Scalar and vector fields

In almost every branch of mathematical physics, we have to deal with physical quantities which extend continuously through regions of 3-space. These regions and their boundaries do not usually have any awkward features; we therefore assume that they can be described mathematically as volume regions and smooth surfaces which satisfy the conditions set out in Chapter 5. We also assume that any curves in physical space can be validly represented as piecewise smooth curves. Volumes, areas of surfaces and lengths of curves can therefore be defined. We can also define integrals of functions along curves, over surfaces and throughout volumes, provided that the region is finite and the functions are piecewise continuous; these definitions are based on analytic theorems established in Appendix A.

In this chapter, we shall be studying the analysis of functions in 3-space which might represent the properties of, for example, fluids, gravitational or electromagnetic fields, stress and strain in solids, or wave functions in atoms, molecules and nuclei. Such a function may be unbounded when the position vector tends to certain points, curves or surfaces; for example the electrical potential of a point charge e at the origin is e/r, which is unbounded near the origin (r = 0). Special care must be taken in the study of functions in regions where they are unbounded; we shall concentrate our attention on the analysis of functions in regions where they are ‘well-behaved’. By ‘well-behaved’ we not only mean that a function is continuous, but also that any derivatives we use exist and are continuous.