Divergence from covariant derivative?

I was reading my GR textbook when I came across the definition of the covariant derivative in coordinate form. Now, to get some experience with it, I decided to play around with it and see if I could construct some definitions from vector calculus with it. Now, for the Laplacian, this works fine, giving $$\nabla^{\mu}\partial_{\mu}\phi=g^{\mu\nu}(\partial_{\nu}\partial_{\mu}\phi-\Gamma^{\lambda}_{\nu\mu}\partial_{\lambda}\phi)$$

This expression works perfectly well for the three-dimensional metrics for spherical, cylindrical, etc.. Note that $$\Gamma^{\mu}_{\nu\sigma}$$ are the Christoffel coefficients describing the Levi-Civita connection. Now, when I try to find, say, a generalized expression for the divergence the same way, $$\nabla_{\mu}V^{\mu}=\partial_{\mu}V^{\mu}+\Gamma^{\mu}_{\mu\lambda}V^{\lambda}$$ this fails every time, save for on a Cartesian metric. Why does one work and not the other?

• What is it that goes wrong? Commented Sep 24, 2019 at 16:05
• @Arthur probably the fact that if you were to grind it out with the Christoffel symbols, the expression does not line up with the accepted defintions of divergence in standard curvilinear coordinates, like what you can find in a vector calculus table. This is because the basis vectors have different defintions than from vector calculus. When you define the basis vectors to be partial derivatives in tensor calculus, they don't line up with the unit vectors used in vector calculus. Commented Sep 24, 2019 at 16:11
• It simply gives an incorrect answer for metrics other than Cartesian. I’m not precisely sure what’s going wrong, to be honest. Commented Sep 24, 2019 at 16:14
• @Tesseract I have an answer in my reply to Arthur Commented Sep 24, 2019 at 16:16

Take 3D spherical coordinates and consider the basis vector $$\partial_\theta$$ that you might find in a GR book. If the definitions for vector calculus stuff were to line up with their tensor calculus counterparts then $$\partial_\theta$$ would have to be a unit vector. But using the defintion of the metric in spherical coordinates,
$$\partial_\theta \partial^\theta = g^{\theta\theta} = \frac{1}{r^2}$$