Is it possible to apply Gauss' law to a Klein bottle?

[TensorCalculus]

TL;DR

Asking about the possibility of applying gauss's law to a klein bottle, topology, differential geometry, parametric equations, electromagnetism

So, I was just doing some practice on Gauss's law, and most of the questions, when I needed to take the surface integral of something, it would be something simple, like a sphere, cylinder or at worst a torus.
Though it's impractical (and probably useless) - it got me wondering, what would happen if I tried to apply Gauss's law to a Klein bottle?

Obviously, this wouldn't work in 3D, the bottle doesn't really enclose a volume in 3D, and I don't even want to think what trying to define a normal vector would look like, but suppose I embed it in a 4d hyperspace, and then take a 3D integral rather than the 2D surface one, would it be possible?
I did a bit of reading around, and people said this would be the equations for a Klein bottle in 4D (for simplicity, v = 1, u = 1):

using this, and some geometry tricks to find a defined volume for said bottle in 4D, would it be feasible to apply gauss's law to the Klein bottle? What about defining the vector field, could I just utilize the electric Tensor in a similar fashion to how you would do in 3D? If so, where should I start and how do you think I could go about achieving this?

NOTE: I'm new here so if there's any more info you want me to provide/I need to do something I haven't done do please tell me :) Also if anyone can tell me how to insert LaTeX that would be much appreciated! This question is completely out of curiosity/ for fun by the way! I wasn't sure whether to put it beginner (probably not...?), intermediate or advanced so I've put it in intermediate for the time being, do tell me if I should move it!

 

[jedishrfu]

In my signature below there's a Link to our site Mathjax/Latex help guide that can help you.

$x = (R+r\cos(u))\cos(v)$
$y = (R+r\cos(u))\cos(v)$
$z = R \sin(u) \cos(\frac{v}{2})$
$w = R \sin(u) \sin(\frac{v}{2})$

%% x = (R+r\cos(u))\cos(v) %%
%% y = (R+r\cos(u))\cos(v) %%
%% z = R \sin(u) \cos(\frac{v}{2}) %%
%% w = R \sin(u) \sin(\frac{v}{2}) %%

replace all % with a #
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