Question

Let f = (ax + by + 4z) i + (x + cz) j + (9y + mx) k where a, b,c, and m are constants.

Answer 1

Let $\mathcal R$ be an arbitrary closed region with boundary the surface $\mathcal S$. By the divergence theorem,

$\displaysytle\iint_{\mathcal S}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_{\mathcal R}\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$

We have

$\nabla\cdot\mathbf f(x,y,z)=\dfrac{\partial(ax+by+4z)}{\partial x}+\dfrac{\partial(x+cz)}{\partial y}+\dfrac{\partial(9y+mx)}{\partial z}=a$

so that the flux satisfies

$\displaystyle a\iiint_{\mathcal R}\mathrm dV=0$

We're assuming $\mathcal R$ is a closed region, and the integral above is its volume, which must be positive. This means we must have $a=0$.