 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/1/a/i/solution

r (ρ, ϕ, z) = (ρ cos ϕ, ρ sin ϕ, z)

. Its coordinate tangent vectors are

r_{ρ} = (cos ϕ, sin ϕ, 0), r_{ϕ} = (- ρ sin ϕ, ρ cos ϕ, 0), r_{z} = (0, 0, 1) .

h_{ρ} = 1, h_{ϕ} = ρ, h_{z} = 1.

Thus the cylindrical coordinate system is orthogonal wherever it is a nonsingular coordinate chart. At

ρ = 0

, the angular tangent vector vanishes and the azimuth is undefined; this is the usual coordinate singularity on the axis, not a failure of the Euclidean geometry.

 New to topics? Read the docs here!