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it is often written in books that the solid angle [itex]\Omega[/itex] subtended by an oriented surface patch can be computed with a surface integral:

[tex]\Omega = \int\int_S \frac{\mathbf{r}\cdot \mathbf{\hat{n}} }{|\mathbf{r}|^3}dS[/tex]

whereris the position vector for the patch dS andnits normal (see also wikipedia).

However I would like to know how to derive this formula from the definition of solid angle, that is:the area of the the projection of a surface on the unit sphere.

I can already see that:

[tex]\frac{\mathbf{r}}{|\mathbf{r}|} \cdot \mathbf{\hat{n}} dS = cos(\theta)dS[/tex]

where [itex]\theta[/itex] is the angle between the position (unit)-vector for dS and the normal vector for dS

Unfortunately I don't understand where that [itex]|\mathbf{r}|^{-2}[/itex] comes from.

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# Solid angle question

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