A First Course in String Theory
The relations and and thus the volumes of -balls and areas of -spheres can also be derived geometrically. As noted above, because a ball of radius is obtained from a unit ball by rescaling all directions in times, is proportional to , which implies .
Also, because a ball is a union of concentric spheres and increasing radius by corresponds to a shell of thickness . Thus, ; equivalently, .
— Wikipedia on Volume of an -ball
— Me@2022-05-18 09:08:11 AM
2022.05.19 Thursday (c) All rights reserved by ACHK