Volume and surface of an n-dimensional sphere ↩

aliases: [Volume of an n-dimensional sphere, Surface area of an n-dimensional sphere]
shorthands: {}
created: 2022-02-22 01:35:31
modified: 2022-04-21 15:37:05

Where the coefficient is the following:

Where is the Gamma function.

Notes

• In high dimensions, most of the volume of the sphere is contained in its outermost layers

This has a significance in statistical mechanics, because in the high dimensional phase space, most of the possible states are the highest energy ones

Examples

1D

Just a line segment:

2D

Circle:

3D

Normal sphere:

4D

4D hypersphere:

5D

Derivation

In dimensions, the volume will be dependent on and it can also be written in integral forms:

While the surface is the derivative of the volume formula with respect to :

What we want to calculate is . For that we need to realize that we can express the -dimensional Gaussian integral in two ways.

1. In the first one we just realize that the integrals are independent, so the solution is the the solution to the one dimensional case raised to the power of :

2. Then we can also express it with spherical coordinates:

We substitute , so :

Where is the Gamma function.