A differential -form can be integrated over an oriented -dimensional manifold. When the -form is defined on an -dimensional manifold with , then the -form can be integrated over oriented -dimensional submanifolds. If , integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of correspond to line integrals, surface integrals, volume integrals, and so on. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.
