I have been kicking around the generalization of spherical harmonics to vector spherical harmonics, and how that might generate the tensor spherical harmonics to all orders of tensor and all parities. I think I got it today! For every spherical harmonic (ell and em), there are three vector spherical harmonics obtained by multiplying by the radial vector, taking the transverse gradient, and taking the transverse gradient and crossing it into the radial direction. I think these can be generalized (using, say, the Ricci calculus) to make the 2-tensors and so on. If I am right, this is a new way to represent tensor fields on the sphere. Use cases: Cosmic backgrounds, and ocean dynamics.
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