I'm struggling with understanding the generalization of a solid angle to higher dimensions. What is the intuition behind a solid angle in general and how does it generalize in higher dimensions?
Edit: One thing that is not very clear to me:
For 3D space $d\Omega = \dfrac{dS}{r^2}$ so the solid angle is the ratio of the area subtended to the square of the radius, how can I write the n dimensional infinitesimal solid angle?
Best Answer
You generalize by generalizing the notation. Let $dS$ be the surface "area" $(n-1)$-form on $\Bbb R^n-\{0\}$ which gives the "area" on the unit sphere. And then you set $d\Omega = \dfrac{dS}{r^{n-1}}$ as an $(n-1)$-form on $\Bbb R^n-\{0\}$. Integrating this over a chunk of hypersurface in $\Bbb R^n$ gives the solid angle it subtends. (And yes, one can write down explicit formulas.)