What is the difference between a sphere and a ball?

[onanox]

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?

 

[mathman]

You have a slight error. The formula is for d-1 dimensional sphere, not d dimensional.

http://en.wikipedia.org/wiki/N-sphere

 

[onanox]

yea, you're right. Schwabl must be counting the dimension its embedded in or something.

 

[HallsofIvy]

It is the difference between a "sphere" and a "ball". A "2-ball" is the a two dimensional disk, which might have equation x^2+ y^2\le r^2, while the "2-sphere" is the surface of a "3-ball" and might have equation x^2+ y^2+ z^2= r^2