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I'm considering the covariant derivative

[tex]\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma_{\mu\nu}^\lambda V^\lambda[/tex]

in spherical coordinates in flat 3D space (x = r cos sin, y = r sin sin, z = r cos; usual stuff).

Now I wrote down the gradient of a scalar functionf, for which I got [itex]\nabla_\mu f = \partial_\mu f[/itex] (having constructed the derivative to reduce to the ordinary partial derivative on scalars) which is of course not correct, it should be something like

[itex] \partial_r f, \frac{1}{r} \partial_\theta f \text{ and } \frac{1}{r \sin\theta} \partial_\phi f[/itex] for the three components.

Same when I try to derive an expression for the divergence of a vector field, and then I need to show that it is the same as the familiar

[tex]\frac{1}{r^2} \frac{\partial}{\partial r} (r^2 V^1) + \frac{1}{r \sin\theta} \frac{\partial}{\partial\theta} (\sin\theta V^2) + \frac{1}{r \sin\theta} \frac{\partial V^3}{\partial \phi}[/tex]

Of course,

[tex]\nabla_\mu V^\nu = \partial_\mu V^\mu + \Gamma_{\mu\mu}^\nu V^\nu[/tex]

(thatmustbe correct) but if plug in the connection coefficients I calculated, I get an otherwise correct result, except there are some factors 1/r and 1/r sin theta missing.

I have a feeling I'm mixing up my coordinate systems here, but how?

I did get the right result for the Laplacian [itex]\nabla^2 f[/itex] by the way ... isn't that odd?

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# Identities for covariant derivative

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