A metric tensor $g$ is a symmetric, non-degenerate, rank 2 covariant tensor. The inverse of a metric tensor is a symmetric, non-degenerate, rank 2 contravariant tensor $g$. The components of $h$ are given by the inverse of the matrix defined by the components of $g$.

InverseMetric(g) calculates the inverse of the metric tensor g.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form InverseMetric(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-InverseMetric.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(x\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}\right)$

$g≔x\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}$

$h≔\mathrm{InverseMetric}\left(g\right)$

$h≔\frac{1}{x}\mathrm{D\_x}\mathrm{D\_x}-\mathrm{D\_y}\mathrm{D\_y}$

$\mathrm{ContractIndices}\left(g\,h\,\left[\left[1\,1\right]\right]\right)$

$\mathrm{dx}\mathrm{D\_x}+\mathrm{dy}\mathrm{D\_y}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\,w\right]\,E\right)$

$\mathrm{frame\; name:\; E}$

$g≔\mathrm{evalDG}\left(\mathrm{du}\&t\mathrm{du}-\mathrm{dv}\&t\mathrm{dw}-\mathrm{dw}\&t\mathrm{dv}\right)$

$g≔\mathrm{du}\mathrm{du}-\mathrm{dv}\mathrm{dw}-\mathrm{dw}\mathrm{dv}$

$\mathrm{InverseMetric}\left(g\right)$

$\mathrm{D\_u}\mathrm{D\_u}-\mathrm{D\_v}\mathrm{D\_w}-\mathrm{D\_w}\mathrm{D\_v}$