proof of general Stokes theorem
We divide the proof in several steps.
(i.e. the term is missing). Hence we have
From the fundamental theorem of Calculus we get
Since and hence have compact support in we obtain
and hence, as wanted
Suppose now that and let be any differential form. We can always write
and by the additivity of the integral we can reduce ourself to the previous case.
When we could follow the proof as in the first case and end up with while, in fact, .
Consider now the general case.
First of all we consider an oriented atlas such that either is the cube or . This is always possible. In fact given any open set in and a point up to translations and rescaling it is possible to find a “cubic” neighbourhood of contained in .
Then consider a partition of unity for this atlas.
From the properties of the integral on manifolds we have
The second integral in the last equality is zero since , while applying the previous steps to the first integral we have
On the other hand, being an oriented atlas for and being a partition of unity, we have
and the theorem is proved.
|Title||proof of general Stokes theorem|
|Date of creation||2013-03-22 13:41:43|
|Last modified on||2013-03-22 13:41:43|
|Last modified by||paolini (1187)|