structural:summarization
Integration on Euclidean space

Let U be an open subset of R. Give R its standard orientation and U the restriction of that orientation. Every smooth n -form ω on U has the form

for some smooth function f : R → R. Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of ω to be the integral of f:

Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, dx ∧ dx must be the negative of the integral of dx ∧ dx. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity.