= Solution
An <oriented atlas> is a <smooth atlas> whose <smooth transition maps> have positive <Jacobian determinants> at every point of their domains. Such an atlas supplies an <orientation of a vector space> on every <tangent space>, and these choices agree on overlaps.
Let $\omega$ be a nowhere-vanishing smooth top-degree <differential form>. In a <coordinate chart> write
$$
\omega=f(x)\,dx^1\wedge\cdots\wedge dx^n.
$$
The coefficient $f$ is smooth and nowhere zero. Restrict the <coordinate chart> to connected neighborhoods; the <intermediate value theorem> makes the sign of $f$ constant on each. If $f$ is negative, replacing $x^1$ by $-x^1$ changes the sign of the coordinate <volume form>. Thus for $n\geq1$ we can choose a covering of <coordinate charts> in each of which the coefficient is positive.
On an overlap of two such <coordinate charts>, the <wedge product of differential forms> transforms by
$$
dx^1\wedge\cdots\wedge dx^n
=\det\left(\frac{\partial x}{\partial y}\right)
dy^1\wedge\cdots\wedge dy^n.
$$
Consequently $f_y=f_x\det(\partial x/\partial y)$. Both coefficients are positive, so the <Jacobian determinant> is positive, as is that of the inverse transition. \b[A nowhere-vanishing top form therefore determines an oriented atlas.] The zero-dimensional case has only zero-dimensional transitions, whose determinant is one, and is automatically orientable. Connectedness of $M$ is not needed for the positive-dimensional construction; only the local sign choice matters.
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