Denoting Indices of Tensors

Miloslav

I was wondering how the indices of tensors work. I do not understand how the indices of tensors in can be used. For example, \eta _{\mu \nu }, the metric tensor, is like a matrix, and x^{u} is a contravector. How does this extend to notations such as T{_{a}}^{bc} and T{_{ab}}^{c}?

Matterwave

use [tex] wraps to make the latex show up.

The index notation is used in 2 ways. 1, it is used to denote the components of a tensor in some (arbitrary) coordinate system. And 2, it is often used to denote the tensor itself. The second way is a shortcut, but, strictly speaking, is an abuse of notation.

Lower indices indicate covariance while upper indices indicate contravariance. The tensors you wrote at the end are respectively once covariant twicce contravariant and twice covariant once contravariant.

Miloslav

Ok. Thanks for the tip about Latex. I was hoping to clarify that tensors are used in a certain context. Thank you for the information, I now understand how it can be used, particularly in the case of the Kronecker delta.

HallsofIvy

In flat, Euclidean, space, in which we can use the Kronecker [itex]\delta[/itex] as metric tensor, there is no distinction between "covariant" and "contravariant" tensors.