Editing Differential form (section)

=== Integration using partitions of unity ===
There is another approach, expounded in {{Harv|Dieudonné|1972}}, which does directly assign a meaning to integration over {{math|''M''}}, but this approach requires fixing an orientation of {{math|''M''}}.  The integral of an {{math|''n''}}-form {{math|''ω''}} on an {{math|''n''}}-dimensional manifold is defined by working in charts.  Suppose first that {{math|''ω''}} is supported on a single positively oriented chart.  On this chart, it may be pulled back to an {{math|''n''}}-form on an open subset of {{math|'''R'''<sup>''n''</sup>}}.  Here, the form has a well-defined Riemann or Lebesgue integral as before.  The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of {{math|''ω''}} is independent of the chosen chart.  In the general case, use a partition of unity to write {{math|''ω''}} as a sum of {{math|''n''}}-forms, each of which is supported in a single positively oriented chart, and define the integral of {{math|''ω''}} to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate {{math|''k''}}-forms on oriented {{math|''k''}}-dimensional submanifolds using this more intrinsic approach.  The form is pulled back to the submanifold, where the integral is defined using charts as before.  For example, given a path {{math|''γ''(''t'') : [0, 1] → '''R'''<sup>2</sup>}}, integrating a {{math|1}}-form on the path is simply pulling back the form to a form {{math|''f''(''t''){{thin space}}''dt''}} on {{math|[0, 1]}}, and this integral is the integral of the function {{math|''f''(''t'')}} on the interval.