Exterior differential forms occur implicitly in all aspects of physics and engineering because they are the natural objects appearing as integrands of line, surface, and volume integrals as well as the n-dimensional generalizations required in, for example, Hamiltonian mechanics, relativity, and string theories. We shall see in this chapter that one does not integrate vectors; one integrates forms. If there is extra structure available, for example, a Riemannian metric, then it is possible to rephrase an integration, say of exterior 1-forms or 2-forms, in terms of a vector integrations involving “arc lengths” or “surface areas,” but we shall see that even in this case we are complicating a basically simple situation. If a line integral of a vector occurs in a problem, then usually a deeper look at the situation will show that the vector in question was in fact a covector, that is, a 1-form! For example (and this will be discussed in more detail later), the strength of the electric field can be determined by the work done in moving a unit charge very slowly along a small path, that is, by a line integral. The electric field strength is a 1-form.

Integration of a pseudoform proceeds in a way that differs slightly from that for a (true) form. We shall consider pseudoforms later on.