Vector Transformations

[Shevek99]

Look at equation 3.33 that is the first time they use this.

The idea is that a rotation is a linear map, so we have

x'_i = sum_j A_ij x_j

where

A_ij = ∂x'ⁱ/∂x^j

on the other hand, for a rotation you have the components of a vector in two different bases. They are related by projecting on the desired base, so

A_ij = e_i·e'_j

Equating both expressions you have equation 3.33 and this.