[tex]\vec F(x,y,z)=\langle x^2,y^2,z^2\rangle\implies\mathrm{div}\vec F(x,y,z)=2x+2y+2z[/tex]
By the divergence theorem,
[tex]\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[/tex]
where [tex]R[/tex] the region with [tex]S[/tex] as its boundary. Convert to spherical coordinates, taking
[tex]\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[/tex]
Then the volume integral is
[tex]\displaystyle\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]
[tex]=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[/tex]
[tex]=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}[/tex]
In this exercise we have to use the divergent theorem to calculate the flow of the given equation, so we will find that:
[tex]\int\limits \int\limits \int\limits_R {divF(x,y,z)} \, dx dy dz= 0[/tex]
So from the given equation, we will find that:
[tex]\int\limits \int\limits_S {F} \, ds = \int\limits \int\limits \int\limits_R {div F(x, y, z) } \, dx dy dz[/tex]
where [tex]R[/tex] the region with [tex]S[/tex] as its boundary. Convert to spherical coordinates, taking:
[tex]\left[\begin{array}{c}x= \rho cos(\theta) sin(\phi) \\y= \rho sin(\theta) sin(\phi) \\z= \rho cos (\phi) \end{array}\right[/tex]
Then the volume integral is:
[tex]\int\limits \int\limits \int\limits_R {divF(x,y,z)} \, dxdydz\\= 2 \int\limits^{2\pi}_0 \int\limits^{\pi}_0 \int\limits^{5}_0 {(x+y+z)\rho ^2 sin(\phi) d(\rho) d(\phi) d(\theta)= 0[/tex]
See more about Divergence Theorem at brainly.com/question/6960786
