Let f = (ax + by + 4z) i + (x + cz) j + (9y + mx) k where a, b,c, and m are constants.a.suppose that the flux of f through any closed surface is 0. which of the constants can be determined?

Let [tex]\mathcal R[/tex] be an arbitrary closed region with boundary the surface [tex]\mathcal S[/tex]. By the divergence theorem,

[tex]\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[/tex]

We have

[tex]\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[/tex]

so that the flux satisfies

[tex]\displaystyle a\iiint_{\mathcal R}\mathrm dV=0[/tex]

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