Let $\omega = f_1 dx_1 +f_2dx_2 + \cdots + f_ndx_n$ be a closed $ C^{\infty}$ $1-$form on $ \mathbb R ^n$. Define a function $g$ by
$\displaystyle{ g(x_1, x_2,\cdots, x_n) = \int_{0}^{x_1} f_1(t,x_2 , x_3, \cdots ,x_n) +\int_{0}^{x_2} f_1(0,t, x_3, x_4, \cdots ,x_n) + \int_{0}^{x_3} f_1(0,0,t, x_4 ,x_5, \cdots ,x_n) + \cdots +\int_{0}^{x_n} f_1(0,0, \cdots ,t) }$
Show that $dg =\omega$.