The discussion revolves around the properties and applications of spherical harmonics, particularly in the context of solving the Laplace equation in spherical coordinates. Participants explore the definition and role of spherical harmonics in separating variables and expanding functions defined on a sphere.
Participants generally agree on the definition and basic properties of spherical harmonics, but there are multiple views regarding their applications and the implications of their use in different contexts.
The discussion does not resolve the broader implications of using spherical harmonics in various applications, nor does it clarify all assumptions related to their definitions and uses.
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.