(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that a line element of form ds^{2}= g_{ab}dX^{a}dX^{b}transforms like a scalar under any general coordinate transform

2. Relevant equations

3. The attempt at a solution

I think I've actually found the solution here, but I can't make sense of it!

http://en.wikipedia.org/wiki/Metric_tensor#Invariance_of_arclength_under_coordinate_transformations"

I am not very confident with matrix notation but I think the first line of this solution is saying

dX'^{a}= (∂X'^{a}/∂X_{b})dX^{b}

Which comes from my textbook definition of how a vector transforms

V'^{a}= (∂X'^{a}/∂X_{b})V^{b}

I don't understand the second line though. Why is there a transposed transformation matrix in there? We have just said (du,dv) = A(du', dv'), and now we seem to be saying (du,dv) = A^{T}(du', dv'). Why have they just thrown that transpose in there? My textbook follows a similar argument, and I don't understand that either :(

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# Show that a line element transforms like a scalar

Can you offer guidance or do you also need help?

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