$(r,s)$ values of the inertia tensor.

Question

I have had brief introduction to tensors in two directions: the physics way as an array of numbers and the math way as multi-linear maps. How do I link them together?

For example, is there a relation between the rank and dimensions with the $(r,s)$ values of a tensor?

I need it specifically for the inertia tensor.

I'm coming here straight from the W.E. Hereaus International School on Gravity and Light, Lecture-03, uploaded on YouTube. Referral to relevant material is welcome.

Answer 1

You have

$$T_{j_1 \cdots j_s}^{i_1 \cdots i_r} = T(\epsilon^{i_1}, \cdots, \epsilon^{i_r}, e_{j_1}, \cdots, e_{j_s})$$ where $\{e_j\}$ is a basis for $V$, $\{\epsilon^i\}$ a canonical cobasis for $V^*$ and $T$ the multi-linear map defining the tensor.

See wikipedia tensor article for more details.