Homework 3—Global Analysis
Due date:17.11.2020
1. For a topological space M denote by C0
(M) the vector space of continuous realvalued
functions f : M → R. Any continuous map F : M → N between
topological spaces M and N induces a map F∗
: C0
(N) → C0
(M) given by
F∗
(f) := f ◦ F : M → R.
(a) Show that F∗
is linear.
(b) If M and N are (smooth) manifolds, show that F : M → N is smooth ⇐⇒
F∗
(C∞
(N)) ⊂ C∞
(M).
(c) If F is a homeomorphism between (smooth) manifolds, show that F is a diffeomorphism
⇐⇒ F∗
is an isomorphism.
2. Suppose M = R3
with standard coordinates (x, y, z). Consider the vector field
ξ(x, y, z) = 2
∂
∂x
−
∂
∂y
+ 3
∂
∂z
.
How does this vector field look like in terms of the coordinate vector fields associated
to the cylindrical coordinates (r, φ, z), where x = r cos φ, y = r sin φ and z =
z? Or with respect to the spherical coordinates (r, φ, θ), where x = r sin θ cos φ,
y = r sin θ cos φ and z = r cos θ?
3. Consider R3
with coordinates (x, y, z) and the vector fields
ξ(x, y, z) = (x2
− 1)
∂
∂x
+ xy
∂
∂y
+ xz
∂
∂z
η(x, y, z) = x
∂
∂x
+ y
∂
∂y
+ 2xz2 ∂
∂z
.
Are they tangent to the cylinder M = {(x, y, z) ∈ R3
: x2
+ y2
= 1} ⊂ R3
with
radius 1 (i.e. do they restrict to vector fields on M)?
4. Suppose M = R2
with coordinates (x, y). Consider the vector fields ξ(x, y) = y ∂
∂x
and η(x, y) = x2
2
∂
∂y
on M. We computed in class their flows and saw that they are
complete. Compute [ξ, η] and its flow? Is [ξ, η] complete?
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