Gauss Law in Arbitrary dimensions

Question

What is the Gauss law in any arbitrary dimension "n" and how can one derive it?

Answer 1

Whether by Gauss law, you mean the dependence of the electrostatic (gravitystatic) potential $\phi(x)$, created by some source on the distance from the source (located at the origin), then you start from the equation for the Green's function of the Laplace operator $\Delta$ for $x \in \mathbb{R}^{n}$: $$ \Delta \phi(x) = q \delta(x) $$ After the Fourier transform one gets: $$ k^2 \phi(k) = q \Rightarrow\phi(x) = \int \frac{d^{n} k}{(2 \pi)^{n}} \frac{q \ e^{i kx}}{k^2} $$ This integral in dimensions $n \geqslant 3$ gives a power law decay for the potential: $$ \phi(x) = \frac{C q}{|x|^{n-2}} \qquad C = const $$ For the $n = 2$ case the Green function gives a logarithm: $$ \phi(x) = C q \ln\frac{|x|}{|x_0|}\qquad C = const $$ And for the $n=1$ case ($\theta(x)$ is the Heavyside function): $$ \phi(x) = q x \theta(x) $$ The static for general distribution of charges follows from the principle of superposition.

Answer 2

The general law is called Stokes' theorem, https://en.wikipedia.org/wiki/Stokes%27_theorem. In three dimentions, Stokes' theorem has two special forms: the Stokes–Cartan theorem, https://en.wikipedia.org/wiki/Kelvin%E2%80%93Stokes_theorem, and Gauss's theorem, https://en.wikipedia.org/wiki/Divergence_theorem.