At any given point in space, an electric or magnetic field possesses two
properties, a *magnitude* and a *direction*. In
general, these properties
vary (continuously) from
point to point. It is conventional to represent such a field in terms
of its components measured with respect to some conveniently chosen set of
Cartesian axes (*i.e.*, the conventional -, -, and -axes). Of course, the
orientation of these axes is *arbitrary*. In other words, different observers
may well choose different coordinate axes to describe the same field.
Consequently, electric and magnetic fields may have different components
according to different observers. We can see that any description of
electric and magnetic fields is going to depend on two seperate things.
Firstly, the nature of the fields themselves, and, secondly, our arbitrary choice
of the coordinate axes with respect to which we measure these fields.
Likewise, Maxwell's equations--the equations which describe the behaviour
of electric and magnetic fields--depend on two separate things. Firstly, the
fundamental laws of physics which govern the behaviour of electric and magnetic
fields, and, secondly, our arbitrary choice
of coordinate axes. It would be helpful if we could easily distinguish those
elements of Maxwell's equations which depend on physics from those which
only depend on coordinates. In fact, we can achieve this by using what mathematicians
call *vector field theory*. This theory enables us to write Maxwell's
equations in a manner which is *completely independent* of our choice
of coordinate axes. As an added bonus, Maxwell's equations look a
lot simpler when written in a coordinate-free manner. In fact, instead of
*eight* first-order partial differential equations, we only require
*four* such equations within the context of vector field theory.