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Recently I encountered a type of covariant derivative problem that I never before encountered:

$$

\nabla_\mu (k^\sigma \partial_\sigma l_\nu)

$$

My goal: to evaluate this term

According to Carroll, the covariant derivative statisfies ##\nabla_\mu ({T^\lambda}_{\lambda \rho}) = {{\nabla(T)_\mu}^\lambda}_{\lambda \rho}\ \ (\dagger)## and also ## \nabla(T \otimes S) = (\nabla T)\otimes S + T \otimes (\nabla S) \ \ (\ddagger)##. We know that ##\partial \sigma## forms a basis, so therefore ##k^\sigma \partial_\sigma ## is a (1,0) tensor

$$

\nabla_\mu (\underbrace{k^\sigma \partial_\sigma}_{\equiv T} l_\nu) = \nabla_\mu (T l_\nu) \stackrel{(\dagger)}{=} \nabla(T l)_{\mu \nu} = [\nabla(T l)]_{\mu \nu} \stackrel{(\ddagger)}{=} [\nabla(T) \otimes l + T\otimes \nabla(l)]_{\mu \nu}

$$

But how do I continue from there? I want to distribute back the ## \mu \nu## and replace ## \otimes## with regular multiplication, but I dont know any algebra rules which applies, also my textbook doesn't help. I know that the tensors are multilinear, but its not quite what I need. How do I continue to evaluate ##\nabla_\mu (k^\sigma \partial_\sigma l_\nu)##?

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# A Trying to understand covariant derivative on tensors

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