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I have an immersion $F: M\to\mathbb R^{n+1}$ with an n-dimensional smooth manifold. The coefficients of the metric $g$ are defined by $g_{ij}(p)=\left\langle \frac{\partial F}{\partial x_i}(p), \frac{\partial F}{\partial x_j}(p) \right\rangle, p\in M$, where $\langle \cdot, \cdot \rangle$ is the standard scalar product on $\mathbb R^{n+1}$.

How can I calculate the coefficients of the inverse matrix $\{g^{ij}\}=\{g_{ij}\}^{-1}$?

Thanks for any help

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  • $\begingroup$ How about the usual formula for the inverse matrix, see e.g. here? Or, you have plenty of choices from Wikipedia. $\endgroup$ Jun 7 '18 at 11:00
  • $\begingroup$ But there is no "explicit" formula for the matrix elements? For fixed n one can find this, but in general? $\endgroup$
    – d.s.
    Jun 7 '18 at 12:20

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