Question

Use the Divergence Theorem to compute the net outward flux of the following field across the given surface S. F = < x^2, y^2, z^2 > ; S is the sphere {(x, y, z) : x^2 + y^2 + z^2 = 25}

Answer 1

$\vec F(x,y,z)=\langle x^2,y^2,z^2\rangle\implies\mathrm{div}\vec F(x,y,z)=2x+2y+2z$

By the divergence theorem,

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz$

where $R$ the region with $S$ as its boundary. Convert to spherical coordinates, taking

$\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

Then the volume integral is

$\displaystyle\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz$

$=2\displaystyle\int_0^{2\pi}\int_0^\pi\int_0^5(x+y+z)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta$

$=2\displaystyle\int_0^{2\pi}\int_0^\pi\int_0^5(\cos\theta\sin\varphi+\sin\theta\sin\varphi+\cos\varphi)\rho^3\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta=\boxed{0}$