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Show that a line element transforms like a scalar

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that a line element of form ds2 = gabdXadXb 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/∂Xb)dXb

    Which comes from my textbook definition of how a vector transforms

    V'a = (∂X'a/∂Xb)Vb

    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) = AT(du', dv'). Why have they just thrown that transpose in there? My textbook follows a similar argument, and I don't understand that either :(
     
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
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