8. Hyperspheres

As reported in wikipedia:  http://en.wikipedia.org/wiki/Hypersphere, a hypersphere is the set of points at a constant distance from a given point called its center.

The "volume" of a hypersphere is given by:

 

where Γ denotes the gamma function.

The "surface area" is instead given by:

 

In three dimensions, the formula for the calculation of volume is:

 

As for the surface is:

For the definition of derivative, as the limit of the quotient, we have:

the surface is the derivative of the volume.

And this in each dimension:     D (V_n) = S_n-1

 

Their relationship in every dimension is remarkably simple:

    V_n / S_n-1 = R / n       e.g.  in  3  dimensions we have:  V₃ / S₂ = R / 3

 

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

For the calculation of hypervolumes and hypersurfaces, formulas contain:

in 2 or 3 dim. appears π,  while in 4 or 5 dim. π²,  6 or 7 dim. π³  and so on.
 

As an example for even number of dimensions, we have:

 

while for odd number of dimensions: