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A bit of googling tells me the upper index means a co-ordinate. However I'm not sure what the lower index is. Overall I have no idea what makes it so special, or how to perform the operation in any way different from the sort of index notation I've already come across in fluid dynamics, wherein we have things like this for the divergence of a tensor:

$$\nabla \cdot A = \frac{\partial}{\partial x_i}A_{ij} $$

I simply don't get what's going on with the sudden separation into upper and lower indices here. Any advice would be appreciated, thanks.