Hi, in the attachment, in the calculation at teh top of page 27,
(i) how do we go from the 2nd to 3rd line? surely this assumes and commute?
(ii)how do we go from the 4th to the 5th line?
Thanks you.

latentcorpse
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 19122010 22:32

boromir9111
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 19122010 22:34
A bit off topic but where is part 1 to this if you don't mind me asking?

latentcorpse
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 19122010 22:36
(Original post by boromir9111)
A bit off topic but where is part 1 to this if you don't mind me asking? 
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 19122010 22:38
(Original post by latentcorpse)
part i of what sorry? 
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 19122010 22:40
oh i don't know. presumably they're available online. the part 3 bit is like a masters course which i am doing so i did my undergrad elsewhere.

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 19122010 22:42
Oh, ok.....thanks for that and sorry for not being of no help at all!

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 20122010 00:45
i) They're just functions. Why shouldn't they commute?
ii) Switch the indices summed over in the first. They're just dummy variables after all. 
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 20122010 01:41
(Original post by SimonM)
i) They're just functions. Why shouldn't they commute?
In a lesscoordinateful way: can be understood as the components of the tensor product . The ordering of the tensor product is fixed by the indices, not the order of the factors  in the sense that . (And yes, in a very real way . In abstract index notation this is just saying that I can twist the indices of a tensor around and it's the same as reversing the order of multiplication to begin with.) 
latentcorpse
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 20122010 01:41
(Original post by SimonM)
i) They're just functions. Why shouldn't they commute?
ii) Switch the indices summed over in the first. They're just dummy variables after all.
from eqn 27?
Also, could you perhaps explain to me where eqn 27 comes from? It gets used loads in these notes and it's bugging me that I don't seem to understand it at all! 
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 20122010 01:52
(Original post by latentcorpse)
Also, could you perhaps explain to me where eqn 27 comes from? It gets used loads in these notes and it's bugging me that I don't seem to understand it at all! 
latentcorpse
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 20122010 12:24
(Original post by Zhen Lin)
That's practically a definition of what it means to partialdifferentiate a function on a manifold. (f is defined on a manifold, F is defined on .) To obtain the derivation, note that you can obtain as a tangent vector by considering the curve which is constant in all coordinates except one and constant velocity in .
So this curve you are talking about is given by eqn 26 but then he just says "the tangent to this curve is ... 
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 20122010 13:41
(Original post by latentcorpse)
I don't understand. Sorry!
So this curve you are talking about is given by eqn 26 but then he just says "the tangent to this curve is ... 
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 20122010 15:05
(Original post by Zhen Lin)
I don't know if I'm misremembering, but I thought you did differential geometry before? Anyway. It's hard to talk about this formally because you'll run into notational difficulty quite fast. But it basically follows by definition  for example, the tangent to the curve x = 0 in Euclidean space, parametrised appropriately, is the standard unit vector .
Anyway, I would quite like to try and get my head around this. So, looking at the notes, it tells us (at the bottom of p15) that we can derive eqn27 using eqn 24 and additionally that we can denote the tangent vector by .
This means that we can look at eqn 24 and replace by .
Now we just need to sort out the RHS.
We have
So we need the second term on the RHS to equal 1 and we'll be done.
I'm not 100% sure why this is  is it because from the defn of at the bottom of p15, we see that the derivative of the th term wrt t is just 1? I think I may be getting confused here....
Thanks again for your help!Last edited by latentcorpse; 20122010 at 15:06. 
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 20122010 15:56
Actually I just realised I wrote down a contradiction. Rather, it's the curve that has tangent vector . But anyway, let me try to give a full derivation. (No pun, haha.)
Firstly, we need to define what we mean by . It's the tangent vector to the curve such that under the chart , is constant in all components except the th component, where it has unit velocity, i.e. the th component of is 1. Now, let be a point on the manifold. Then, by definition, (pardon the difference in notation) is the differential operator such that .
We can further expand the RHS using the charts and the chain rule: write , and then . So, writing for the th component of , we have, by the chain rule, (summation convention; is the th component of ). But , so . So .
Of course, there are certain details being swept under the rug here. For instance, there are many curves where the tangent vector at 0 acts on functions exactly the same way as . When you do the construction of the tangent space, you quotient out all these equivalent operators so that the remaining tangent vectors act on functions in a unique way. And then this only defines the tangent space (and its action on functions) at one point  so you have to repeat this construction everywhere on the manifold, and then define a vector field in order to get . And even then, in the general case this vector field is only defined on one patch on the manifold, because in general a chart is only defined on one patch on the manifold.Last edited by Zhen Lin; 20122010 at 16:00. 
latentcorpse
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 20122010 21:08
(Original post by Zhen Lin)
Actually I just realised I wrote down a contradiction. Rather, it's the curve that has tangent vector . But anyway, let me try to give a full derivation. (No pun, haha.)
Firstly, we need to define what we mean by . It's the tangent vector to the curve such that under the chart , is constant in all components except the th component, where it has unit velocity, i.e. the th component of is 1. Now, let be a point on the manifold. Then, by definition, (pardon the difference in notation) is the differential operator such that .
We can further expand the RHS using the charts and the chain rule: write , and then . So, writing for the th component of , we have, by the chain rule, (summation convention; is the th component of ). But , so . So .
Of course, there are certain details being swept under the rug here. For instance, there are many curves where the tangent vector at 0 acts on functions exactly the same way as . When you do the construction of the tangent space, you quotient out all these equivalent operators so that the remaining tangent vectors act on functions in a unique way. And then this only defines the tangent space (and its action on functions) at one point  so you have to repeat this construction everywhere on the manifold, and then define a vector field in order to get . And even then, in the general case this vector field is only defined on one patch on the manifold, because in general a chart is only defined on one patch on the manifold.
A couple of things:
(i) why do we evalutate at ?
(ii) Just trying to get this chain rule thing to work out and having some trouble:
Now by taking the th component of , we can get the that we want but I don't see how we get the in the first term? 
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 21122010 04:03
Moreover, you can't extract components like that, since the multivariate part of the expression is embedded inside. The key point is that is just the vector with 1 in the th component and 0 everywhere else. This is by construction of the curve.
I'm afraid this is very difficult to explain clearly. Either you get bogged down in notation and lose sight of the essential intuition, or you focus on the intuitive ideas and don't get a good idea of what is formally happening...Last edited by Zhen Lin; 21122010 at 04:04. 
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 21122010 13:01
(Original post by Zhen Lin)
As opposed to...? , so it's not in the domain of . On the other hand .
is fixed, so you can't use it as a summation index. Recall the chain rule from multivariable calculus: .
Moreover, you can't extract components like that, since the multivariate part of the expression is embedded inside. The key point is that is just the vector with 1 in the th component and 0 everywhere else. This is by construction of the curve.
I'm afraid this is very difficult to explain clearly. Either you get bogged down in notation and lose sight of the essential intuition, or you focus on the intuitive ideas and don't get a good idea of what is formally happening...
we have
and then by the chain rule we can write
now i realise that is just the th component of but to me it appears that there should be an term, no?
how do we combine the and the to give just a single ? 
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 21122010 14:11
(Original post by latentcorpse)
ok. i get the 1st point now but as for the 2nd perhaps this will make my problem a bit clearer:
we have
and then by the chain rule we can write
now i realise that is just the th component of but to me it appears that there should be an term, no?
how do we combine the and the to give just a single ?
(Also, and mean different things. Please distinguish between them.) 
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 21122010 18:40
(Original post by Zhen Lin)
Your expression is incorrect in two ways: firstly, is fixed and cannot appear as a summation index; secondly, it's just , not . The latter is an invalid expression  and are both functions of the type so cannot be composed. (I realise my definition of conflicts with the way it's used in your lecture notes. In the lecture notes, is the th component of .)
(Also, and mean different things. Please distinguish between them.)
also, surely is the th component of and not the th component of since 2 lines before eqn 24, it says ?
And finally, what do you take to mean if is composition? Do you use it just for multiplication?
Thanks and sorry for draggin this out! 
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 21122010 19:30
(Original post by latentcorpse)
ok. so i don't understand why can't appear as a summation index  surely it's summed over in eqn 24?
I have never, ever seen it used for function composition.
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