1987 Putnam A5

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

让

$$

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

\右)。

$$

证明或反驳向量值函数的存在

$$

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

$$

具有以下属性：

(i) $M,N,P$ 对于所有的都有连续偏导数

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

(ii) $\mathrm{Curl}\,\vec{F} = \vec{0}$ 对于所有 $(x,y,z) \neq (0,0,0)$；

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