MathDB

Let

\vec{G}(x,y) = \left( \frac{-y}{x^2+4y^2}, \frac{x}{x^2+4y^2},0 \right).

Prove or disprove that there is a vector-valued function

\vec{F}(x,y,z) = (M(x,y,z), N(x,y,z), P(x,y,z))

with the following properties:
(i)

M,N,P

have continuous partial derivatives for all

(x,y,z) \neq (0,0,0)

; (ii)

\mathrm{Curl}\,\vec{F} = \vec{0}

for all

(x,y,z) \neq (0,0,0)

; (iii)

\vec{F}(x,y,0) = \vec{G}(x,y)

Problem Statement