hypersphere_integrals


hypersphere_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

The surface of the unit hypersphere in M dimensions is defined by

        sum ( 1 <= i <= m ) x(i)^2 = 1
      

The integrands are all of the form

        f(x) = product ( 1 <= m <= m ) x(i) ^ e(i)
      
where the exponents e are nonnegative integers. If any exponent is an odd integer, the integral will be zero. Thus, the "interesting" results occur when all exponents are even.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

hypersphere_integrals is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

ball_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

circle_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

cube_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

disk01_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

hexagon_integrals, a MATLAB code which returns the exact value of the integral of a monomial over the interior of a hexagon in 2D.

hyperball_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

hypercube_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

hypersphere_integrals_test

hypersphere_properties, a MATLAB code which carries out various operations for an M-dimensional hypersphere, including converting between Cartesian and spherical coordinates, stereographic projection, sampling the surface of the sphere, and computing the surface area and volume.

hypersphere_monte_carlo, a MATLAB code which uses the Monte Carlo method to estimate the integral of a function over the surface of the unit hypersphere in M dimensions.

line_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

polygon_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

pyramid_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

simplex_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

sphere_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

square_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

tetrahedron_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

triangle_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

wedge_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

Reference:

  1. Gerald Folland,
    How to Integrate a Polynomial Over a Sphere,
    American Mathematical Monthly,
    Volume 108, Number 5, May 2001, pages 446-448.

Source Code:


Last revised on 01 February 2019.