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)[reply]

¨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)[reply]

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)[reply]

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)[reply]

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

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′.^[1]

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:^[1]

{\boldsymbol {A\cdot B}}=A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}\ ,

{\boldsymbol {A\times B}}=(A_{2}B_{3}-A_{3}B_{2})\ {\boldsymbol {i}}+(A_{3}B_{1}-A_{1}B_{3})\ {\boldsymbol {j}}+(A_{1}B_{2}-A_{2}B_{1})\ {\boldsymbol {k}}\ .

Unit bivectors { e_ℓ } are related to the imaginary i=√-1 using the connection (for example):^[2]

({\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.^[3]

^ ^a ^b Philippe Boulanger, Michael A. Hayes (1993). "Chapter 2: Bivectors". Bivectors and waves in mechanics and optics. CRC Press. ISBN 0412464608.
^ John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 1846289963.
^ Hermann Grassmann, Gert Schubring, ed. (1996). Hermann Günther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar. Springer. p. 246 ff. ISBN 0792342615.

Brews ohare (talk) 19:04, 21 December 2009 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

[Hayes-1] Philippe Boulanger, Michael A. Hayes (1993). "Chapter 2: Bivectors". Bivectors and waves in mechanics and optics. CRC Press. ISBN 0412464608.

[Vince-2] John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 1846289963.

[Grassman-3] Hermann Grassmann, Gert Schubring, ed. (1996). Hermann Günther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar. Springer. p. 246 ff. ISBN 0792342615.

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