While studying the famous Ampere's law, I came up with the following vector field $F$ and a surface $S$ lying in $R^3$. (In terms of physics, $F$ is the current density of some current in a circuit, and $S$ is the surface)
$F$ is zero everywhere except at $(0,0,0)$, where $F(0,0,0)=(0,0,c)$.
$S$ is a piecewise plane, which abruptly changes like a step function at $x=0$.
Specifically,
$$ F(x,y,z) = \cases{ (0, 0, c) & $(x, y, z) = (0, 0, 0)$ \cr (0, 0, 0) & otherwise} $$
$$ S(t,y) = \cases{ (t,y,0) & $t< 0$ \cr (0,y,t) & $0\le t < 1$ \cr (t-1, y, 1) & $1\le t$. } $$
Then here's my question;
- What is the flux of F on S? That is, $\oint_S \overline{F}\cdot d\overline{s}$?
I am inclined to say that it is undefined, as S is not differentiable at the point of interest, $(t=0,y=0)$.
However, if that is the case, what about other non-smooth surfaces?
When applying Gauss's law, we often set up a "piecewise" Gauss surface, for example, a cylinder surface. Then we calculate the flux for the top, bottom and the side surfaces, and add them up.
- In this sense, for the first problem I proposed, shouldn't I say that the total flux is $c+c=2c$? One c for each plane $t<0$ and $1\le t$?
Or is it just that choosing such a non-smooth surface as a Gauss surface is not a correct thing to do in a rigorous sense?
Thank you in advance.
P.S. Could you recommend what material or topics I should look into to understand the related concepts? I want to understand electromagnetism under rigorous mathematics, but it has been hard for me to find good materials/explanations.