Say I have a metric representation $g_{\mu\nu}$ in a coordinate system $x$ and I want to find the representation of the metric in a new set of coordinates $y = y(x)$. I know how to do this if you are given $x(y)$, as in this post.
$g_{\mu' \nu'} = \frac{\partial x^{\mu}}{\partial y^{\mu'}} \frac{\partial x^{\nu}}{\partial y^{\nu'}} g_{\mu \nu}$ -------- $(1)$
But what if I'm only given $y=y(x)$, and it's tricky to figure out $x=x(y)$? Is there a method that uses partial derivatives $\frac{\partial y^{\mu'}}{\partial x^{\mu}}$ instead? Or is $(1)$ the only way?
Answer
The $\partial x^\mu/\partial y^{\mu'}$ are just the components of the Jacobian matrix, and the Jacobian of an inverse transformation is equal to the inverse of the original Jacobian. Find the Jacobian matrix that underlies $y = y(x)$, invert it, and you should have the correct components.
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