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