# Dictionary:Gauss’s theorem

The total flux ${\displaystyle \phi }$ through any closed surface is equal to ${\displaystyle 4\pi k}$ times the source strength m enclosed by the surface
${\displaystyle \phi =4\pi km=\iint {\textbf {g}}\cdot {\textbf {ds}}=\iint \nabla U\cdot {\textbf {ds}}}$
${\displaystyle =\iiint \nabla \cdot {\textbf {g}}dv=4\pi k\iiint \rho dv}$.
Here, ds is a vector surface element and dv a volume element. (The ${\displaystyle 4\pi k}$ 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 ${\displaystyle \rho }$, and the potential U. ${\displaystyle \phi }$ may be electrical flux if m is electrical charge. In the mks system, ${\displaystyle \phi }$ is in webers if m is in coulombs and ${\displaystyle k\approx 9\times 10^{9}}$, or ${\displaystyle \phi }$ may be gravitational flux if m is mass, in which case ${\displaystyle k=-\gamma }$, where ${\displaystyle \gamma }$ is the gravitational constant. Or ${\displaystyle \phi }$may be magnetic flux if m is magnetic pole strength.