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 $\nabla^i \delta_k^m$?