Coordinate Transformation Groups

Coordinate transformations in physics form a group, or more properly, can be split up into several named groups and subgroups. It is beyond the scope of this short review to introduce you to all of subtleties and joys of group theory in physics (one could write a whole book on this alone - or two or three books!) so we will just move as directly as possible to two or three examples that should already be somewhat familiar to the reader.

Let us define the position vector (in any coordinate frame or coordinate system, but for now we will think only of $\mathbb{R}^3$ , real Euclidean space in three dimensions) to be denoted by $\vr$ (dressed with indices or primes as need be). For example, in Cartesian coordinates:

$$\vr = x \hx + y \hy + z \hz$$

The displacement vector - or any general difference of position vectors - is an enormously useful object in physics in general and electrodynamics in particular. We will use a special notation for it that simplifies certain formulas that occur quite often in Electrodynamics (following Griffiths):

$$\vrcurs = \vr - \vr\ ' = (x - x')\hx + (y - y')\hy + (z - z')\hz$$

It is also essential for the definition of differentiation, manifolds, the construction of calculus and the calculus-based entities of physics such as velocity $\vv$ or acceleration $\va$ , but for the moment we will not worry about any of this.

Note Well! A vector is defined to be a dimensioned object that transforms like a displacement vector when the underlying coordinate frame is transformed! More on this later, but first, let's look at some specific Lie groups.