Homogeneous Harmonic Polynomials of Three Variables

In this section we consider the vector spaces of homogeneous harmonic 

To calculate the dimension of the vector space H for every nonnegative integer f we will use the Fundamental Theorem of Linear Algebra (Proposition 2.5), which we repeat here if T is a linear transformation from a finitedimensional vector space V to a finite-dimensional vector space W, then we have 

Proposition 7.1 Suppose f is a nonnegative integer. Then the dimension of the vector space ZZ of homogeneous harmonic polynomials of degree f in three variables is -F 1. 

Consider the vector spaces of homogeneous polynomials of degree (. and P3 of homogeneous polynomials of degree f — 2 in three variables. (Sticklers for rigor should define Pf. = Pf .= 0. ) Let denote the restriction of the Laplacian = -f 9 -j- to P. By I xercise 2.21 we know that the image of the linear transformation Vf lies in P.  

Our goal is to calculate the dimension of the kernel of V, since this kernel consists precisely of the harmonic functions in. From Section 2.2 we know that the dimension of P is (f -I- l)(f -f 2). So, by the Fundamental Theorem of Linear Algebra (Proposition 2.5) it suffices to calculate the dimension of the image of the the linear transformation V.  

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Physicists are familiar with many special functions that arise over and over again in solutions to various problems. The analysis of problems with spherical symmetry in P often appeal to the spherical harmonic functions, often called simply spherical harmonics. Spherical harmonics are the restrictions of homogeneous harmonic polynomials of three variables to the sphere S. In this section we will give a typical physics-style introduction to spherical harmonics. Here we state, but do not prove, their relationship to homogeneous harmonic polynomials a formal statement and proof are given Proposition A. 2 of Appendix A. 

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