Throughout this book so far we have talked about fields as functions of position only. In most physical applications of this theory the fields are also functions of time. To physicists fields are dynamical things that both experience and effect interactions with other systems, so the study of their development in time is of great importance.

As the time dependence of any physical quantity is usually governed by some basic dynamical law and the laws are different in different branches of physics, it might be thought that the ideas of vector analysis would be too general and all embracing to have anything to say about the time dependence of fields. To some extent this is true and we cannot here go far into dynamical problems. But there are two aspects of time dependence that can be very profitably discussed as aspects of vector analysis. It is fitting to conclude any study with a chapter on these topics, particularly because they provide an excellent illustration of how our general ideas can actually be used in a real physical context.

In this chapter we shall talk of scalar and vector fields ø(t, r) and f(t, r) and, in addition to the differentiation symbolised by ▽, we shall frequently form derivatives of the fields with respect to time: ∂ø/∂t and ∂f/∂t. We assume that differentiation with respect to time does not alter the vector character of a field: like ø itself, ∂ø/∂t is a scalar and, like f, ∂f/∂t is a vector.