Traceless Symmetric Tensor Approach to Legendre Polynomials and Spherical Harmonics

In these notes I will describe the separation of variable technique for solving Laplace’s equation, using spherical polar coordinates. The solutions will involve Legendre polynomials for cases with azimuthal symmetry, and more generally they will involve spherical harmonics. I will construct these solutions using traceless symmetric tensors, but in Lecture Notes 8 I describe how the solutions in this form relate to the more standard expressions in terms of Legendre polynomials and spherical harmonics. (Logically Lecture Notes 8 should come after these notes, although they were posted first.) If you are starting from scratch, I think that the traceless symmetric tensor method is the simplest way to understand this mathematical formalism. If you already know spherical harmonics, I think that you will find the traceless symmetric tensor approach to be a useful addition to your arsenal of mathematical methods. The symmetric traceless tensor approach is particularly useful if one needs to extend the formalism beyond what we will be doing — for example, there are analogues of spherical harmonics in higher dimensions, and there are also vector spherical harmonics that are useful for expanding vector functions of angle. Vector spherical harmonics are used in the most general treatments of electromagnetic radiation, although we will not be introducing them in this course.