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).

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