Second Homework DRAFT
Curves and surfaces, MATH2040
February 13, 2021
This is the second homework sheet for the course curves and surfaces, it is due February
26th.
I would like to remind you that any solutions you hand in for this (and future) home-
work sheet(s) must be written by you, and may not be copied from anyone else. You
are encouraged to discuss these problems with classmates, the TA, and the instructor,
but with no one else. Please refer to the course outline, the honesty declaration, and
the university guidelines for more information.
This is a draft, more exercises will be added later.
Definition 1. If S ⊆ R3 is a subset, then a chart (of rank 2) of S is a pair (U,ϕ), where
• U is an open subset of R2,
• ϕ is a map ϕ : U → S.
This data is subject to the following conditions:
• The map ϕ is a homeomorphism onto its image.
• When viewed as a map into R3, the map ϕ : U → R3 is smooth and regular.
Warning: This definition of charts is specific to regular surfaces in R3. See any intro-
ductory book on smooth manifolds for the general definition.
Problem 1. Let S ⊆ R3 be a subset. Let Λ be an index set. Then, let {(Uk, ϕk)}k∈Λ
be a collection of charts of S, with the property that, for every point p ∈ S, there exists
a k ∈ Λ and an x ∈ Uk, such that ϕk(x) = p. (The collection {(Uk, ϕk)}k∈Λ is called an
atlas of S.)
a) Prove that S is a regular surface.
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b) Assume now that T ⊆ R3 is a regular surface. Prove that there exists a collection
of charts {(Uk, ϕk)}k∈T , such that the above condition holds, i.e. for every point p ∈ T ,
there exists a k ∈ T and an x ∈ Uk such that ϕk(x) = p.
Problem 2. Let I ⊆ R be an open interval. Let f : I → R be a smooth function, with
f(t) > 0 for all t ∈ I. We consider the surface of revolution
Rf = {(x, y, z) ∈ R3 | z ∈ I, x2 + y2 = f(z)2}.
a) Find charts for Rf , which exhibit it as a (smooth) regular surface. (And conclude
that, indeed, Rf is a regular surface; see Problem 1.)
b) Prove that Rf is a regular surface by using the regular value theorem.
Problem 3. Let S be a regular surface, let f : S → R be a map, and let p ∈ S be a
point. Prove that the following statements about f are equivalent:
1. There exists an open neighbourhood V ⊆ R3 of p, and a smooth map F : V → R,
such that F restricts to f , i.e. F |S∩V = f |S∩V .
2. There exists a chart ϕ : U → S of S such that p ∈ ϕ(U) and such that the map
f ◦ ϕ : U → R is smooth.
3. If ϕ : U → S is any chart of S such that p ∈ ϕ(U), then f ◦ ϕ : U → R is smooth.
HINT: First, show that 1 is equivalent to 2. For this, you might want to review the
proof of the fact that transition functions are smooth. That 2 implies 3 can be proved
by using the fact that transition funtions are smooth. That 3 implies 2 follows from a
fairly simply argument.
Problem 4. Let S1, S2 be regular surfaces, and let f : S1 → S2 be a smooth map. Let
p ∈ S1, and prove that if the differential of f at p, i.e. dpf : TpS1 → Tf(p)S2 is invertible,
then f is a local diffeomorphism at p.
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