MAT 237: Multivariable Calculus with Proofs
Assignment # 5
Due June 17, 11:59 PM EST
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page 1
MAT237
1. Show that the set S = {(x, y) ∈ R2|(x− 1)2 + 5y2 = 10} is a one-dimensional manifold right from the
definition given in class (without using either the inverse or implicit function theorems).
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MAT237
2. a) Let U , V , and W be open subsets of Rn. If F : U → V and G : V →W are diffeomorphisms, show
that ϕ := G ◦ F : U →W is a diffeomorphism.
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MAT237
b) Let U and V be open subsets of Rn. Let F : U → V be a diffeomorphism and S ⊆ U a subset. Fix
p⃗ ∈ S. Show that v⃗ ∈ Tp⃗S if and only if (DF (p))(v⃗) ∈ TF (P⃗ )F (S).
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MAT237
3. Show that Sn−1 is an (n− 1)-dimensional manifold using the implicit function theorem.
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MAT237
4. Define the cylindrical coordinate map F : R3 → R3 by F (ρ, θ, ϕ) = (ρ cos θ sinϕ, ρ sin θ sinϕ, ρ cosϕ).
Prove that F is a local diffeomorphism for all points of R3 except those points which satisfy ρ ̸= 0 and
ϕ not an integer multiple of π.
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