3

Use the Divergence Theorem to compute the net outward flux of:

$$ F = \langle x^2, y^2, z^2 \rangle $$ $S$ is the sphere: $$ \{(x,y,z): x^2 + y^2 + z^2 = 25\} $$

First, I took: $$ \nabla \cdot F = 2x + 2y + 2z $$

Then, I tried setting up the triple integral with spherical coordinates, but it is just not working out for me.

Michael Hardy
  • 1
dendritic
  • 315

1 Answers1

2

Converting to spherical coordinates, we have

$$\nabla \cdot \vec F=2(x+y+z)=2(r\sin \theta \cos \phi+r\sin \theta \sin \phi +r\cos \theta)$$

Thus,

$$\int_V \nabla \cdot \vec F dV=\int_0^{2\pi}\int_0^{\pi}\int_0^5 2(r\sin \theta \cos \phi+r\sin \theta \sin \phi +r\cos \theta)\,r^2\sin \theta \,dr\,d\theta\,d\phi=0$$

since $\int_0^{2\pi}\sin \phi d\phi=\int_0^{2\pi}\cos \phi d\phi=0$ and $\int_0^{\pi}\cos \theta \sin \theta d\theta=0$

Mark Viola
  • 179,405
  • So you just simply had to substitute the x, y, and z spherical coordinates.. It seems relatively simple, yet I'm always struggling with this section. Thank you. – dendritic Aug 07 '15 at 01:46
  • You're very welcome. It was my pleasure. As a suggested exercise, see if you can recover the result using the surface integral. – Mark Viola Aug 07 '15 at 01:47
  • Sure, will definitely try that out! – dendritic Aug 07 '15 at 01:48