Is it always possible to construct a function whose jacobian determinant is equal to some desired scalar function $f$?

Question

Suppose we have an arbitrary smooth function $f:\Bbb R^m \rightarrow \Bbb R$.

Is it always possible to construct a set of functions $y_1(x_1,...,x_m),\ldots, y_m(x_1,...,x_m)$ such that $\det(\frac{\partial y_i}{\partial x_j})=f$ given such $f$?

edit: vocabulary and indices, corrections from John Hughes

edit: and partials, correction from Tony K

Answer 1

Let $F(x_1, \ldots, x_m) = \int_0^{x_1} f(u, x_2, \ldots, x_m) ~ du$. Then $\partial F/\partial x_1 = f$.

Let $y_1 = F$, and for $i = 2, \ldots, m$, let $$ y_i(x_1, \ldots, x_m) = x_i, $$ so that the partials of those functions are particularly nice. Then you're done.

By the way,

1. I'm assuming you meant to write $\frac{\delta y_i}{\delta x_j}$, i.e., one of the two $i$s in your expression should have been a $j$.

2. I'm assuming that by "construct a series of equations" you meant "produce a set of functions."