1986 Putnam A5

Suppose $f_1(x), f_2(x), \dots, f_n(x)$ are functions of $n$ real

variables $x = (x_1, \dots, x_n)$ with continuous second-order partial

derivatives everywhere on $\mathbb{R}^n$. Suppose further that there are

constants $c_{ij}$ such that

$$

\frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i}

= c_{ij}

$$

for all $i$ and $j$, $1\leq i \leq n$, $1 \leq j \leq n$. Prove that

there is a function $g(x)$ on $\mathbb{R}^n$ such that $f_i + \partial

g/\partial x_i$is linear for all$i$,$1 \leq i \leq n$. (A linear

function is one of the form

$$

a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n.)

$$

假设 $f_1(x), f_2(x), \dots, f_n(x)$ 是 $n$ 实数的函数

具有连续二阶部分的变量 $x = (x_1, \dots, x_n)$

$\mathbb{R}^n$ 上到处都是导数。进一步假设有

常量 $c_{ij}$ 使得

$$

\frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i}

= c_{ij}

$$

对于所有 $i$ 和 $j$，$1\leq i \leq n$，$1 \leq j \leq n$。证明

$\mathbb{R}^n$ 上有一个函数 $g(x)$ 使得 $f_i + \partial

g/\partial x_i$对于所有$i$、$1 \leq i \leq n$ 都是线性的。 (线性

函数是形式之一

$$

a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n。)

$$