Concepts

Spherical Harmonics: angular basis functions on a sphere

Legendre Polynomials: system of complete and orthogonal polynomials

Laplacian Operator: $(\nabla \cdot \nabla)$ — divergence of the gradient of a scalar field

• Gradient: direction of steepest ascent
• Divergence: local rate of outward spread
• Laplacian $> 0$: point is lower than its neighbours
• Laplacian $< 0$: point is higher than its neighbours

Spherical Harmonic Basis

spherical_harmonics.mp4

Basis Function: $Y_{\ell m}(\theta, \phi) = (-1)^m N_{\ell m} P_{\ell m}(\cos\theta) e^{im\phi}$

Normalisation Factor: $N_{\ell m} = \sqrt{\frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!}}$

References

• Spherical Harmonics