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I have a question regarding the contravariant components of the metric tensor in spherical coordinates. I have calculated the covariant components as $g_{rr}=1$, $g_{\theta\theta}=r^2\sin\phi$, and $g_{\phi\phi}=r^2$ (all others are 0).

My question is how would I get the contravariant components from here? I am told that $g^{ij}$ is such that $g^{ij}g_{jk}=\delta^i_k$ but have no clue how to proceed. Thanks, in advance. Eduardo

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First, I think you should have $g_{\theta\theta}=r^2\sin^2\phi$.

All this tensor notation is telling you to do is to find the inverse of the $3\times 3$ matrix $\big[g_{ij}\big]$. Since the matrix is diagonal, it's easily done!

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