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Is the comment at the end of the first section accurate? "Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three dimensional space." If an n-dimensional surface is embedded in (n+1)-dimensional space then the partial derivatives will be vectors in (n+1)-space and there will be n of them. The cross product of a set of n vectors in (n+1)-space is well defined so the formula could still be grammatically sensible. It seems likely that it would still be correct as well. Is there a more general theorem we could reference? —Preceding unsigned comment added by 24.4.97.134 (talk) 07:28, 16 June 2009 (UTC)
I'm not pleased with the section on surface integrals of vector fields. It talks about these things before giving the definition. This is thoroughly confusing to people who have just read the section on surface integrals of scalar functions. They will have no idea what the "surface integral of a vector field" means, and then they will have a definition "derived" for them based on facts derived from a non-existent definition. Revolver 18:31, 17 January 2006 (UTC)
Am I wrong, or is the notation used for surface integrals? My understanding of path and surface integrals, so I'm not sure exactly how this notation is used, but I had thought that it might even be used for both surface and path integrals. If so, it should be noted what say "" means. Fresheneesz 07:13, 9 February 2006 (UTC)
If we think of S as made of some material, and for each x in S the number f(x) is the density of material at x, then the surface integral of f over S is the mass per unit thickness of S. May S be thought of as a hollow object, such as a bubble or a vase? --Abdull 16:45, 30 May 2006 (UTC)
to prove integral (A.dS) = integral (del.A dV) Let A (Ax,Ay,Az)
RHS =integral (pd/dx . i + pd/dy .j + pd/dz .k)(Ax i + Ay j + Az k)dV
= int(int(int((pd Ax/dx + pd Ay/dy + pd Az/dz).dx.dy.dz))
= int(int(int (pd Ax. dy.dz + pd Ay.dx.dz + pd Az.dx.dy)))
= int(int (Ax.dy.dz + Ay.dx.dz + Az.dx.dy)
= int (A.dS)
Hi, This article gives definitions in terms of cross products, which means that they are only applicable in 3 dimensional space. The article does mention this restriction, but it would be good to also have a section that gives the/a generalisation to n-dimensional spaces. Cheers, Nathanielvirgo (talk) 15:58, 29 July 2009 (UTC)
How come that this article does not link to the area element??? And I hope someone can write something to relate them. Jackzhp (talk) 02:38, 19 March 2011 (UTC)
For someone who doesn't already know what is going on, the notation is very very confusing. Some letters are bold, some are not (this is probably vector x scalar quantity difference), but what are the dimensions of domains/codomains? It can be guessed and then verified to "sort of make sense" if one has some intuition, but in general, I find that the weakest point of the article.
It would be beneficial if someone who understands the topic added the domains/codomains or explanations of what are f,x,v,n,u,v,r (which seems to be x from the example just before!) and other symbols used in the first two sections. Thanks 94.112.136.34 (talk) 14:08, 24 June 2014 (UTC)
The whole area of advanced issues reads like original research and is completely unreferenced.I think we need to find a source for it . — Preceding unsigned comment added by Loneather (talk • contribs) 18:18, 16 December 2017 (UTC)
There is no definition of what f refers to when subscripted by either x or y. Please define. Jim Bowery (talk) 00:07, 6 April 2022 (UTC)