A First Course in String Theory
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Show that
~~~
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Geometric proof
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
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— Me@2022-05-18 09:08:11 AM
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2022.05.19 Thursday (c) All rights reserved by ACHK