Advanced Engineering Mathematics, 7th Edition

14.3 Spherical Coordinates

INTRODUCTION

In this section we continue our examination of boundary-value problems in different coordinate systems. This time we are going to consider problems involving the heat, wave, and Laplace’s equation in spherical coordinates.

Laplacian in Spherical Coordinates

As shown in FIGURE 14.3.1, a point in 3-space is described in terms of rectangular coordinates and in spherical coordinates. The rectangular coordinates x, y, and z of the point are related to its spherical coordinates r, θ, and ϕ through the equations

(1)

By using the equations in (1) it can be shown that the Laplacian ∇²u in the spherical coordinate ...