Spherical Harmonics in p Dimensions

The following sections are included:

• Contents
• Preface
• Acknowledgements
• List of Symbols

Many important equations in physics involve the Laplace operator, which is given by

In this chapter, we spend some time developing our skills in performing calculations in the p-dimensional Euclidean space , i.e. the vector space of p-tuples of real numbers endowed with the Euclidean metric, and exercising our abilities in visualizing a p-dimensional space for arbitrary natural number p. We will use the majority of the results we obtain here in the development of our main subject, but some topics we discuss just out of pure interest or to improve our intuition…

This chapter provides the reader with several useful facts from the theory of orthogonal polynomials that we will require in order to prove some of the theorems in our main discussion. In what follows, we will assume the reader has taken an introductory course in linear algebra and, in particular, is familiar with inner product spaces and the Gram-Schmidt orthogonalization procedure. If this is not the case, we refer him or her to [8].

We will begin by developing some facts about a special kind of polynomials. We will then define a spherical harmonic to be one of these polynomials with a restricted domain, as hinted at in Section 1.1. After discussing some properties of spherical harmonics, we will introduce the Legendre polynomials. And once we have produced a considerable number of results, we will move on to an application of the material developed to boundary value problems.

The following sections are included:

• Solutions to Problems of Chapter 2
• Solutions to Problems of Chapter 3
• Solutions to Problems of Chapter 4

The following sections are included:

• Bibliography
• Index