I have gone through and kind of see how ds^2=g_ij*dx^i*dx^j where g_ij=dy^m/dx^i*dy^m/dx^j. How does one show the inverse metric tensor (g^ij) is equal to dx^i/dy^m*dx^j/dy^m?
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I can guess that $\displaystyle x^n$ are the coordinates in some coordinate system but what is $\displaystyle y^m$? A different coordinate system?
yes the x and y are of two different sytems (unbarred and barred) where dy^m=dy^m/dx^j*dx^j
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