# Definition The harmonic [[Polynomial|polynomials]] are polynomials satisfying the [[Laplace equation]]: $ \nabla^2 f = 0 $ # Vector Space The subset $H_l(\mathbb{R}^n) \subset P_l(\mathbb{R}^n)$ of Harmonic polynomials of degree $l$ in $n$ dimensional [[real coordinate space]] can be shown to be a [[vector subspace]] of the polynomials of the same degree over the same coordinate space and [[field]]. This vector space is intimately related to the [[Spherical Harmonics|spherical harmonics]], which are just the restriction of harmonic polynomials to the unit sphere.