Why Does Index Position in Kronecker Delta and Tensors Not Always Matter?

[helpcometk]

Homework Statement

kroenecker delta has one upper and one lower index. except from uppper index and lower index,we have 2 slots for the upper index and 2 slots for the lower index(for a 2 index tensor).
Why krenecker delta left or right slot index position doesn't matters?
Also why 2 two index tensors may not be equal because of the peculiarity of left and right slot index position?

Homework Equations

The Attempt at a Solution

 

[clamtrox]

Why krenecker delta left or right slot index position doesn't matters?

That's just because of the definition. Tensors that behave like this are called symmetric.

Also why 2 two index tensors may not be equal because of the peculiarity of left and right slot index position?

I don't understand what this question means.

 

[dextercioby]

Because of the general definition of a tensor as a multilinear mapping, the order in which the vector spaces and their duals are chosen really matters, especially when the bundle of tensors is defined for a metric manifold (as is the case in relativity).

You will study electrodynamics and discover that the mixed components of the field tensor are all different, that is

$$F_{\mu}^{~~\nu} \neq F^{\nu}_{~~\mu}$$

for arbitrary index values.