Scalar fields

Many physical quantities may be suitably characterised by scalar functions of position in space. Given a system of cartesian axes a scalar field ø can be represented as ø = ø(r), where r is the position vector defined in chapter 2. There is no real difference between this way of referring to a scalar field and the alternative statement ø = ø(x, y, z), except that in this latter form one is definitely committed to a particular set of cartesian coordinates for r, while the form ø(r) can be taken to refer to any coordinate axes – or indeed to any other equivalent way of defining the position vector r of a point.

In dealing with functions of a single variable, x say, the universal standby that helps one visualise the function is the simple graph, fig. 1. This notion is familiar enough to need no explanation and is so closely associated with the function itself that it is sometimes difficult to remember that the graph is not the function. But when we come to functions of more than one variable, things become rather different. Let us go one step at a time and first consider a function of two variables f(x, y). The values over which x and y vary cover all or part of the xy-plane – a plane where before we had a line – and the function f(x, y) may then be thought of as plotted in a third dimension which we may call z; z = f(x, y).