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 (Vn) = Sn-1
Their relationship in every dimension is remarkably
simple:
Vn / Sn-1 = R / n e.g. in 3 dimensions we
have: V3 / S2 = R / 3
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:
