1. Homework 9
Definition 1.1. A subset C of a real vector space V is said to be convex if for any x, y ∈ C, the line segement xy = {tx + (1 − t)y : t ∈ [0, 1]} is contained in C.
Definition 1.2. Let E be open subset of Rn. A function f : E → Rn is locally invertible if for any x ∈ E, there exist an open neighborhood U of x with U ⊆ E and an open neighborhood V of y in Rn so that f : U → V is a bijection.
(1) Let U be an open subset of Rn and f : U → Rn be a function. Suppose that there exists c > 0 so that
kf (x1) − f (x2)k ≥ ckx1− x2k
for any x1, x2 ∈ U. Prove or disprove that f : U → f (U ) is a homeomorphism and that V = f (U ) is also open? If assume further that f is C1, is f a C1- diffeomorphism?
(2) Let D(0, r) = {x ∈ Rn: kxk ≤ r}. Let f : D(0, R) → Rn be a map with (a) kf (x) − f (y)k ≤ 13kx − yk for x, y ∈ D(0, r) and
(b) kf (0)k ≤ 23r.
Prove that there is a unique x ∈ D(0, r) such that f (x) = x.
(3) Let U be any open convex subset of Rnand f : U → R be a C1-function. Show that for any x, y ∈ U, there exists z ∈ xy so that
f (y) − f (x) = Df (z)(y − x).
(4) Consider the equations (x4+ y4)/x = u and sin x + sin y = v. Near which points can we solve for x and y in terms of u, v?
(5) Let f : R2 → R2 be the function f (x, y) = (x2− y2, 2xy). We set u(x, y) = x2− y2 and v(x, y) = 2xy for x, y ∈ R2.
(a) Show that f is locally invertible at all points of R2\ {(0, 0)}.
(b) Compute ∂x/∂u, ∂x/∂v, ∂y/∂u, ∂y/∂v.
(6) Consider the following system of equations:
xu + yv2 = 0 xv3+ y2u6 = 0.
Are they uniquely solvable for u, v in terms of x and y near x = 1, y = −1, u = 1 and v = −1? Near x = 0, y = 1, u = 0, v = 0? Compute ∂u/∂x at x = 1, y = −1 and at x = 0, y = 1 if it exists.
(7) In the system
3x + 2y + z2+ u + v2= 0 4x + 3y + z + u2+ v + w + 2 = 0 x + z + w + u2+ 2 = 0,
discuss the solvability for u, v, w in terms of x, y, z near x = y = z = 0, u = v = 0, w = −2.
(8) Show that the equations
x2− y2− u3+ v2+ 4 = 0 2xy + y2− 2u2+ 3v4+ 8 = 0
determine functions u(x, y), v(x, y) near x = 2, y = −1 such that u(2, −1) = 2 and v(2, −1) = 1. Compute ∂u/∂x.
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