Surface area of a d-sphere

  • #1
15
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I'm trying to follow Schwabl Thermodynamics, and I found the following equation for the surface area of a unit d-sphere:
$$ \int d\Omega_d = \frac{2 \pi^{d/2}}{\Gamma(d/2)} $$

But this formula clearly fails for d=1:
should be $$\pi$$
and d=2:
should be $$ 4 \pi $$. What gives?
 

Answers and Replies

  • #3
15
0
yea, you're right. Schwabl must be counting the dimension its embedded in or something.
 
  • #4
HallsofIvy
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Homework Helper
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It is the difference between a "sphere" and a "ball". A "2-ball" is the a two dimensional disk, which might have equation [tex]x^2+ y^2\le r^2[/tex], while the "2-sphere" is the surface of a "3-ball" and might have equation [tex]x^2+ y^2+ z^2= r^2[/tex]
 

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