Surface Integral for Mass and Moments

Given the density function $G(x,y,z)$ on a surface S, the total mass of the surface is given by

$$\iint \_SG(x,y,z)d\sigma$$

with $d\sigma$ being the Surface Integral Differential

Surface (Flux) Integral

With $\vec{n}$ being the normal unit vector to the surface $S$

$$\iint \_SF\cdot \vec{n}d\sigma$$

Gives the flux of the field $F$ across $S$

Mass and Moments

The following integrals give the mass and moment for thin shells i.e. surfaces $M=\iint \_S\delta d\sigma$ $M\_yz=\iint \_Sx\delta d\sigma$ $I\_L=\iint \_Sr(x,y{)}^2\delta \sigma$ with r being distance to the line/axis

Orientation

If the direction of $r\_u\times r\_v$ is different from $-(r\_u\times r\_v)$ such that we can pick one, the surface is orientable. A Möbius band is not orientable