Surface discontinuities

In the discussion so far we have talked of fields being irrotational or solenoidal without paying much attention to the regions in space where these fields exist and have such properties. All actual fields in physics exist only in confined regions of space and very often one must think of fields as terminating on outer boundary surfaces. Or conditions may change discontinuously within regions of interest – an electric field may exist partly in air, partly in glass. In this way, boundary surfaces enter. Also it may happen that the curl or divergence of a vector field – or indeed the gradient of a scalar field – is only different from zero in some confined region. This may be a narrow boundary layer which for practical purposes can be treated as a surface or it may lie effectively on a curve or at a point. Very often the general nature of the field is decisively determined by discontinuities on surfaces or by singularities on lines and points. The study of such behaviour will be the theme of this chapter.

We begin by discussing discontinuities of vector fields across surfaces. It is rather natural to find that we need to distinguish between the behaviour of a field component normal to a surface of discontinuity and one tangential to it. We shall see shortly that the behaviour of the former is related to div f and that of the latter to curl f. We shall put these statements into quantitative form.