Simplest mixed tensor example What is a simple example of a mixed tensor, (very) preferably obeying the following:


*

*Is of valence $\left( {1 \atop 1} \right)$

*Does not involve calculus

*Has an intuitive meaning and usage, e.g: within physics. In other words, is not 'artificial'.


In case not all the above can be satisfied, maybe examples with different combinations can be found.

The simplest example from physics that I can think of is the moment of inertia 2-tensor $\mathbf{I}$ in mechanics. The moment of inertia measures how the mass of a system is distributed in space. It relates the angular velocity vector $\mathbf{\omega}$ of a rotating system to its angular momentum vector $\mathbf{L}$ by
$$
\mathbf{L} = \mathbf{I}\mathbf{\omega}
$$
or, in component notation,
$$
L^i = I^i_j\omega^j
$$
This tensorial relation indicates that angular momentum is not, in general, in the same direction as angular velocity and thus not along the axis of rotation! For example, if you throw a brick into the air, giving it some spin, its angular momentum won't be along its axis of rotation unless the brick is spinning around one of its three symmetry axes. In a coordinate system along these symmetry axes, the moment of inertia tensor is diagonal.
