1. Homework 10
(1) Let U be an open subset of Rn and V be an open subset of Rm. Prove that U × V is an open subset of Rm+n.
(2) Let U be an open subset of R2and F : U → R be a C1-function. Let p = (x0, y0) ∈ U.
Assume that F (p) = c and Fy(p) 6= 0. Use the inverse function theorem to prove that there exists a C1 function
f : (x0− δ, x0+ δ) → (y0− δ, y0+ δ)
such that the open rectangle (x0− δ, x0+ δ) × (y0− δ, y0+ δ) is contained in U and y0 = f (x0) and F (x, f (x)) = c for each x ∈ (x0 − δ, x0+ δ). Here δ > 0. Consider the map φ : U → R2 defined by φ(x, y) = (x, f (x, y)). Apply the inverse function to φ at p. (This is in quiz 9 and will be on the final exam).
(3) Let (M, d) and (N, ρ) be metric spaces. Recall that a function f : M → N is continuous at x0 in M if for any > 0 there exists δ = δx0,> 0 such that
ρ(f (x), f (x0)) < whenever d(x, x0) < δ.
If f is continuous at every point of M, we say that f : M → N is continuous. Prove that f : M → N is continuous if and only if for any open subset V of N,
f−1(V ) = {x ∈ M : f (x) ∈ V } is an open subset of M.
(4) Let M2(R) ∼= R4 be the space of all 2 × 2 real matrices. Equip M2(R) with the Euclidean topology.
(a) Prove that the map
m : M2(R) × M2(R) → M2(R), (A, B) 7→ AB is continuous. Here we identify M2(R) × M2(R) with R8.
(b) Let G = {A ∈ M2(R) : det A 6= 0} be the set of all 2 × 2 real invertible matrices.
Prove that G is an open subset of M2(R). (You may use the result obtained in Problem 3). (Since m is continuous, its restriction m : G × G → G is also continuous).
(c) Prove that the map
a : G → G, A → A−1. is continuous.
(5) Let f be a real valued smooth function on an open subset U of Rn and p ∈ U. Let Sp(1) be the unit sphere {vp∈ TpRn: kvpkp = 1} centered at 0p of radius 1 in TpRn. Consider the extremum of the function φ : Sp(1) → R defined by φ(vp) = |dfp(vp)|.
(a) Prove that |φ(vp)| ≤ k∇f (p)kp for all vp∈ Sp(1).
(b) Prove that the function φ : Sp(1) → R attains its maximum and minimum when vp is parallel to ∇f (p).
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