# How is the second-order covariant derivative of a scalar computed?

What is second-order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a scalar function of $r$?

For Cartesians, I know that the covariant derivatives reduce to partial derivatives. However, since this is in polar coordinates...

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$$\nabla_j f=\partial_jf$$
$$\nabla_i \nabla_j f=\nabla_i \partial_jf=\partial_i \partial_j f-\partial_k f~\Gamma^{k}_{ij}$$