sphere_integrals

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.

The surface of the unit sphere in 3D is defined by

        x^2 + y^2 + z^2 = 1

The integrands are all of the form

        f(x,y,z) = x^a y^b z^c

where the exponents 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 information on this web page is distributed under the MIT license.

Languages:

sphere_integrals is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

matlab_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface or interior of some geometric object, including a line, quadrilateral, box, circle, disk, sphere, ball and others.

sphere_monte_carlo, a MATLAB code which uses the monte carlo method to estimate the integral of a function over the surface of the unit sphere in 3d.

Reference:

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

Source Code:

i4vec_uniform_ab.m returns a scaled pseudorandom I4VEC.
monomial_value.m evaluates a monomial.
sphere01_area.m returns the surface area of the unit sphere in 3D.
sphere01_monomial_integral.m monomial integrals on the surface of the unit sphere in 3D.
sphere01_sample.m samples points uniformly from the surface of the unit sphere in 3D.

Last revised on 25 March 2019.