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*[Moderator's note: Thread spun off from previous thread due to topic change.]*

This thread brings a pet peeve I have with the notation for covariant derivatives. When people write

##\nabla_\mu V^\nu##

what it looks like is the result of operating on the component ##V^\nu##. But the components of a vector are just scalars, so there's no difference between a covariant derivative and a partial derivative.

My preferred notation (which I don't think anybody but me uses) is:

##(\nabla_\mu V)^\nu##

The meaning of the parentheses is this: First you take a covariant derivative of the vector ##V##. The result is another vector. Then you take component ##\nu## of that vector.

Then with this notation, you can substitute ##V = V^\nu e_\nu## to get ##V## in terms of basis vectors, and you get:

##\nabla_\mu V = \nabla_\mu (V^\nu e_\nu) = (\nabla_\mu V^\nu) e_\nu + V^\nu (\nabla_\mu e_\nu)##

With my notation, the expression ##(\nabla_\mu V^\nu)## is just ##\partial_\mu V^\nu##. So we have (after relabeling the dummy index ##\nu## to ##\sigma## on the last expression)

##\nabla_\mu V = (\partial_\mu V^\nu) e_\nu + V^\sigma (\nabla_\mu e_\sigma)##

Taking components gives:

##(\nabla_\mu V)^\nu = \partial_\mu V^\nu + V^\sigma \Gamma^\nu_{\mu \sigma}##

where ##\Gamma^\nu_{\mu \sigma}## is just defined to be equal to ##(\nabla_\mu e_\sigma)^\nu## (component ##\nu## of the vector ##\nabla_\mu e_\sigma##).

With the usual notation, you would have something like:

##\nabla_\mu (e_\nu)^\sigma = \Gamma^\sigma_{\mu \nu}##

which is, to me, clunky and weird.