{j=1}^{k}d\beta _{j}}$ Sławomir Biały (talk) 19:59, 28 March 2011 (UTC)

## On change of coordinate system

Volume Element is needed for integration. The volume element at any specific point is needed for integration. When change the coordinate system, the volume element in the original system can be just replaced by the one in the new system, no other adjustment is needed. The volume element in the whole space is the product of that in orthogonal subspaces. Jackzhp (talk) 19:47, 28 March 2011 (UTC)

What is this supposed to mean? Could you give a reference? Sławomir Biały (talk) 20:01, 28 March 2011 (UTC)
You obviously are unfamiliar with the basics. Let me spell it for you: ${\displaystyle dV=dxdydz}$. In spherical coordinates ${\displaystyle dV=\rho ^{2}sin\phi d\theta d\phi d\rho }$. Look it up on Wolfram, you might learn something useful for a change. Tach123 (talk) 05:23, 14 August 2013 (UTC)
I was not asking you, and this does not seem to be what the original poster meant. My reading of the post was that there is no need to modify by a Jacobian change of variables—which is obviously wrong. If you think that I am unfamiliar with the basics, then you have been inattentive in our conversation above (and in your reading of the article). I have designed and taught very successful courses on the basics of this material to thousands of students at a number of universities, both at the undergraduate and graduate levels. I assure you, I am very well versed in the basics. Finally, I would like to invite you please to remain polite during our friendly chat. ! Sławomir Biały (talk) 14:44, 14 August 2013 (UTC)

## Differential volume element

Is that the same thing? ᛭ LokiClock (talk) 21:53, 21 September 2011 (UTC)