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Find a vector field $v$ on $\mathbb{R}^n$ with wich you can calculate the volume of every open subset with a smooth edge $\Omega\subset \mathbb{R}^n$ using the flow of the vector field through the edge $\partial \Omega$.

Can someone help me with this. I have a feeling that this should be kind of easy, but I cant get it...

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  • $\begingroup$ Do you know the divergence theorem/stokes theorem in $n$ dimensions? $\endgroup$
    – muaddib
    Jul 9, 2015 at 16:56
  • $\begingroup$ We only did it for 3 dimensions.. $\endgroup$
    – kaos
    Jul 9, 2015 at 16:58
  • $\begingroup$ Is this a question from the class, or something from self study (trying to determine what you can use). $\endgroup$
    – muaddib
    Jul 9, 2015 at 17:00
  • $\begingroup$ Its from the class. But we are pretty much left on our own with the problems. So its not unusual to have to use something we didnt do in class in order to solve the problems. $\endgroup$
    – kaos
    Jul 9, 2015 at 17:05
  • $\begingroup$ Well I suggest solving it in three dimensions with the Divergence theorem and then thinking about how that solution works in $n$ dimensions with "Stokes theorem$ applied to manifolds with boundaries. Maybe there is some way I am not thinking of that doesn't require that machinery. $\endgroup$
    – muaddib
    Jul 9, 2015 at 17:07

1 Answer 1

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Any field with divergence identically equal to $1$ will work. The are infinitely many such fields of the form $$ ax\vec \imath+by\vec\jmath + cz\vec k $$ where $a+b+c=1$. Some are a little nicer than others.

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