It will be convenient to state these equations in the slightly modified form which is due to Lorentz. Space is referred to the fixed rectangular axis system , as in subarticle 6.1. It will be necessary to explain a few small points of nomenclature mid notation.
A vector is a directed physical quantity; for example, the electric force at point is a vector. This example also shows that we have to conceive vectors which have analogous significations at different points of space. Such a vector is the electric force which may have a distinct magnitude and direction at each point of space, but expresses at all points one definite physical fact. Such a vector will be a function of its position, that is to say, of the coordinates of the point () of which it is that characteristic vector.
Let () be any such vector. Then and and are each of them functions of () and also of the time , i.e. they are functions of . We shall assume that our physical quantities are differentiable, except possibly at exceptional points.
Let ) stand for (), and analogously and ) for the vector
Finally if () be another vector at the same point, then stands for what is called the 'vector product' of the two vectors, namely the vector
It is evident that ) can be expressed in the symbolic form
The vector equation is an abbreviation of the three equations
Let ) be the electric force at ), and let ) be the magnetic force at the same point and time. Also let be the volume density of the electric charge and ) its velocity; and let ) be the ponderomotive force: all equally at ). Finally let be the velocity of light in vacuo.
Then Lorentz's form of Maxwell's equation is
It will be noted that each of the vector equations (3), (4), (5) stands for three ordinary equations, so that there are eleven equations in the five formulae.