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A5

Part of 1987 Putnam

Problems(1)

Putnam 1987 A5

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8/5/2019
Let G⃗(x,y)=(−yx2+4y2,xx2+4y2,0). \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 F⃗(x,y,z)=(M(x,y,z),N(x,y,z),P(x,y,z)) \vec{F}(x,y,z) = (M(x,y,z), N(x,y,z), P(x,y,z)) with the following properties:
(i) M,N,PM,N,P have continuous partial derivatives for all (x,y,z)≠(0,0,0)(x,y,z) \neq (0,0,0); (ii) Curl F⃗=0⃗\mathrm{Curl}\,\vec{F} = \vec{0} for all (x,y,z)≠(0,0,0)(x,y,z) \neq (0,0,0); (iii) F⃗(x,y,0)=G⃗(x,y)\vec{F}(x,y,0) = \vec{G}(x,y).
Putnam