Talk:Isotropic quadratic form

Simplicity
I'm a dilettante at math. As I understand it, a quadratic form is a polynomial. Can somebody contribute a plainly-worded explanation of "isotropic" in "isotropic quadratic form"? Also, can somebody explain how this is connected to analytic geometry, if at all, in two or three dimensions? Unfree (talk) 19:36, 31 July 2009 (UTC)

Even Characteristic?
Typically in the literature I've seen isotropic refers to the properties of a bilinear form.


 * a vector w is isotropic if b(w,w)=0.
 * a subspace W is totally isotropic if b(u,w)=0 for all u,w in W

I've seen


 * a vector w is singular if q(w)=0
 * a subspace W is totally singualar if q(w)=0 for all w in W

In odd characteristic singular=isotropic and totally singular = totally isotropic. However, in even-characteristic they are distinct concepts. Unfortunately, there are many books and articles that are a little careless with these definitions because they are only concerned with the odd-characteristic case. —Preceding unsigned comment added by Somethingcompletelydifferent (talk • contribs) 15:11, 11 April 2010 (UTC)

Field theory
The examples 3, 4, and 5 are in fact results in comparative field theory. In the case of the real number field, example 1 is significant as pseudo-Euclidean space used to discuss special relativity. An expansion of example 1 would provide an existence proof for isotropic quadratic forms.Rgdboer (talk) 00:45, 30 November 2012 (UTC)

After 6 months the "examples 3, 4, 5" have been moved to a section "Field theory". No references are given, perhaps the demonstrations are not hard. For now, a new section, Hyperbolic plane, has been introduced. Anisotropic details have been moved to lede, out of "examples".Rgdboer (talk) 21:47, 2 June 2013 (UTC)

Pseudo-Euclidean space
Possibly, it is excessively detailed for “pseudo-Euclidean space”, but for this article the subsection is rather topical. Can it be integrated here? Incnis Mrsi (talk) 14:32, 6 June 2013 (UTC)