Question

Use a computer algebra system to plot the vector field F(x, y, z) = sin(x) cos2(y)i + sin3(y) cos4(z)j + sin5(z) cos6(x)k in the cube cut from the first octant by the planes x = π/2, y = π/2, and z = π/2. Then compute the flux across the surface S of the cube.

Answer 1

By the divergence theorem,

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_{[0,\pi/2]^3}\mathrm{div}\vec F\cdot\mathrm dV$

We have

$\mathrm{div}\vec F(x,y,z)=\cos x\cos^2y+3\sin^2y\cos y\cos^4z+5\sin^4z\cos z\cos^6x$

Then the volume integral is

$\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{\pi/2}\mathrm{div}\vec F\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac{19\pi^2}{64}}$