A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form α in the open unit ball in R^n. Since α is a top-dimensional form, it can be written α = f d x_1 ⋀...⋀d x_n and so integral_B α = integral_B f d μ, where the integral is the Lebesgue integral. On a manifold M covered by coordinate charts U_i, there is a partition of unity ρ_i such that 1.ρ_i has support in U_i and 2. sum ρ_i = 1.

de Rham cohomology | Stokes' theorem | submanifold | top-dimensional form