Find the coordinate transformation given the metric

[Whitehole]

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?

 

[Orodruin]

You are missing the off diagonal terms in your relations between the metric components, but yes. That would be the general approach.

 

[Whitehole]

You are missing the off diagonal terms in your relations between the metric components, but yes. That would be the general approach.

I keep forgetting that off diagonal term, I've edited my post. So generally I need to isolate the variable then integrate?

 

[Orodruin]

By identifying the terms, you should get a system of partial differential equations which you can solve.

 

[Whitehole]

By identifying the terms, you should get a system of partial differential equations which you can solve.

That is quite clear but maybe there are other ways to solve this problem, I doubt if A. Zee (author of the book I'm reading) wanted his readers to solve it this way.