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We present in a unified and self-contained manner the coordinate-freeapproach to spherical harmonics initiated in the mid 19th century by JamesClerk Maxwell, William Thomson and Peter Guthrie Tait. We stress thepedagogical advantages of this approach which leads in a natural way to manyphysically relevant results that students find often difficult to work outusing spherical coordinates and associated Legendre functions. It is shown howmost physically relevant results of the theory of spherical harmonics - such asrecursion relations, Legendre's addition theorem,surface harmonics expansions,the method of images, multipolar charge distributions, partial wave expansions,Hobson's integral theorem, rotation matrix and Gaunt's integrals - can beefficiently derived in a coordinate free fashion from a few basic elements ofthe theory of solid and surface harmonics discussed in the paper.

The coordinate-free approach to spherical harmonics