Preliminary Tools

Abstract

In this chapter we introduce the basic mathematical tools. We start with some words on notation in the three-dimensional Euclidean space ℝ³. The most important differential operators in ℝ³ will be mentioned briefly. We shall give the representation of the gradient and the Laplace operator and split them into a radial and an angular part. In this context, certain differential operators on the unit sphere Ω in ℝ³ are defined, including the surface gradient, the surface curl gradient, the surface divergence, the surface curl, and the Beltrami operator. Although we rely on coordinate-free representations throughout this book, for the convenience of the reader, these operators will be discussed in the particular system of polar coordinates. We then turn to the characterization of function spaces of scalar and vector-valued functions on the unit sphere. Radial basis functions are specified on the sphere. An uncertainty principle is shown to be the appropriate tool for qualifying as well as quantifying space and momentum (frequency) properties of spherical functions. Finally, closure and completeness properties will be explained for systems of harmonic functions outside a sphere, for example, outer harmonics.