'The Road to Reality' Internet Forum - View topic - Exercise [11.14]

FAQ • Search • Login

It is currently 04 Sep 2010, 12:35

View unanswered posts | View active topics

Exercise [11.14]

Page 1 of 1

[ 3 posts ]

Print view

Previous topic | Next topic

Exercise [11.14]
Author	Message
Sameed Zahoor Joined: 12 Mar 2008, 10:57 Posts: 69 Location: India	Exercise [11.14] If p is odd, $P\wedge{P}=-P\wedge{P}$ (from Exercise [11.13])or $2(P\wedge{P})=0$ or $P\wedge{P}=0$
12 Jul 2008, 09:52

DimBulb Joined: 07 May 2009, 16:45 Posts: 54	Re: Exercise [11.14] Doesn't $P \wedge P = 0$ no matter what? An example where P is of even grade 2: $P = p(\boldsymbol\eta_1 \wedge \boldsymbol\eta_2)$ $P \wedge P = p(\boldsymbol\eta_1 \wedge \boldsymbol\eta_2) \wedge p(\boldsymbol\eta_1 \wedge \boldsymbol\eta_2)$ $= p^2 (\boldsymbol\eta_1\wedge \boldsymbol\eta_2 \wedge \boldsymbol\eta_1 \wedge \boldsymbol\eta_2)$ $= -p^2 (\boldsymbol\eta_1\wedge \boldsymbol\eta_1 \wedge \boldsymbol\eta_2 \wedge \boldsymbol\eta_2) = -p^2( 0 \wedge 0) = 0$ This example could be expanded to cover an element of any grade. Furthermore, for any and made up of wedge products of $\boldsymbol\eta$ s, all the $\boldsymbol\eta$ s would have to be different or the wedge product will be zero. That is, if they share a common $\boldsymbol\eta_i$ , you could rearrange the wedge product to get $\boldsymbol\eta_i \wedge \boldsymbol\eta_i$ in the chain which equals , and $0 \wedge anything = 0$ , doesn't it? Or does it?
19 Jul 2009, 23:31

DimBulb Joined: 07 May 2009, 16:45 Posts: 54	Re: Exercise [11.14] Er, no wait. " . . .the general element of the algebra of grade r need not be a simple wedge product . . ., but can be a sum of such expressions." So, if $P = p_1(\boldsymbol\eta_1 \wedge \boldsymbol\eta_2) + p_2(\boldsymbol\eta_3 \wedge \boldsymbol\eta_4)$ then $P \wedge P = -p_1^2( 0 \wedge 0) + p_1p_2(\boldsymbol\eta_1 \wedge \boldsymbol\eta_2 \wedge \boldsymbol\eta_3 \wedge \boldsymbol\eta_4) + p_2p_1(\boldsymbol\eta_3 \wedge \boldsymbol\eta_4 \wedge \boldsymbol\eta_1 \wedge \boldsymbol\eta_2) - p_2^2( 0 \wedge 0)$ $= 2p_1p_2(\boldsymbol\eta_1 \wedge \boldsymbol\eta_2 \wedge \boldsymbol\eta_3 \wedge \boldsymbol\eta_4)$
20 Jul 2009, 00:03

Display posts from previous: Sort by

Page 1 of 1

[ 3 posts ]

Who is online

Users browsing this forum: No registered users and 1 guest

You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Powered by phpBB © phpBB Group.
Designed by Vjacheslav Trushkin for Free Forums/DivisionCore.