What is wrong with this flux integral?

[james weaver]

I think the issue is how I parameterize my vector field, but not quite sure. In case you were wondering, this is problem # 27, chapter 16.7 of the 8th edition of Stewart. Thanks for any help.

 

[robphy]

Possibly helpful: https://www.physicsforums.com/help/latexhelp/

 

[Orodruin]

Your problem is that you add an additional $r$ in your surface element. This is presumably because you think that in polar coordinates you need to use $r\,dr\,d\theta$ for the area element, but the $r$ is already accounted for in the cross product $\vec r_r \times \vec r_\theta$. (Consider the flux of $\hat j$ through $y=0$ and $x^2+z^2 \leq 1$, which should clearly be -up to a sign depending on normal direction- $\int r \, dr\, d\theta$ over the same ranges of $r$ and $\theta$ as you have.)

Generally, the surface element as parametrized by $t$ and $s$ is given by
$$
d\vec S = \vec r_t \times \vec r_s \, dt\, ds
$$