Question

Let F(x, y, z) = 3xi+ 2yj and let σ be the cube with opposite corners at (0, 0, 0) and (5, 5, 5), oriented outwards. Find the flux of the flow field F across σ.

Answer 1

Use the divergence theorem,

$\displaystyle\iint_{\partial\sigma}\mathbf F(x,y,z)\cdot\mathrm d\mathbf S=\iiint_\sigma\mathrm{div}\mathbf F(x,y,z)\,\mathrm dV$

We have

$\mathrm{div}\mathbf F(x,y,z)=\dfrac{\partial(3x)}{\partial x}+\dfrac{\partial(2y)}{\partial y}+\dfrac{\partial0}{\partial z}=5$

so that the flux across $\sigma$ is equal to 5 times the volume of the cube. The cube itself has edge length 5, so its volume is $5^3=125$, making the flux $5^4=\boxed{625}$.