1 Orthogonal curvilinear coordinates

The results shown in Section 28.2 have been given in terms of the familiar Cartesian ( x , y , z ) coordinate system. However, other coordinate systems can be used to better describe some physical situations. A set of coordinates u = u ( x , y , z ) , v = v ( x , y , z ) and w = w ( x , y , z ) where the directions at any point indicated by u , v and w are orthogonal (perpendicular) to each other is referred to as a set of orthogonal curvilinear coordinates . With each coordinate is associated a scale factor h u , h v or h w respectively where h u = x u 2 + y u 2 + z u 2 (with similar expressions for h v and h w ). The scale factor gives a measure of how a change in the coordinate changes the position of a point.



Two commonly-used sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates . These are similar to the plane polar coordinates introduced in HELM booklet  17.2 but represent extensions to three dimensions.

1.1 Cylindrical polar coordinates

This corresponds to plane polar ( ρ , ϕ ) coordinates with an added z -coordinate directed out of the x y plane. Normally the variables ρ and ϕ are used instead of r and θ to give the three coordinates ρ , ϕ and z . A cylinder has equation ρ = constant.

The relationship between the coordinate systems is given by

x = ρ cos ϕ y = ρ sin ϕ z = z

(i.e. the same z is used by the two coordinate systems). See Figure 20(a).

Figure 20:

{ Cylindrical polar coordinates}

The scale factors h ρ , h ϕ and h z are given as follows

h ρ = x ρ 2 + y ρ 2 + z ρ 2 = ( cos ϕ ) 2 + ( sin ϕ ) 2 + 0 = 1

h ϕ = x ϕ 2 + y ϕ 2 + z ϕ 2 = ( ρ sin ϕ ) 2 + ( ρ cos ϕ ) 2 + 0 = ρ

h z = x z 2 + y z 2 + z z 2 = 0 2 + 0 2 + 1 2 = 1

1.2 Spherical polar coordinates

In this system a point is referred to by its distance from the origin r and two angles ϕ and θ . The angle θ is the angle between the positive z -axis and the line from the origin to the point. The angle ϕ is the angle from the x -axis to the projection of the point in the x y plane.

A useful analogy is of latitude, longitude and height on Earth.

A sphere has equation r = constant.

The relationship between the coordinate systems is given by

x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ . See Figure 21.

Figure 21:

{ Spherical polar coordinates}

The scale factors h r , h θ and h ϕ are given by

h r = x r 2 + y r 2 + z r 2 = ( sin θ cos ϕ ) 2 + ( sin θ sin ϕ ) 2 + ( cos θ ) 2 = 1

h θ = x θ 2 + y θ 2 + z θ 2 = ( r cos θ cos ϕ ) 2 + ( r cos θ sin ϕ ) 2 + ( r sin θ ) 2 = r

h ϕ = x ϕ 2 + y ϕ 2 + z ϕ 2 = ( r sin θ sin ϕ ) 2 + ( r sin θ sin ϕ ) 2 + 0 = r sin θ