Index Notation for Rank-2 Tensor with Summation

[Niles]

Homework Statement

I have the following rank-2 tensor
$$T = \nabla \cdot \sum_{i}{c_ic_ic_i}$$
I would like to write this using index notation. According to my book it becomes
$$T_{ab} = \partial_y \sum_{i}{c_{ia}c_{ib}c_{iy}}$$
Question: The change $\nabla \rightarrow \partial_y$ and $c_i \rightarrow c_{ia}$ I agree with. However, it is not clear to me why my book uses the same index $y$ for $\partial_y$ as it does for $c_{iy}$. Why are we allowed to do that?

 

[HallsofIvy]

The indices, which in 3 dimensions would be "1", "2", and "3", typically correspond components in the direction of the "x", y", and "z" axes. Apparently your book is allowing "i", "a", and "b" to mean any of the directions but the "y" refers specifically to the direction of the y axis.

 

[Niles]

But is it true that

$$\nabla \cdot \sum_{i}{c_ic_ic_i} \leftrightarrow \partial_y \sum_{i}{c_{ia}c_{ib}c_{iy}}$$

in general? IMO the last $c_{iy}$ should be a $c_{iq}$, i.e. some index different from y.

 

[Niles]

Note that it is a rank-2 tensor, not a 3D tensor as I originally wrote

 

[Niles]

I get it now... thanks