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Exercise [14.05]
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Joined: 28 Oct 2009, 19:06
Posts: 7
Exercise [14.05]
Hi. I'm new to this forum having only found it a few days ago whilst browsing. I've been reading Road to Reality for about 18 months and have reached chapter 14. I'm really stuck on exercise 14.05.

I can sort of intuitively see the answer. If we take a vector field and the two connections to be the coordinate connection and the covariant derivative (del), then the difference represents the total change in the components of the vector field as we move along an infinitesmal segment of a curve on an n-manifold from one point P to a neighbouring point Q minus the changes in the components resulting from parallel transporting the vector at P along this segment to Q, so that we are comparing these two values at Q. For an infinitesmally small displacement along the curve the net changes in components will be linear functions of the components of the vector field at P , ie the ksi^i, and the changes in the coordinates ie the delta_x^j.

So we get (indices following ^ are superscript, those following _ are subscript)

dx^k[partial d(ksi^i) by d(x^j) - del_j(ksi^i)] is proportional to (ksi^m)*(dx^k)

where the "constant" of proportionality is the gamma^i_j,m.

Sorry about the maths text but I've not done any Latex, yet.

If anybody's done this exercise I'd welcome any comments please. My problem is that I don't see how to express the gamma^i_j,m in terms of partial derivatives of the coordinates in moving from P to Q along the curve.

I've got several other questions relating to the text in earlier chapters, is it OK to post these queries in this Exercise Discussion area?

31 Oct 2009, 19:05
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Joined: 07 Jun 2008, 08:21
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Re: Exercise [14.05]