Dictionary:Gauss’s theorem

{{#category_index:G|Gauss’s theorem}} The total flux $\phi$ through any closed surface is equal to $4\pi k$ times the source strength m enclosed by the surface

$\phi =4\pi km=\iint {\textbf {g}}\cdot {\textbf {ds}}=\iint \nabla U\cdot {\textbf {ds}}$ $=\iiint \nabla \cdot {\textbf {g}}dv=4\pi k\iiint \rho dv$.

Here, ds is a vector surface element and dv a volume element. (The $4\pi k$ is often deleted in the mks system.) k is a constant that depends on the units of measure. This can also be expressed in terms of the flux density or field strength g, the source density $\rho$, and the potential U. $\phi$ may be electrical flux if m is electrical charge. In the mks system, $\phi$ is in webers if m is in coulombs and $k\approx 9\times 10^{9}$, or $\phi$ may be gravitational flux if m is mass, in which case $k=-\gamma$, where $\gamma$ is the gravitational constant. Or $\phi$may be magnetic flux if m is magnetic pole strength.

Also called Gauss's law: The equality between the surface and volume integrals involving g is also called the divergence theorem (q.v.). This theorem postulates the inherent nonuniqueness of potential fields.