On Jun 20, 4:00*am, "yogesh_gharote" <yogesh_ghar...@yahoo.com> wrote:
<<<material snipped>>

> * * * * *Remarkably for a n-sphere as it is generally called , u get hyper
> surface area = derivative of hypervolume , again theory of beta n gamma
> functions is involved in this , so once u know the volume of a hypersphere
> , u differentiate it wrt r to get hypersurface area.
> * * * * * The formula for a n-sphere hypervolume is
> * * * * *Vn(r) = (gamma(1/2))^n * r^n/ gamma(1/2*n +1)
> &
> * * * * *Sn(r) = d/dr(Vn(r))


The volume of a sphere in n-dimensional space is
proportional to r^n, and, as Yogesh states, we have
to use the theory of beta and gamma functions to
find the value of the constant of proportionality. But
the basic result asked for by Tim and stated as

Sn(r) = d/dr(Vn(r))

above should be intuitively obvious even if one does
not know the value of the constant of proportionality.
Consider the thin shell of thickness dr formed by
scooping out the sphere of radius r from the inside of
of a sphere of radius r+dr. The volume of this shell
is Vn(r+dr) - Vn(r). But since the shell has surface
area approximately Sn(r) and thickness dr, its volume
is approximately Sn(r)dr. Equate the two expressions,
divide both sides by dr, take limits as dr goes to 0.
Finicky epsilon-delta folks are advised to use the
approach once espoused by Julius Kusuma in
this newsgroup ("The most rigorous arguments shall
be proved by the most vigorous hand waving") as
needed to establish the result.

Incidentally, I don't understand what Tim meant when
he wrote

>integrate the volume of the sphere that forms the
>surface of the hypersphere as one of the dimensions
>varies from -r to r

The antiderivative of 2\pi r is \pi r^2, the antiderivative
of 4\pi r^2 is (4/3)\pi r^3, and so on and so forth, and
there is no need to let "one of the dimensions" vary
from -r to + r.

Hope this helps

--Dilip Sarwate