Hello. I have a skewed elliptical coordinate system (I use alternative elliptical coordinates). And I need to determine connections between coordinates in order to fetch a covariance matrix of those coordinates. How can I do it?

And one more question: if I will do QR- decomposition of the metric tensor matrix it will produce an orthogonal system of bases vectors. But what does it mean? And what the matrix R mean if Q provides a new orthogonal basis? I confused a little.