# Definition # Differential Equation Given a [[Harmonic Polynomial]] of degree $l$ in $\mathbb{R}^3$ over the complex numbers $\mathbb{C}$, it can be written as $ f(x,y,z) = \sum_{\substack{i,j,k\\i+j+k = l}} c_{ijk} x^iy^jz^k $ or in spherical coordinates $ f(r,\theta,\phi) = r^l Y(\theta,\phi) $ for some function $Y(\theta,\phi)$, which is the restriction of the harmonic polynomial $f$ to the unit sphere. Writing the [[Laplace equation]] in spherical coordinates: $ \Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2}(\sin\theta)^{-1} \frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2}(\sin\theta)^{-2} \frac{\partial^2}{\partial \phi^2}f = 0 $ You can show that $ \left[(\sin\theta)^{-1} \frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial \theta}\right) + (\sin\theta)^{-2} \frac{\partial^2}{\partial \phi^2}\right]Y(\theta,\phi) = -l(l+1)Y(\theta,\phi) $ which is the defining differential equation of spherical harmonics. # Vector Space The spherical harmonics form a [[vector space]] in 1-1 correspondence with the vector space of [[Harmonic Polynomial|harmonic polynomials]]. $ \tilde{H}_l(\mathbb{R}^n) \leftrightarrow H_l(\mathbb{R}^n) $