Hello, I'm trying to get a hang of the defintion presented in Arfken - Mathematical Methods for Physicists for 3 dimensional rotations (a setup for an introduction to tensors). That being said I'm a Physicist and I'd like a component approach if possible to the explanation. They give:

The first equation I'm not really confused about, it's just representing one vector in another basis. It shows that the coefficient in front of each component is just the projection of the primed unit vector onto the unprimed vector. They do a graphical derivation. What I have a problem with is their next step and the justification:

So I can justify to myself why this should be true, in English the dot product above and the partial derivative is saying the same thing. The change in the [itex]i^{th}[/itex] primed coordinate relative to the [itex]j^{th}[/itex] unprimed coordinate.

I'm not sure why they are all of a sudden using [itex]x_i[/itex] to represent the coordinates and also why it is okay to represent the dot product in this way. They give an explanation: