Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 25H / a / i

The coordinate tangent vectors are

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.

 Ancestors (11)

There are no discussions about this article yet.

There are currently no matching articles.