 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/25h/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

ϕ_{u} = (- f sin u, f cos u, 0), ϕ_{v} = (f^{'} cos u, f^{'} sin u, g^{'}),

where

f, f^{'}, g^{'}

are evaluated at

v

ϕ_{u} \times ϕ_{v} = (f g^{'} cos u, f g^{'} sin u, - f f^{'}),

and therefore

∥ ϕ_{u} \times ϕ_{v} ∥ = f (v) f^{'} (v)^{2} + g^{'} (v)^{2} > 0.

The positivity follows because

f (v) > 0

and the profile is a regular curve. Hence

d ϕ

has rank two everywhere, so

ϕ

is an immersion. The angle interval

(0, 2 π)

and the absence of profile self-intersections make

ϕ

injective with a continuous local inverse. It is therefore an embedded surface parametrization, and its image is a regular surface.

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