In his book Multivariate Calculus and Geometry, Sean Dineen explains the chain rule as follows:
I understood well enough his point but, then he goes on to introduce an unknown and unexplained $\omega$ to explain something having to do with the unit basis:
This has nothing to do with ordinals. Here $\omega$ is a vector, and $\omega_j$ is its $j$th coordinate. It's entirely analogous to saying $a = (a_1, a_2, a_3)$, which we could also write as $a = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$ if you're familiar with that notation. In multiple dimensions, instead of using letters we use $e_i$ to denote the $i$th standard basis vector, so we would write $a = a_1 e_1 + a_2 e_2 + a_3 e_3 = \sum_i a_i e_i$. Your book is doing exactly the same thing but with $\omega$.
I hope this helps ^_^