What's the mathematical formulation for frames for reference?

Question

In this lecture I've come across an interesting formulation of kinematics and reference frames. When I studied kinematics, the idea of what is meant by a 'frame of reference' was never quite fully esplained, and I've never had the notion of taking time derivatives with respect to various frames of ref. impressed upon me. The course uses this textbook which I unfortunately wasn't able to get as of yet. The course is intended for engineers, so precise mathematical foundations are unnecessary for them, but I find this idea of derivatives being taken w.r.t. different frames very intriguing. Surely it must be a part of a more rigorous formulation of kinematics, but so far I had no luck finding anything of the sort. I would appreciate any reference!