area of the -sphere
The area of the unit -sphere (or hypersphere) is the same as the totalsolid angle it subtends at the origin. To calculate it, consider the followingintegral
Switching to polar coordinates we let and theintegral becomes
The first integral is the integral over all solid angles and is exactly what wewant to evaluate. Let us denote it by . With the change of variable, the second integral can be evaluated in terms of the gamma function:
We can also evaluate directly in Cartesian coordinates:
where we have used the standard Gaussian integral .
Finally, we can solve for the area
If the radius of the sphere is and not , the correct area is.
Note that this formula works only for . The first few special casesare
-
, hence (in this case, thearea just counts the number of points in );
-
, hence (this is the familiar resultfor the circumference
of the unit circle);
-
, hence (this is thefamiliar result for the area of the unit sphere);
-
, hence ;
-
, hence .