{"id":694,"date":"2014-01-21T14:39:10","date_gmt":"2014-01-21T19:39:10","guid":{"rendered":"http:\/\/www.curtisbright.com\/bln\/?p=694"},"modified":"2014-01-21T14:39:10","modified_gmt":"2014-01-21T19:39:10","slug":"volume-of-a-hypersphere","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2014\/01\/21\/volume-of-a-hypersphere\/","title":{"rendered":"Volume of a hypersphere"},"content":{"rendered":"

The volume of a hypersphere with radius $R$ in $n$ dimensions is given by the expression1<\/a><\/sup><\/p>\n

\\[ V_n(R) = \\frac{\\pi^{n\/2}}{(n\/2)!} R^n . \\]<\/p>\n

We will show this by induction on $n$. The base cases can be checked directly, where we make use of\u00a0polar coordinates<\/a> in two dimensions:<\/p>\n

\\begin{gather*}
\nV_1(R) = \\int_{-R}^R\\newcommand{\\d}{\\,\\mathrm{d}}\\d x = 2R = \\frac{\\pi^{1\/2}}{(1\/2)!} R \\\\
\nV_2(R) = \\iint\\limits_{x_1^2+x_2^2\\leq R^2}\\d x_2\\d x_1 = \\int_0^{2\\pi}\\int_0^R r\\d r\\d\\theta = 2\\pi\\biggl[\\frac{r^2}{2}\\biggr]_0^R = \\pi R^2
\n\\end{gather*}<\/p>\n

Suppose the formula holds in dimension $n-2$. Using this, we will show that the formula holds in dimension $n$:<\/p>\n

\\begin{align*}
\nV_n(R) &= \\int\\limits_{x_1^2\\leq R^2}\\;\\int\\limits_{x_1^2+x_2^2\\leq R^2}\\;\\int\\limits_{x_1^2+x_2^2+x_3^2\\leq R^2}\\dotsi\\int\\limits_{x_1^2+\\dotsb+x_n^2\\leq R^2}\\d x_n\\dotsm\\d x_1 \\\\
\n&= \\int\\limits_{x_1^2\\leq R^2}\\;\\int\\limits_{x_1^2+x_2^2\\leq R^2}\\;\\int\\limits_{x_3^2\\leq R^2-x_1^2-x_2^2}\\dotsi\\int\\limits_{x_3^2+\\dotsb+x_n^2\\leq R^2-x_1^2-x_2^2}\\d x_n\\dotsm\\d x_1 \\\\
\n&= \\int\\limits_{x_1^2\\leq R^2}\\;\\int\\limits_{x_1^2+x_2^2\\leq R^2}V_{n-2}\\Bigl(\\sqrt{R^2-x_1^2-x_2^2}\\Bigr)\\d x_2\\d x_1 \\\\
\n&= \\frac{\\pi^{n\/2-1}}{(n\/2-1)!}\\iint\\limits_{x_1^2+x_2^2\\leq R^2}\\sqrt{R^2-x_1^2-x_2^2}^{n-2}\\d x_2\\d x_1 \\\\
\n&= \\frac{\\pi^{n\/2-1}}{(n\/2-1)!}\\int_0^{2\\pi}\\int_0^R\\sqrt{R^2-r^2}^{n-2}r\\d r\\d\\theta \\\\
\n&= \\frac{2\\pi^{n\/2}}{(n\/2-1)!}\\biggl[-\\frac{1}{n}\\sqrt{R^2-r^2}^n\\biggr]_0^R \\\\
\n&= \\frac{\\pi^{n\/2}}{(n\/2)!} R^n
\n\\end{align*}<\/p>\n

By induction, the formula holds for all positive integers $n$.<\/p>\n

\n
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    \n
  1. As one might expect, the factorial with a noninteger argument is simply notation for the gamma function<\/a>, i.e.,\u00a0$n!:=\\Gamma(n+1)$. ↩<\/a><\/span><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    The volume of a hypersphere with radius $R$ in $n$ dimensions is given by the expression1 \\[ V_n(R) = \\frac{\\pi^{n\/2}}{(n\/2)!} R^n . \\] We will show this by induction on $n$. The base cases can be checked directly, where we … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/694"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=694"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/694\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=694"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=694"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}