PROBLEM SET VI
DUE FRIDAY,  APRIL
Exercise . Consider the open subset GL(Rn ) ⊂ Lin(Rn , Rn ) of the isomorphisms Rn . Rn . Prove that the function I : GL(Rn ) . GL(Rn ) given by
I(ϕ) := ϕ−1 is differentiable at any point ϕ ∈ GL(Rn ); compute its derivative
Dϕ I.
Exercise . (A) If M ∈ Matn×n (R), show that the limit
R
∑
1 r
M
R→∞
m!
r=0
lim
exists; call the resulting matrix exp(M).
(B) Prove that if M, N ∈ Matn×n (R) such that MN = NM, then exp(M +
N) = exp(M) exp(N) and that exp(0) = id.
(C) Show that if N is an invertible matrix, then
exp(NMN−1 ) = N exp(M)N−1 .
(D) Show that for any matrix M ∈ Matn×n (R), the matrix exp(M) is invertible.
(E) Finally, show that exp de nes a differentiable map Matn×n (R) . GLn (R),
and compute its derivative.
Exercise . Suppose γ : R . Rm+1 a differentiable curve such that for any
t ∈ R, one has γ(t) · γ ′ (t) = 0. Show that the image of γ lies on a sphere
{
}
m
∑
Sm (r) := (x0 , x1 , . . . , xm ) ∈ Rm+1 x2i = r2
i=0
for some radius r ≥ 0.
Exercise . Consider the function F : R3 .
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 nd open sets U, V ⊂ R3 such
that F de nes a homeomorphism U . V with a differentiable inverse.
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DUE FRIDAY,  APRIL
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?
De nition. Suppose X ⊂ Rm and Y ⊂ Rm two subsets, and suppose α a
nonnegative real number; then a map g : X . Y is said to be Lipschitz if there
is a nonnegative real number M such that for any x, y ∈ X,
|g(x) − g(y)| ≤ M|x − y|.
Exercise⋆ . Suppose U ⊂ Rm a convex open subset, and suppose that g : U . Rn
is a continuous function that is differentiable at every point of U. Show that g
is Lipschitz if and only if
sup
|Dx g(ξ)| < ∞.
x∈U, ξ∈Sm−1
Deduce that if g is continuously differentiable, then its restriction to any compact subset K ⊂ U is Lipschitz.