PROBLEM SET V
DUE FRIDAY,  MARCH
Exercise . Show that no continuously differentiable function Rm .
Exercise . Consider the function F : R3 .
Rn with m > n can be injective.
R3 given by the formula
F( ρ, θ, φ) := ρ(cos θ sin φ, sin θ sin φ, cos φ).
Show that F is continuously differentiable, and find open sets U, V ⊂ R3 such that F defines a homeomorphism
U . V with a differentiable inverse.
Exercise⋆ . Fix real numbers λ > 0 and y0 , and consider the curves γ : R . R3 and η : R . R3 given by
γ(t) := (1, t, 2 arctan(exp(λt))) and η(t) := (1, y0 , t). Using F from the previous problem, at what angle does the
curve F ◦ γ intersect the curve F ◦ η?
Exercise . Suppose G : Rm × Rn . Rn a function, continuously differentiable in an open neighborhood of a
point (x, y) ∈ Rm × Rn . Suppose, in addition, that G(x, y) = 0, and the n × n matrix
)
(
∂Gi
∂yj
ij
is invertible. en show that there exist: an open neighborhood U of x in Rm , an open neighborhood of y in Rn , and
a differentiable function s : U . V with the property that for any u ∈ U, the point s(u) ∈ V is the unique point
with the property that G(u, s(u)) = 0.
is result permits us to do a certain amount of geometry on subsets of Euclidean space defined by systems of
equations.
Exercise . Consider the cylinder
K = {(x, y, z) ∈ R3 | x2 + y2 − y = 0}.
Let C be the intersection K ∩ S2 (1) with the 2-sphere of radius 1. At any point (x, y, z) ∈ C except for (0, 1, 0),
write an equation for the line tangent to C. What goes wrong at (0, 1, 0)? If we take K ∩ S2 (r) for r ̸= 1, do we still
have such a “bad” point? Explain.
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