A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system. Source: Internet
Darboux integrals have the advantage of being simpler to define than Riemann integrals. Source: Internet
First, one shows that the second definition is equivalent to the definition of the Darboux integral; for this see the article on Darboux integration. Source: Internet
Fix ε, and choose a partition such that the lower and upper Darboux sums with respect to this partition are within of the value s of the Darboux integral. Source: Internet
It would be described by Darboux as “one of the most handsome discoveries of modern Geometry”. Source: Internet
Now we will show that a Darboux integrable function satisfies the first definition. Source: Internet