# Why does Gauss's law apply to any shape of a closed surface?

What seems to incredibly bother me is why Gauss's law applies to any shape of a closed surface. Moreover, the fact that the electric flux is proportional to the enclosed charge is by many sources simply proven by using a point particle that is enclosed by a sphere. Hence the electric flux is proportional to the enclosed charge for any closed surface, which to me isn't obvious by just using a specific proof that includes merely a sphere.
Furthermore I have watched various videos on youtube, including a lecture given by Walter lewin, and visited numerous websites, however, all sources didn't resolve my confusion. Lastly, why does Gauss's law also work for any collection of randomly distributed charges? Many sources state the any collection of charges can be thought of a collection of separate point charges, and since the electric fields of point charges should be added vectorially, they can also be be considered to be one total charge that generates one net electric field. Does this imply that the closed surface integral, used in Gauss's law, can be divided into separate closed surface integrals? Like so:

$$\int\sum\ \vec E\bullet d\vec A = \int\ \vec E1\bullet d\vec A +\int\ \vec E2\bullet d\vec A + \ldots +\int\ \vec Ei\bullet d\vec A$$ Where every seperate line integral includes the electric field of a single point charge.

Now my mathematical toolkit is relatively limited, thus, please do not utilize complicated mathematical equations that originate from divergence, differential equations, etc.

• It isn't clear to me how a law like Gauss's law can be "proved". Where do you want to start from? It's quite simple to show that Gauss's law holds for any configuration of charges if you believe Coulomb's law and the principle of superposition. So from where would you like such a proof to start? Also, remember that while the law is true always, it isn't always useful to calculate the field: that only happens when the charge configuration has some special symmetries. Oct 12, 2020 at 9:46