91Ó°ÊÓ

An incompressible fluid is contained within a volume \(V\) with surface \(S\) and \(\boldsymbol{u} \cdot \boldsymbol{n}=0\) on \(S\). Using the divergence theorem, show that $$ \iiint_{V} \boldsymbol{u} \cdot \boldsymbol{\nabla} \phi d V=0 $$ for any differentiable scalar field \(\phi\).

The magnetic field \(B\) in an electrically conducting fluid moving with velocity \(u\) obeys the magnetic induction equation $$ \frac{\partial \boldsymbol{B}}{\partial t}=\boldsymbol{\nabla} \times(\boldsymbol{u} \times \boldsymbol{B}) $$ Show that the total flux of magnetic field through a surface enclosed by a streamline of the flow (a closed curve which is everywhere parallel to \(\boldsymbol{u}\) ) is independent of time.