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'area of the $n$-sphere'
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| Title of object: |
area of the $n$-sphere |
| Canonical Name: |
AreaOfTheNSphere |
| Type: |
Derivation |
| Created on: |
2003-07-23 02:20:18 |
| Modified on: |
2004-02-13 11:39:08 |
| Classification: |
msc:51M05 |
Revision comment (for changes between this and next version):
| Changes for correction #4320 ('d^nx?'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
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% define commands here
\def\sse{\subseteq}
\def\bigtimes{\mathop{\mbox{\Huge $\times$}}}
\def\impl{\Rightarrow}
\def\R{\mathbb{R}} |
Content:
The area of $S^n$ the unit $n$-sphere (or hypersphere) is the same as the total
solid angle it subtends at the origin. To calculate it, considere the following
integral
I(n) = \int_{\R^{n+1}} e^{-\sum_{i=1}^{n+1} x_i^2}\, d^n x.
Switching to polar coordinates we let $r^2=\sum_{i=1}^{n+1} x_i^2$ and the
integral becomes
I(n) = \int_{S^n} d\Omega \int_{0}^{\infty} r^{n} e^{-r^2}\, dr.
The first integral is the integral over all solid angles and is exactly what we
want to evaluate. Let us denote it by $A(n)$. The second integral can be
evaluated with the change of variable $t=r^2$:
I(n)/A(n) = \frac{1}{2}\int_0^\infty t^{\frac{n-1}{2}} e^{-t}\, dt
= \frac{1}{2}\Gamma\left(\frac{n+1}{2}\right).
We can also evaluate $I(n)$ directly in Cartesian coordinates:
I(n) = \left[ \int_{-\infty}^\infty e^{-x^2}\, dx \right]^{n+1}
= \pi^{\frac{n+1}{2}},
where we have used the standard Gaussian integral $\int_{-\infty}^\infty
e^{-x^2}\, dx = \sqrt{\pi}$.
Finally, we can solve for the area
\[ A(n) = \frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}. \]
If the radius of the sphere is $R$ and not $1$, the correct area is
$A(n)R^{n}$.
Note that this formula works only for $n\ge0$. The first few special cases
\begin{itemize}
\item[$n=0$] $\Gamma(1/2)=\sqrt{\pi}$, hence $A(0)=2$ (in this case, the
area just counts the points number of points in $S^0=\{+1,-1\}$);
\item[$n=1$] $\Gamma(1)=1$, hence $A(1)=2\pi$ (this is the familiar result
for the circumference of the unit circle);
\item[$n=2$] $\Gamma(3/2)=\sqrt{\pi}/2$, hence $A(2)=4\pi$ (this is the
familiar result for the area of the unit sphere);
\item[$n=3$] $\Gamma(2)=1$, hence $A(3)=2\pi^2$;
\item[$n=4$] $\Gamma(5/2)=3\sqrt{\pi}/4$, hence $A(4)=8\pi^2/3$.
\end{itemize} |
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