ABSTRACT
We saw in the last chapter how k-forms provide a generalization to ℝ n of the notions of scalar and vector fields in ℝ3, and how the differential operator d provides a generalization of the operators grad, curl, and div. Now we define the integral of a k-form over a k-manifold; this concept provides a generalization to ℝ n of the notions of line and surface integrals in ℝ3. Just as line and surface integrals are involved in the statements of the classical Stokes’ theorem and divergence theorem in ℝ3, so are integrals of k-forms over k-manifolds involved in the generalized version of these theorems.
