Spherical and cylindrical coordinates

Many problems in mathematical physics exhibit a spherical or cylindrical symmetry. For example, the gravity field of the Earth is to first order spherically symmetric. Waves excited by a stone thrown into water are usually cylindrically symmetric. Although there is no reason why problems with such a symmetry cannot be analyzed using Cartesian coordinates (i.e. ( x, y, z )-coordinates), it is usually not very convenient to use such a coordinate system. The reason for this is that the theory is usually much simpler when one selects a coordinate system with symmetry properties that are the same as the symmetry properties of the physical system that one wants to study. It is for this reason that spherical coordinates and cylindrical coordinates are introduced in this section. It takes a certain effort to become acquainted with these coordinate systems, but this effort is well spent because it makes solving a large class of problems much easier. Introducing spherical coordinates In Figure 4.1 a Cartesian coordinate system with its x -, y -, and z -axes is shown as well as the location of a point r . This point can be described either by its x -, y -, and z -components or by the radius r and the angles θ and ϕ shown in Figure 4.1. In the latter case one uses spherical coordinates. Comparing the angles θ and ϕ with the geographical coordinates that define a point on the globe one sees that ϕ can be compared with longitude and θ can be compared with colatitude , which is defined as ( latitude – 90 degrees).