MATH 247, Spring 2022 Assignment 3

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Assignment 3

MATH 247, Spring 2022


Problem 1

Let D ⊂ RN , and let f : D → R, g : D → RM  be uniformly continuous.

(a) Is it true that fg is uniformly continuous?

(b) What happens if at least one of f or g is bounded?

 

Problem 2

Let K ⊆ RN  be a compact set and let f : K → RM  be continuous and one-to-one. Prove that the inverse function f−1  : f(K) → K is continuous.  Show by example that this claim is false if we replace K with a non-compact connected set.

 

Problem 3

Prove the contraction mapping principle: Let ∅  D ⊂ RN  be a closed set and let f : D → D be a function. Assume that there exists r ∈ (0, 1) such that

∥f(x) − f(y)∥ ≤ r∥x − y∥    (x,y ∈ D).

Then f has a unique fixed point x∗ ∈ D . That is, there is a unique x∗ ∈ D such that f(x∗ ) = x∗ . (Hint:  Fix any x0   ∈ D and consider the sequence (xn )n∈N  given by xn  = f(xn−1).  Prove that x∗ = limn xn  exists and f(x∗ ) = x∗ . Why is x∗ the unique point with this property?)

 

Problem 4

Compute the Jacobians of the functions f(x,y) =  (sin(xy), cos(xy),x2y2 ) and g(x,y) = x3  − 3x2  + y2 .  Find the equation of the plane tangent to the graph of the equation z = g(x,y) at (x0 ,y0 ) = (1, 1).



Problem 5

Let F = (F1 , . . . ,FN ) : RN  → RN  be a C1  vector field. Recall that F is called a gradient field if there exists a (necessarily C2 ) function f : RN  → R such that F = ∇f . Prove that F is a gradient field if and only if

∂Fi          ∂Fj

∂xj         ∂xi

(Hint: Define

f(x) = 0 1 (x1 , . . . ,xN ) · (F1 (tx), . . . ,FN (tx))dt

.)

 

Problem 6

Let X ∈ MN (R) and let ∥X∥ denote the Euclidean norm of X (viewed as an vector in RN2).

(a) Prove that ∥XY ∥ ≤ ∥X∥∥Y ∥ for all X,Y ∈ MN (R).

(b) Fix n ∈ N and consider function f : Mn (R) → MN (R) given by f(X) = Xn . Prove that f is differentiable and that for any X ∈ MN (R), the Jacobian Df(X) : Mn (R) → MN (R) is


given by

n−1

(Df)(X)(H) =      Xi HXn−1−i

i=0

(H ∈ MN (R)).


(Hint: Use (a) to make the arguments from class on Wednesday June 8 rigorous.)

 

Problem 7

Let C ⊆ RN  be a non-empty open convex set and let f : C → R be differentiable. Prove the mean value theorem: For any x,y ∈ C, there exists a point z on the line segment connecting x to y such that

f(y) − f(x) = (∇f)(z) · (y − x).

You may use the one-variable version of the mean value theorem without proof.

 

Problems from the textbook

• None this time





















2022-06-24