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

Show that we can always write this line-element

ds^2=Adx^2+Bdy^2+Cdxdy ; which A B and C are any real function of x and y

to this form ds^2 = du^2 + dv^2

2. Relevant equations

3. The attempt at a solution

I try to solve it in two way

First , I write total derivative of u as du = (du/dx)dx + (du/dy)dy and then dv = (dv/dx)dx + (dv/dy)dy

and I square both of du and dv then I can write du^2 + dv^2 = ()dx^2 + ()dy^2 + ()dxdy

but I'm not sure this is correct.

Another way to do,I have to find a matrix transform that can diagonalize my metric tensor from the first line-element. I found that, I have to find eigenvalue of these metric and then eigenfunction after that ,I compose its eigenfunction into a new metric (called U) then by follow this equation U^-1 A U = D when U^-1 is U inverse

A is the original metric

D is diagonal metric

but I have a little confused because my metric has component as an any function that make me a difficult to find metric U

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# Homework Help: Diagonalized matrix

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