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Hypersphere in 4 dimensions, I am having problem with finding the surface area of it. please help.
I know that surface area will have 3 dimensions in 4 dimensional space, I am having trouble to evaluate it
Circumference of circle= $2\pi r$
Surface area of sphere = $4\pi r^2$
Now, Surface area (or whatever, it will be having dimensions of a volume) of hypersphere = ??
The Wikipedia article gives the volume of the 4-dimensional hypersphere, with radius $r$, to be $$V = \frac{\pi^2}{2}r^4$$
The surface area can be found by differentiating with respect to $r$:
$$A = \frac{\mathrm{d}V}{\mathrm{d}r} = 2\pi^2r^3$$
The article also gives a proof of how to calculate the volume, and hence surface area.
Another little known way to get this answer is to double integrate $\sqrt{1 + \nabla f \cdot \nabla f}$, where $f = \sqrt{R^2 - x^2 - y^2}$.
For hyper "areas" of geometric objects in $n$ dimensions, just integrate the above formula $n-1$ times, where $f = f (x_1, x_2, ...,x_{n-1})$.
