Hi, everyone I was playing with the coordinate transformations and metric tensors to get a feeling of how it all behaves, and got stuck with some basic problem I am hoping you can help me with. So, I have defined a coordinate system (s,t), with the s axis going along the x axis in the cartesian coordinates, and t axis going along the y=x line: s = x-y t = y*sqrt(2) with inverse transformation: x = s + t/sqrt(2) y = t/sqrt(2) If I am differentiating correctly, the metric tensor in these coordinates looks like: 1 (2+sqrt(2))/2 (2+sqrt(2))/2 1 g11 = g22 = 1, g21=g12 = (2+sqrt(2))/2 Now, I pick a point (3,1) in cartesian coordinates, and transform it to my new frame, and get the contravariant coordinates as (2, sqrt(2)). So far so good. What I am trying to do is find out what its covariant coordinates are going to be. I think, that covariant coordinates are supposed to be the lengths of orthogonal projections of the vector on the respective axes. From basic geometry, I get (3, 2*sqrt(2)). The problem is that when I try to multiply my metric tensor by the contravariant vector, I get a different answer - (3+sqrt(2), 2+2*sqrt(2)) Clearly, there is something I am doing wrong here, but I can't figure out what it is :( Can somebody please help me spot the problem? Thanks a lot for your help!