I am trying to do exercise 3.2 of Carol Sean'sSean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives.
The covariant derivative is the ordinary derivative for a scalar,so
$$D_\mu f = \partial_\mu f$$
Which is different from
$${\partial f \over \partial r}\hat{\mathbf r} + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta} + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}$$
Also, for the divergence, I used
$$\nabla_\mu V^\mu=\partial_\mu V^\nu + \Gamma^{\mu}_{\mu \lambda}V^\lambda = \partial_r V^r +\partial_\theta V^\theta+ \partial_\phi V^\phi + \frac2r v^r+ \frac{V^\theta}{\tan(\theta)} $$
Which didn't work either.
(Wikipedia: ${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}$).
I was going to try
$$(\nabla \times \vec{V})^\mu= \varepsilon^{\mu \nu \lambda}\nabla_\nu V_\lambda$$
But I think that that will not work. What am I missing?
EDIT: The problem is that the ortonormal basis used in vector calculus is different from the coordinate basis.
