# Covariant differentiation commutes with contraction?

1. Jun 4, 2010

### kiddokiddo

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
For example, $$\nabla_{i}T^{jk}_{kl}$$.

3. 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!

Last edited: Jun 4, 2010
2. Jun 4, 2010

### 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 $\nabla^i \delta_k^m$?