Talk:Interior product

Not the right map?

The definition given for interior product is

$i_{v}\omega (u_{1},\ldots ,u_{p-1})=\omega (v,u_{1},\ldots ,u_{p-1}).$

but the thing on the RHS is not a (p-1)-form! Shouldn't it be

$i_{v}\omega (u_{1},\ldots ,u_{p-1})=\omega (v,u_{2},\ldots ,u_{p-1}).$

? I'm going to change it... feel free to correct me if I'm wrong. Trevorgoodchild 20:10, 1 March 2007 (UTC)

Thanks for pointing out how unclear this article is: however, ω is a p-form, so that $i_{v}\omega$ is a (p-1)-form, so the formula was correct. I'll try and rephrase it to make it a bit clearer. Geometry guy 20:40, 1 March 2007 (UTC)

Ok, my mistake. I wonder, though, if there's a way to put the p-form on the LHS and the (p-1)-form on the RHS so that the definition matches the direction of the mapping. I.e., something like this but less ugly:

$i_{v}\left(\omega (u_{1},\ldots ,u_{p})\right)=(i_{v}\omega )(v,u_{2},\ldots ,u_{p})$ Trevorgoodchild 11:50, 2 March 2007 (UTC)

Well, you could write

$i_{v}\left(\omega (\cdot ,u_{2},\ldots ,u_{p})\right)=(i_{v}\omega )(u_{2},\ldots ,u_{p})$

but I'm not sure it is very helpful. Definitions don't usually match the direction of the mapping, e.g., y=f(x) defines y as a function of x, but the direction of the mapping f is from the set of xs to set of ys. Geometry guy 12:09, 2 March 2007 (UTC)

(Somewhat) fork from exterior algebra

Hey all,

I just discovered this article while working on a sandbox revision of exterior algebra (at User:Silly rabbit/Sandbox/Exterior algebra.) Would anyone object to a merge from that article to this one? I realize that some things will have to change: This article focuses on forms, while exterior algebra focuses on the general algebraic beast. All in all, it may be more trouble than it's worth. Still, I would rest easier if this article also treated the case of a general exterior algebra. In fact, it may even clarify some of the above confusion. See my axiomatic characterization of the interior product in the exterior algebra article. Silly rabbit 19:03, 5 June 2007 (UTC)

alternative notation

Couldn't we mention that some people are writing $X\llcorner \omega$ for $i_{X}\omega$? -- JanCK (talk) 18:01, 6 December 2007 (UTC)

I took no answer for a yes. I hope that is okay. Oh, by the way, is $\llcorner$ sometimes extended further than vector fields, so that one may formally write $(X\wedge Y)\llcorner \omega :=X\wedge (Y\llcorner \omega )$ or $(X\llcorner Y)\llcorner \omega :=X\llcorner (Y\llcorner \omega )$? -- JanCK (talk) 03:52, 20 December 2007 (UTC)

why \Omega ^p?

Why don't we write $i_{X}:\Lambda ^{p}V^{*}\to \Lambda ^{p-1}V^{*}$ for $X\in V$. I guess that differential forms are the most common example but one could define Interior product in correspondence with any exterior product on any vectorspace, couldn't one? -- JanCK (talk) 04:35, 20 December 2007 (UTC)

Typography

I moved this rather long remark from the article here. It doesn't seem to fit in the article. Silly rabbit (talk) 22:17, 6 March 2008 (UTC)

[Note that the use of the TeX encoding \llcorner, as in the formula just given, can be viewed as an artefact of the TeX installation and fonts used here. Unicode distinguishes the character code 2A3C for 'INTERIOR PRODUCT' from 231E 'BOTTOM LEFT CORNER', and the character illustrated in the Unicode tables for interior product is more clearly not just a corner than above. This sort of encoding problem is common for mathematical symbols, and there is often quibbling over the exact visual form used.]

That seems reasonable enough to move the comment; it isn't about the mathematical structure. But it could perhaps be made clear that the actual visual form of the symbol used in the page, and the TeX encoding employed, are not the 'standard' forms according Unicode, which is as near a standards organization for characters as there is. Unicode now seems to cover both codes and shapes too, after AFII (The Association for Font Information Interchange) gave its material to Unicode. Maybe, I just think that the fact that what's there (coded as \llcorner) isn't a matter of standard usage, though probably common enough. It is likely the result of a decision based on expediency (it looks about right and there is a known TeX control sequence to use). Perhaps there's a way of handling such matters which could be used over the math and science pages generally. There are variations of notation and conventions all over the place; from an encyclopedia people should be alerted to this.70.90.41.249 (talk) 14:02, 7 March 2008 (UTC)