Let $x(t, \phi) = (t\cos(\phi), t\sin(\phi),\log(\phi))$ describe a surface.
Calculate the metric tensor $g_{ij}$ and $\tilde{g}_{ij}$
Using the first fundamental form I get for $g_{ij}=$
$\begin{pmatrix}
1+\frac{1}{t^2} &0 \\
0 & t^2
\end{pmatrix}$
However, I don't know how to express $\tilde{g}_{ij}$, if I am not mistaken it should be the covariant form of $g_{ij}$.
I'm confused on how to derive $\tilde{g}_{ij}$, any help is appreciated.
Edit: I'm not sure how to interpret this definition of a covariant vector:
A tensor having a transformation law that mimics that of the scalar field gradient
$$A^{'}_\mu (x^{'}) = \frac{ \partial x^{\nu} }{ \partial {x^{'}}^{\mu} }A_{\nu}(x) $$