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...
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.