The parameters which measure change in dynamical systems have a unique importance: they describe both the layout and the development of a system. Space (position) and time are the most familiar parameters, but there are other possibilities, such as Fourier modes.

In the previous chapter, it was seen how the unification of spatial and temporal parameters, in electromagnetism, led to a tidier and deeper form of the Maxwell equations. It also made the equations easier to transform into other relativistic frames. In the covariant approach to physics one is concerned with what does and does not change, when shifting from one perspective to another, i.e. with the properties of a system which are dependent and independent of the circumstances of observation. In a continuous, holonomic system, this is summarized by two independent concepts: parameter spaces and coordinates.

• Parameter space (manifold). This represents the stage for physical reality. A parameter space has coordinate-independent properties such as topology and curvature.

• Coordinates. These are arbitrary labels used to mark out a reference scheme, or measurement scheme, in parameter space. There is no unique way to map out a parameter space, e.g. Cartesian or polar coordinates. If there is a special symmetry, calculations are often made easier by choosing coordinates which match this symmetry.

Coordinates are labels which mark a scale on a parameter space. They measure a distance in a particular direction from an arbitrary origin. Clearly, there is nothing fundamental about coordinates: by changing the arbitrary origin, or orientation of measurement, all coordinate labels are changed, but the underlying reality is still the same.