Index notation of tensors and mnemonics

Question

I've been trying to learn to manipulate tensors but I've got probably too comfortable with all the matrices in my Linear Algebra course, that it gets really difficult beyond rank-3 tensors. So, ultimately, I've to move on to the index notation, and it is causing some difficulty for me.

So, are there any mnemonics, intuitive visualisations inside one's head, resources etc. that might help me in making the transition a bit more comfortably. Or is there no other option than just practicing it?

Answer 1

Expressions like

$v^se_s$,

$f_t\ e^t$,

$g^{st}e_s\otimes e_t$,

$F_{st}e^s\wedge e^t$,

$\Omega^{stu}b_s\wedge b_t\wedge b_u$,

$R^s{}_{jkl}\partial_s$,

$A^{ij}{}_{klm}e_i\otimes e_j\otimes e^k\otimes e^l\otimes e^m$,

indicate multi-indexed linear combinations for each of the up-and-down couple-repeated of them.