I am new to calculus and I am puzzled at the following question. I will explain where my thoughts are heading, and any feedback would be really appreciated.
Find the Flux of field $(-y,x,5)$, which leaves the surface of sphere $x^2+y^2+z^2=25$ below $z=4$ (with "leaving" we mean "moving away from the origin").
Naturally, I would want to use the formula $Φ=\iint_σF·\mathbf ndS=\iiint_GdivFdV$, but when I calculate the divergence of formula $(-y,x,5)$, I get $0$, which makes little sense. Could anybody shed light on why this is, where my mistake is, and how to proceed?
Many thanks!
You are almost there, but unfortunately you've missed a crucial element. When are you supposed to use the formula with divergence to calculate the flux? The surface has to enclose a volume. In your case, the flux is leaving the sphere of radius $5$, centered at the origin, but only below $z=4$. That means only a part of the spherical shell (does not completely enclose a volume). But, like I've said, not all is lost. Let's complete the surface by adding the disk at $z=4$. Now we know that the total flux is $0$, But whatever goes out through the spherical part MUST be going in through the part that we added. Now the integral of the flux through the disk is trivial, since the normal is $(0,0,1)$. Then $\mathbf F\cdot\mathbf n=5$. The disk has radius $3$. The surface of the disk is then $\pi3^2$, and when you multiply with $5$ you get $45\pi$. Now all you need is to figure out the sign.