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## Homework Statement

Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##.

## Homework Equations

##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy##

##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar x^2 + (1 + (b\bar y + c\bar x)^2) d\bar y^2 + 2(a\bar x + c\bar y)(b\bar y + c\bar x) d\bar xd\bar y##

## The Attempt at a Solution

My idea was to use the relations

##g_{\bar x\bar x} = g_{xx} (\frac{\partial {x}}{\partial {\bar x}})^2 + g_{yy} (\frac{\partial {y}}{\partial {\bar x}})^2 + 2g_{xy} (\frac{\partial {x}}{\partial {\bar x}}\frac{\partial {y}}{\partial {\bar x}}) ##, etc

Since the components ##g_{ij}##'s of the metric are given, maybe I can isolate the x's and y's, but I'm not sure if this is correct and I can't seem to isolate and integrate. Can anyone give me any hint?

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