MathDB

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)

\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.)

A5