For an orientable surface S and a fixed vector v, prove that...

Question

Prove that $$2\iint_S v\cdot n dS = \int_{\partial S}(v\times r) dS$$ where $r=(x,y,z)$ and $n$ is the unit normal vector for $S$.

Answer 1

The idea is to use Stokes theorem. Basically we want to show that : $$ rot (v \times r) = 2v $$ I will write $v = (v_1,v_2,v_3)^T$ $$ v \times r = (-v_3\ y+v_2\ z, v_3\ x-v_1\ z, -v_2\ x+v_1\ y)^T $$ Then apply the $rot$ operator and remembering that $v$ is constant : $$ \nabla \times (-v_3\ y+v_2\ z, v_3\ x-v_1\ z, -v_2\ x+v_1\ y)^T \overset{(*)}{=} (2v_1,2v_2,2v_3)^T $$ $(*)$ I'll let you do this boring but easy computation !

Putting our result and Stokes together : $$ \iint_{\Sigma} rot(v \times r) \cdot dn = \int_{\partial\Sigma} 2v\cdot dl $$