Covariant differentiation commutes with contraction?

[kiddokiddo]

Homework Statement

I've been reading a textbook on tensor analysis for a while. The book uses the conclusion of "covariant differentiation commutes with contraction" directly and I searched around and found most people just use the conclusion without proof.

Homework Equations

For example, \nabla_{i}T^{jk}_{kl}.

The Attempt at a Solution

I believe it can be interpreted in two ways. First, form the variant T^{jk}_{kl} with two free indices j, l and apply \nabla_{i} to that tensor; Or, apply \nabla_{i} to the tensor T^{jk}_{ml} and contract m and k. If the two interpretations lead to the same result, it should then be proved.

Any help is appreciated!

 

[diazona]

Whenever you contract something, there's a delta tensor involved. Try writing that out explicitly and using the product rule. What do you know about \nabla^i \delta_k^m?