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 ball_volume [2013/09/08 18:05]nikolaj ball_volume [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/12 10:56 nikolaj 2013/09/09 16:58 nikolaj 2013/09/08 18:05 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:51 nikolaj 2013/09/08 17:51 nikolaj 2013/09/08 17:50 nikolaj 2013/09/08 17:50 nikolaj 2013/09/08 17:50 nikolaj 2013/09/08 17:49 nikolaj 2013/09/08 17:47 nikolaj created Next revision Previous revision 2013/09/12 10:56 nikolaj 2013/09/09 16:58 nikolaj 2013/09/08 18:05 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:52 nikolaj 2013/09/08 17:51 nikolaj 2013/09/08 17:51 nikolaj 2013/09/08 17:50 nikolaj 2013/09/08 17:50 nikolaj 2013/09/08 17:50 nikolaj 2013/09/08 17:49 nikolaj 2013/09/08 17:47 nikolaj created Line 1: Line 1: ===== Ball volume ===== ===== Ball volume ===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $p\in \mathbb N$ | + | @#55CCEE: context ​    | @#55CCEE: $p\in \mathbb N$ | - | @#FFBB00: $V_p:​\mathbb R_+\to \mathbb R_+$ | + | @#FFBB00: definiendum ​| @#FFBB00: $V_p:​\mathbb R_+\to \mathbb R_+$ | - | @#FFBB00: $V_p(r):​=\beta^p(B_0(r))$ | + | @#FFBB00: definiendum ​| @#FFBB00: $V_p(r):​=\beta^p(B_0(r))$ | ==== Discussion ==== ==== Discussion ==== Line 10: Line 10: For all $a\in \mathbb R^p$, the volume of the ball $B_a(r)$ is the same and given by For all $a\in \mathbb R^p$, the volume of the ball $B_a(r)$ is the same and given by - ^ $V_p(r)= \pi^{p/2}\ \Gamma(p/​2+1)^{-1}\ ​r^p = \frac{\pi}{2}^{p/​2}\ \Pi(p/​2) ​r^p$ ^ + ^ $V_p(r)= \pi^{p/2}\ \Gamma(p/​2+1)^{-1}\ r^p$ ^ === Reference === === Reference === Wikipedia: [[http://​en.wikipedia.org/​wiki/​Volume_of_an_n-ball|Volume of an n-ball]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Volume_of_an_n-ball|Volume of an n-ball]] ==== Parents ==== ==== Parents ==== - === Requirements === - [[Lebesgue-Borel measure]], [[Open ball]] === Context === === Context === - [[Archimedes'​ constant π]], [[Archimedes'​ constant τ]] + [[Lebesgue-Borel measure]], [[Open ball]] 