# Show that a line element transforms like a scalar

1. Nov 6, 2011

### ck99

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