Covariant derivative

Question

I would very much appreciate some help in The following:

What is 2nd 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...

Answer 1

The covariant derivative of a scalar is just its gradient because scalars don't depend on your basis vectors:

$$\nabla_j f=\partial_jf$$

Now it's a dual vector, so the next covariant derivative will depend on the connection. Assuming the Levi-Civita connection, i.e. the Christoffel symbols, the covariant derivative will be:

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