2.3 Coordinate Systems: Cartesian, Cylindrical and Spherical

Position and displacement vectors, the Cartesian, cylindrical and spherical coordinate systems with their scale factors and differential elements, the conversion of vector components between them, and the gradient, divergence, curl and Laplacian written in curvilinear coordinates.

18 solved examples and exercises2 interactive tools

In this section you will learn

how to locate a field point and a source point with position vectors, and why the displacement vector $R = r - r^{'}$ points from the source to the field point,
what an orthogonal coordinate system needs (unit vectors and scale factors) and how the differential length, area and volume follow from them,
how to set up line, surface and volume integrals in Cartesian coordinates, including the direction of travel and the orientation of a surface,
how cylindrical coordinates $(ρ, ϕ, z)$ work: conversions, unit vectors that change from point to point, the scale factor $ρ$ , and the elements $ρ d ϕ$ , $ρ d ρ d ϕ$ , $ρ d ρ d ϕ d z$ ,
how to convert the components of a vector between Cartesian and cylindrical coordinates,
how a point is located by its distance $r$ and two angles $θ$ and $ϕ$ , and what the coordinate surfaces (sphere, cone, half-plane) look like,
how to find the spherical unit vectors and scale factors from the position vector, and why the unit vectors change from point to point,
the differential length, surface and volume elements, in particular $d V = r^{2} sin θ d r d θ d ϕ$ ,
how to convert the components of a vector between Cartesian, cylindrical and spherical coordinates,
how to set up and evaluate integrals over spheres and balls,
why the nabla operator cannot simply be rewritten with $a_{r}, a_{θ}, a_{ϕ}$ and derivatives with respect to $r, θ, ϕ$ ,
how the gradient follows from $df = \nabla f \cdot d ℓ$ in any orthogonal system,
how the divergence and the curl follow from their meanings (flux per unit volume, circulation per unit area) applied to small curvilinear boxes and loops,
the Laplacian in cylindrical and spherical coordinates, and
how to use the one-page formula sheet with the scale factors $h_{1}, h_{2}, h_{3}$ .

Every problem of electrostatics asks the same three questions: where is the charge, where do we want the field, and how far apart and in which direction are the two points? Position vectors answer them. Then, because real charges are spread along wires, over plates and through volumes, we have to add up the contributions of many small pieces, that is, to integrate along lines, over surfaces and through volumes. Such integrals are easy only when the coordinate system fits the shape of the object: Cartesian coordinates for boxes and plates, cylindrical coordinates for wires, coaxial cables and anything else with an axis of symmetry. This section develops these two systems and spherical coordinates (for points, spheres and shells) from a single idea, the scale factor, practises the integrals we shall need, and finally writes the gradient, divergence, curl and Laplacian in curvilinear coordinates.

2.3 Coordinate Systems: Cartesian, Cylindrical and Spherical

Contents of Section 2.3