Talk:P-vector

Bivectors
Need more work on bivectors here; if there is a lot more to write about on p-vectors, then maybe the bivector stuff should be shifted to a new article.---Mpatel 08:50, 27 July 2005 (UTC)

¨Mistake
Yep,

There must be a mistake here :

$$F \in \Lambda ^2 V$$ in terms of a basis $$e_a \wedge e_b$$ of $$\Lambda ^2 V$$

$$e_a \wedge e_b$$ is not a basis of $$\Lambda ^2 V$$ because these 2-vectors are not linearly independant. If we take $$e_1 \wedge e_2$$ and $$e_2 \wedge e_1$$ these 2 2-vectors are linked by the relation $$e_1 \wedge e_2$$ = - $$e_2 \wedge e_1$$.

A good basis for $$\Lambda ^2 V$$ should be $$e_a \wedge e_b$$ with $$1 \leq a < b \leq n$$

Lemma
Shouldn't the name of the lemma be "multivecotor" instead of "p-vector"? --Trigamma 15:23, 4 November 2007 (UTC)
 * Having a closer look at the article, it should rather be "bivecors", for the article doesn't say anything non-trivial about p-vecors with p \ne 2. —Preceding unsigned comment added by Trigamma (talk • contribs) 15:32, 4 November 2007 (UTC)

Page Content Too Obtuse
Honestly, the entire entry for P-Vector makes no sense to anyone that is not a specialist in tensorial math. And if someone is a specialist in tensor math, why would they be reading this section. This section needs a rewrite into a form that actually conveys a description without a complete reliance on other abstract concepts. —Preceding unsigned comment added by 74.249.92.155 (talk) 16:37, 24 January 2009 (UTC)

Seems that way to me too; also no in-line references. Brews ohare (talk) 19:23, 21 December 2009 (UTC)

Bivector
Here is a proposal for a new section:


 * Bivector in three-dimensional Euclidean space

If a and b are any vectors in a real three-dimensional Euclidean vector space, then the complex vector A given by:


 * $$\boldsymbol A = \boldsymbol a + i \boldsymbol b \, $$

is a bivector, where i = √−1. Two bivectors A = a + i b and A′ = a′ + i b′ are equal if and only if a = a′ and b = b′.

Given an orthonormal triad of real unit vectors i, j, k,


 * $$\boldsymbol A = A_1 \ \boldsymbol i + A_2\ \boldsymbol j + A_3\  \boldsymbol k \  $$
 * $$ = (a_1 + i\ b_1)\ \boldsymbol i + (a_2 + i\ b_2)\ \boldsymbol j + (a_3 + i\ b_3)\  \boldsymbol k \ . $$

Introducing another bivector B, dot and cross products are:


 * $$\boldsymbol {A \cdot B} = A_1B_1 + A_2B_2 +A_3B_3 \, $$


 * $$\boldsymbol {A \times B} = (A_2B_3 - A_3B_2)\ \boldsymbol i +(A_3B_1 - A_1B_3)\ \boldsymbol j +(A_1B_2 - A_2B_1) \ \boldsymbol k \ . $$

Unit bivectors { eℓ } are related to the imaginary i=√-1 using the connection (for example):


 * $$ (\boldsymbol {e_1} \wedge \boldsymbol {e_2})^2 = -1 \, $$

connecting this discussion to that above, which does not use the imaginary i. See also these historical remarks. Brews ohare (talk) 19:04, 21 December 2009 (UTC)


 * Sorry, I didn't notice this on the talk page before removing this section or I would have come here first. But those are not bivectors. I had a look at the Boulange+ Hayes source before reverting and what they seem to be describing is something completely different: a way of packaging two vectors as a single object, which you can do stuff with but it's not a bivector, it just happens to have the same name. Perhaps this is clearest as their A+iB objects are six dimensional, bivectors in three dimensions are only three dimensional - they correspond to pseudovectors in 3D. P85 of the second ref above is talking about bivectors in ℝ2, It's later (and not in the online preview) they talk about bivectors in 3D.
 * I've got some thoughts on how to expand it in line with my own understanding, using refs I have here. The bivector stuff should be easy to fix with good examples in 2, 3 and 4 dimensions, and there's probably something interesting to add about trivectors too. --JohnBlackburne (talk) 19:29, 21 December 2009 (UTC)

I hope no too tendentiously, I changed the redirection from bivector to p-vector and wrote a starting article for bivector at Bivector in accordance with some objections here that the general case was too abstruse for the uninitiated. Perhaps you could take a look at that and make any necessary changes. Brews ohare (talk) 21:09, 21 December 2009 (UTC)


 * I've added some stuff and clarified some other stuff there. Still a lot more that could be added here and there - if we've two articles we should try and keep duplication to a minimum or one or the other will be nominated for removal/merge. I like that you've used David Hestenes's book as reference, it's one of the two books I have on GA and a remarkable work. --JohnBlackburne (talk) 21:54, 21 December 2009 (UTC)

Intro too narrow; maybe a name change is appropriate?
The restriction of the term p-vector to refer only to differential geometry seems too narrow. It also has meaning in linear algebra. For example, a bivector also is called sometimes a 2-vector, and is not restricted to differential geometry. Perhaps the use of p-vector as an adjective to describe a field, as in bivector field, is more appropriately restricted to differential geometry? Should the name of this article be changed to p-vector field? Brews ohare (talk) 16:55, 27 December 2009 (UTC)