Covariant Derivatives and Swapping Indices Okay,there's a covariant derivative of a rank 2 tensor. Swapping any indices gives a different tensor. Can we associate any physical significance to the swapping? For example, if I have a velocity vector $u$$_i$ and its covariant derivative is $u$$_i,_j$ , will swappinng indices $i$ and $j$ have any consequences on say, the direction of the resulting tensor( Covariant derivative of the velocity vector is a tensor)?

Ok, we have the covariant derivative in the $k$ direction for a rank one tensors, contravariantly $$u^i{}_{;k}=u^i{,k}+\Gamma^i{}_{sk}u^s,$$ and covariantly $$u_{i;k}=u_{i,k}-\Gamma^s{}_{ik}u_s,$$ which indicate a the tensor varies in direction $k$.

Then for rank two tensor we have the possibilities $$A_{ij}\ ,\ A^i{}_j\ ,\ A^{ij}\ ,\ A_iB_j\ ,\ A^iB_j\ ,\ A^iB^j.$$

Then, for example $$A_{ij;k}=A_{ij,k}+\Gamma^s{}_{ik}A_{sj}+\Gamma^s{}_{jk}A_{is},$$ or $$(A_iB_j)_{;k}=(A_iB_j)_{,k}+\Gamma^s{}_{ik}A_sB_j+\Gamma^s{}_{jk}A_iB_s,$$ tell you how tensor $A_{ij}$ or $A_iB_j%$ vary in direction $k$.