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Chapter 12 (and 10, I guess)  vectors, covectors, duals http://www.roadtoreality.info/viewtopic.php?f=22&t=1421 
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Author:  flashbang [ 20 Aug 2009, 07:42 ] 
Post subject:  Chapter 12 (and 10, I guess)  vectors, covectors, duals 
I have already posted in DimBulb's Topic "Chap 10,12. There are vectors. Then there are vectors." about my struggle with covectors, and I think I am just about understanding it (although I still cannot see how to do exercise [10.9] using chain rules as Penrose suggests). Anyway, in section 12.3 Penrose talks about scalars, vectors, and covectors (ok), says how covectors are the duals of vectors (hmm, ok), then at the bottom of page 223 (in my edition), he says "These relations define covectors as dual objects to vectors." I can't quite see what he means when he says "defines"  is he saying that one can use these relations to convert vectors into covectors? He certainly seems to be saying that, since at the top of the next page he says "When we take the dual of the space of covectors..." as if there is a welldefined procedure for the process of "taking the dual of a space of vectors". In other words, he seems to be saying that if I have a set of vectors then I can "automatically" for the space of covectors , and yet this does not seem possible to me. So what is this procedure? Have I missed something? Perhaps I am simply not thinking about it in the right way. Just flicking through chapter 12 before I started to read it I knew it was going to be tricky, and I have not been able to get past page 225. 
Author:  flashbang [ 20 Aug 2009, 07:45 ] 
Post subject:  Re: Chapter 12 (and 10, I guess)  vectors, covectors, duals 
Also, Penrose refers to as a vector field, and as a differential operator. I understand the second name, and I can sort of see that it is a vector (with components a, b or etc), but in what way is this single entity a vector field? Are we to regard it as a space of vectors as we vary the components a, b? I am sure I have missed something fundamental here, so I apologise for being thick! 
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