Home >> Science & Nature >> Nature

How to Derive the Integral Volume of a Hypersphere

Just a circle is the set of all points in a two-dimensional plane equidistant from a central point and a sphere is the set of all points in three dimensions equidistant from a central point, in mathematics there exist analogous structures, called hyperspheres, in dimensional spaces greater than three that are the set of all points equidistant from a central point. Consequently, just as the the integral volume of a sphere in three dimensions can be derived with calculus, so can the integral volumes of these higher-dimensional figures.

Instructions

- 1
  Define the coordinate system that will be used in the problem. Though any coordinate system can be made to work, a variation on spherical polar coordinates works best. As an example, in an n-dimensional space, define r as the distance to the center point, theta as the azimuthal angle and phi1, phi2, ... phi(n-2) as angular coordinates ranging from 0 to pi radians.
- 2
  Write out the basic volume integral over the entire hypersphere. This will be the integral from 0 to some radius R for r, and over the entirety of the possible angles for each angular coordinate, 0 to 2pi for theta and 0 to pi for the remaining variables. The multiple integrals are taken of 1 across the volume element.
- 3
  Replace the volume element with the appropriate terms computed from the Jacobian determinant. For example, for a hypersphere in four dimensions, it will be:
  
  r^3 sin^2(phi1) sin(phi2) dr dphi1 dphi2 dtheta.
  
  For more help computing the Jacobian, see the appropriate resource link.
- 4
  Write down the final answer after taking each integral in succession. In our example of the four-dimensional hypersphere the final answer is:
  
  (pi^2 / 2) * radius^4.

Nature