The Theory of Vector Fields

The notion of a vector field comes from physics: a temperature field, a velocity field and an electric field all have a vector at every point of space. Put the integral tools of the previous chapters into the setting of a vector field, and a powerful physical intuition appears at once.

The core of this chapter is the three operators — the gradient, the divergence and the curl — each of which characterises a rate of change of the vector field: the gradient describes the rate of change of a value, the divergence describes the degree to which the vector field spreads out at a point, and the curl describes the tendency to rotate. A field with a potential (the curl vanishes and the domain is simply connected) means that the vector field can be written as the gradient of some scalar function, and problems of this kind only require solving one set of equations.

Flux and circulation then give the integral meaning of these three operators: the integral of the divergence is the flux through a closed surface, and the integral of the curl is the circulation along a closed curve. Together with Gauss’s theorem and Stokes’s theorem from the previous chapter, this yields the divergence theorem and Stokes’s theorem, which are the counterparts of the fundamental theorem of calculus in three dimensions.