$$\require{cancel}$$
These notes are intended to cover only mechanics, and therefore I resist the temptation to cover here special relativity and electromagnetism. I point out only that in many ways this misses many of the most exciting parts of special relativity, and indeed it was some puzzles with electromagnetism that led Einstein to formulate the theory of special relativity. One proceeds as we have done with mechanical quantities; that is, we have to define carefully what is meant by each quantity and how in principle it is possible to measure it, and then see how it transforms between frames in such a manner that the laws of physics – in particular Maxwell’s equations - are the same in each. One such transformation that is found, for example, is $$\textbf{E'} = \gamma(\textbf{E} + \textbf{u}\times\textbf{B})$$ so that what appears in one frame as an electric field appears in another at least in part as a magnetic field. The Coulomb force transforms to a Lorentz force; Coulomb’s law transforms to Ampères law.