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Surface area of a d-sphere

  1. Oct 1, 2014 #1
    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?
  2. jcsd
  3. Oct 2, 2014 #2


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  4. Oct 2, 2014 #3
    yea, you're right. Schwabl must be counting the dimension its embedded in or something.
  5. Oct 4, 2014 #4


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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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