Definition:Solid Angle/Subtend

Definition
Let $S$ be a surface oriented in space.

Let $P$ be a point in that space.

The solid angle subtended by $S$ at $P$ is equal to the surface integral:


 * $\ds \Omega = \iint_S \frac {\mathbf {\hat r} \cdot \mathbf {\hat n} \rd S} {r^2}$

where:
 * $\mathbf {\hat r} = \dfrac {\mathbf r} r$ is the unit vector corresponding to the position vector $\mathbf r$ of the infinitesimal surface $\d S$ with respect to $P$
 * $r$ is the magnitude of $\mathbf r$
 * $\mathbf {\hat n}$ represents the unit normal to $\d S$.

Also presented as
This can also be presented as:


 * $\ds \Omega = \iint_S \frac {\mathbf {\hat r} \cdot \rd \mathbf S} {r^2}$

where in this case $\d \mathbf S$ is the vector area of the infinitesimal surface $\d S$.