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单词 AreaOfTheNsphere
释义

area of the n-sphere


The area of Sn the unit n-sphere (or hypersphereMathworldPlanetmath) is the same as the totalsolid angle it subtends at the origin. To calculate it, consider the followingintegral

I(n)=n+1e-i=1n+1xi2dn+1x.

Switching to polar coordinatesMathworldPlanetmath we let r2=i=1n+1xi2 and theintegral becomes

I(n)=Sn𝑑Ω0rne-r2𝑑r.

The first integral is the integral over all solid angles and is exactly what wewant to evaluate. Let us denote it by A(n). With the change of variablet=r2, the second integral can be evaluated in terms of the gamma functionDlmfDlmfMathworldPlanetmathΓ(x):

I(n)/A(n)=120tn-12e-t𝑑t=12Γ(n+12).

We can also evaluate I(n) directly in Cartesian coordinatesMathworldPlanetmath:

I(n)=[-e-x2𝑑x]n+1=πn+12,

where we have used the standard Gaussian integral -e-x2𝑑x=π.

Finally, we can solve for the area

A(n)=2πn+12Γ(n+12).

If the radius of the sphere is R and not 1, the correct area isA(n)Rn.

Note that this formula works only for n0. The first few special casesare

  • n=0

    Γ(1/2)=π, hence A(0)=2 (in this case, thearea just counts the number of points in S0={+1,-1});

  • n=1

    Γ(1)=1, hence A(1)=2π (this is the familiar resultfor the circumferenceMathworldPlanetmath of the unit circle);

  • n=2

    Γ(3/2)=π/2, hence A(2)=4π (this is thefamiliar result for the area of the unit sphere);

  • n=3

    Γ(2)=1, hence A(3)=2π2;

  • n=4

    Γ(5/2)=3π/4, hence A(4)=8π2/3.

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更新时间:2025/5/4 22:26:20