Spherical harmonics are functions defined on a sphere, specifically dependent on the angles θ and φ, and do not include the radial component r. They serve as eigenfunctions of the angular part of the Laplace operator, enabling the separation of angular and radial components in problems. Additionally, spherical harmonics are utilized for series expansion of functions defined on the surface of a sphere, such as analyzing cosmic microwave background (CMB) temperature variations.
PREREQUISITESMathematicians, physicists, and engineers interested in solving problems involving spherical coordinates, as well as researchers analyzing cosmic microwave background data.
Because they are functions of ##\theta## and ##\phi## by definition. They are functions defined on a sphere and are the eigenfunctions of the angular part of the Laplace operator.joshmccraney said:Hi PF!
When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of ##\phi,\theta## but not ##r##. Why don't they include the ##r## component?
Thanks!
joshmccraney said:Why don't they include the rr component?
Orodruin said:Because they are functions of θ\theta and ϕ\phi by definition.
In this particular instance, yes. Generally they have other uses as well - such as series expanding any function defined on the surface of a sphere (eg, the CMB temperature variations).Vanadium 50 said:Right. They are useful because they allow you to separate out the angular and radial parts of the problem.