Ratio of Surface to Volume for an n dimensional sphere

 The problem is to find the surface area and volume of an n dimensional sphere.  See these references:

1. Statistical Mechanics, by R K Pathria, (2nd Edition) Elsevier Publication, Appendix-C, page 504

2. Infinite Powers: The story of Calculus, The language of the Universe, by Steven Strogatz, Atlantic Booka, London,

The volume element 

\[ V_n = \int \int \cdots \int \prod_{i=1}^{n} dx_i \]

here \( x_1, x_2, x_3 , x_4,  x_5 \cdots x_n  \) etc are orthogonal cartesian coordinates components. and its understood that the sphere is defined by and the limit of integration is contrained via

\[ 0 \le \sum_{i=1}^n x_i^2  \le R^2\]

here R is the radius of the n dimensional sphere, whose volume and surface we want to find.

The volume would be proportional to \(  R^n \) 

\[ V_n = C_n R^n \]

here \( C_n \) is constant not function of R and it depends on geometry of the space. I shall remember that in 2D, surface area means, perimeter of circle and volume means area of the circular disk, as the 2 dimensional sphere is actually a disc. Imagine that you are in a planet where everything is projection or shadow of 3D object. Thus a football will look like a disc.

Also \[ dV_n = n C_n R^{n-1} dR = S_n (R) dR\]

From sem-3, Gamma function integral,

\[ \int_\infty^\infty e^{-x^2}dx = \sqrt{\pi} \]

Thus 

\[ \pi^{n/2} =  \int_\infty^\infty  \int_\infty^\infty \cdots  \int_\infty^\infty exp\left( x_1^2  \right) exp\left( x_2^2 \right)  exp\left( x_3^2 \right)  \cdots exp\left( x_n^2 \right)  \prod_{i=1}^{n} dx_i  \]

\[ \pi^{n/2} =  \int_\infty^\infty  \int_\infty^\infty \cdots  \int_\infty^\infty  exp\left( x_1^2 +x_2^2 + \cdots + x_n^2  \right)   \prod_{i=1}^{n} dx_i  \]

\[ \pi^{n/2} =  \int_\infty^\infty exp\left( r^2 \right)   dV_n  \]

\[ \pi^{n/2} =  \int_\infty^\infty exp\left( r^2 \right)   n C_n r^{n-1} dr  \]

\[ \pi^{n/2} = n C_n  \int_\infty^\infty exp\left( r^2 \right)  r^{n-1} dr  \]

\[ \pi^{n/2} = \frac{n}{2} C_n  \Gamma(\frac{n}{2}) \]

 You should get \( C_n \) from the above equation. 

 The final results are:

For surface area:

\[ S_n = \frac{2 \pi^{n/2} R^{n-1}}{\Gamma(n/2)}  \]

For Volume:

\[ V_n = \frac{ \pi^{n/2} R^{n}}{(n/2)!}  \]

And one interesting result is 

\[ \frac{S_n}{V_n} = \frac{n}{R} \]

 

**From this post, I have learnt how to write LaTeX in Blog html format.