Gaussian surface

For a closed surface flux integral, especially for Gauss’ law:

$${\iint }_S\vec{F}\cdot dS$$

where $S$ is closed, we sometimes call $S$ a Gaussian surface. This corresponds to the left-hand side of the divergence theorem.

Choosing a surface

On any segment of the closed surface, one of the following three conditions should be satisfied:

• $\vec{F}\perp {\vec{n}}_{S}d\vec{S}$
• $\vec{F}=0$
• Condition 1 and $\vec{F}\cdot {\vec{n}}_S$ is uniform on the surface.

Our simplest surfaces are cylinders, spheres, and planes.

See also

• Amperian loop, the Stokes’ theorem analogue