Vector Analysis

MATH-UA 224 – VECTOR ANALYSIS
HOMEWORK 5 (DUE 4/14)
(1) If A is a subset of a metric space X. Prove that the closure A is closed.
(2) Let f : R
3 → R
2 be of class C
1
; write f in the form f(x, y1, y2). Assume that
f(1, 2, 2) = 0 and
∇f(1, 2, 2) =
1 2 2
3 2 1
(a) Show there is a function g : B → R
2 of class C
1 defined on an open set B ⊂ R
such that
f(x, g1(x), g2(x) = 0
for x ∈ B and g(1) = (2, 2).
(b) Find (∇g)(1).
(c) Discuss the problem of solving the equation f(x, y1, t2) = 0 for an arbitrary pair
of unkowns in terms of the third, near the point (1, 2, 2).
1

Vector Analysis

MATH-UA 224 – VECTOR ANALYSIS
HOMEWORK 5 (DUE 4/14)
(1) If A is a subset of a metric space X. Prove that the closure A is closed.
(2) Let f : R
3 → R
2 be of class C
1
; write f in the form f(x, y1, y2). Assume that
f(1, 2, 2) = 0 and
∇f(1, 2, 2) =
1 2 2
3 2 1
(a) Show there is a function g : B → R
2 of class C
1 defined on an open set B ⊂ R
such that
f(x, g1(x), g2(x) = 0
for x ∈ B and g(1) = (2, 2).
(b) Find (∇g)(1).
(c) Discuss the problem of solving the equation f(x, y1, t2) = 0 for an arbitrary pair
of unkowns in terms of the third, near the point (1, 2, 2).
1