Question

Verify that the Divergence Theorem is true for the vector field F on the region E. F(x, y, z) = z, y, x , E is the solid ball x2 + y2 + z2 ≤ 64 For your answer, put in the flux across the boundary of E with respect to the outward normal.

Answer 1

$\vec F(x,y,z)=\langle z,y,x\rangle$

has divergence 1, so by the divergence theorem, the flux of $\vec F$ across the boundary of $E$ is exactly the volume of $E$,

$\displaystyle\iiint_E\mathrm dV=\int_0^\pi\int_0^{2\pi}\int_0^8\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

or simply $\dfrac43\pi8^3=\boxed{\frac{2048\pi}3}$.