14
$\begingroup$

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the differential of its fiberwise integral is, up to a sign, the fiberwise integral of the restriction of the form to the boundary.

Is there a citeable reference for a proof of this result? For the sake of being definite, let's say that “citeable” means indexed by Mathematical Reviews or Zentralblatt, or available on arXiv.

Of course, for such a well-known result I would very much prefer an older, classical reference as opposed to something recent on arXiv.

Citeable reference for a statement of this theorem without proof do exist, the earliest one that I am aware of is Greub-Halperin-van Stone I, Chapter VII, Problem 4. References which do not include proofs will not suffice for my purposes.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Try Propositions 3.4.54 of these notes. (It is indexed by Math Rev. and Zentralblatt.) $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 9, 2015 at 14:07
  • 1
    $\begingroup$ Yes, but this is the version without boundary. (In fact, I'm not sure if the name “Stokes theorem” can be applied to the case of an empty boundary.) $\endgroup$
    – Dmitri Pavlov
    Commented Apr 9, 2015 at 14:17
  • $\begingroup$ Sorry. I misqouted the theorem. It is Theorem 3.4.54 of the same notes. Unfortunately, I a also left the proof as an exercise. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 9, 2015 at 14:30
  • 1
    $\begingroup$ Yes, my point exactly. I know several other references that leave it as an exercise, but for some strange reason none of them prove it, even though it's not difficult. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 9, 2015 at 14:36

0

Reset to default

You must log in to answer this question.