Preface

The main topic of this course is Maxwell's equations. These are a set of eight first-order partial differential equations which constitute a complete description of electric and magnetic phenomena. To be more exact, Maxwell's equations constitute a complete description of the behaviour of electric and magnetic fields. Students entering this course should be quite familiar with the concepts of electric and magnetic fields. Nevertheless, few can answer the following important question: do electric and magnetic fields have a real physical existence, or are they merely theoretical constructs which we use to calculate the electric and magnetic forces exerted by charged particles on one another? As we shall see, the process of formulating an answer to this question enables us to come to a better understanding of the nature of electric and magnetic fields, and the reasons why it is necessary to use such concepts in order to fully describe electric and magnetic phenomena.

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 $x$-, $y$-, and $z$-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.