Editing Differential form (section)

=== Integration on Euclidean space ===
Let {{math|''U''}} be an open subset of {{math|'''R'''<sup>''n''</sup>}}.  Give {{math|'''R'''<sup>''n''</sup>}} its standard orientation and {{math|''U''}} the restriction of that orientation.  Every smooth {{math|''n''}}-form {{math|''ω''}} on {{math|''U''}} has the form
<math display="block">\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n</math>
for some smooth function {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}}.  Such a function has an integral in the usual Riemann or Lebesgue sense.  This allows us to define the integral of {{math|''ω''}} to be the integral of {{math|''f''}}:
<math display="block">\int_U \omega\ \stackrel{\text{def}}{=} \int_U f(x)\,dx^1 \cdots dx^n.</math>
Fixing an orientation is necessary for this to be well-defined.  The skew-symmetry of differential forms means that the integral of, say, {{math|''dx''<sup>1</sup> ∧ ''dx''<sup>2</sup>}} must be the negative of the integral of {{math|''dx''<sup>2</sup> ∧ ''dx''<sup>1</sup>}}.  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.