Mathematical Physics Vol 1

4.6 Examples

191

4.6.8 Generalised orthogonal systems Curvilinear coordinates

Exercise 142 Determine the coordinate surfaces and coordinate lines for a) cylindrical and b) spherical coordinates.

Solution a) The cylindrical coordinate system ( ρ , ϕ , z ) . The coordinate surfaces are

ρ = c 1 coaxial cylinders, with the centre of the base on the z axis, ϕ = c 2 planes passing through the z axis, z = c 3 planes normal to the z axis. The coordinate lines are the intersection of the surfaces ρ = c 1 and ϕ = c 2 , yielding straight lines ( z - axis), the intersection of the surfaces ρ = c 1 and z = c 3 , yielding circles, the intersection of the surfaces ϕ = c 2 and z = c 3 , yielding straight lines ( ρ > 0). b) The spherical coordinate system ( r , θ , ϕ ) . The coordinate surfaces are

r = c 1 concentric spheres, with the center on the z axis, θ = c 2 cone, with the vertex at the coordinate origin, ϕ = c 3 planes, passing through the z axis.

The coordinate lines are the intersection of the surfaces r = c 1 and θ = c 2 , yielding circles, the intersection of the surfaces r = c 1 and ϕ = c 3 , yielding semi-circles, the intersection of the surfaces ϕ = c 3 and θ = c 2 , yielding lines ( r > 0).

Exercise 143 Express the cylindrical coordinates in terms of Cartesian coordinates.

Solution Let us start with the transformations expressing Cartesian coordinates in terms of cylindrical coordinates x = ρ cos ϕ (4.215) y = ρ sin ϕ (4.216) z = z . (4.217)