Coordinate invariance of the Riemannian volume form ($\sqrt{\det g_x}d{x}^1\wedge \cdots \wedge d{x}^n=\sqrt{\det g_y}d{y}^1\wedge \cdots \wedge d{y}^n$)

For positive coordinate charts put $J=\partial y/\partial x$. The metric tensor transforms by $g_x=J^T{g}_{y}J$, giving $\det g_x=(\det J{)}^2\det g_y$. The top wedge product of differential forms transforms by $d{y}^1\wedge \cdots \wedge d{y}^n=(\det J)d{x}^1\wedge \cdots \wedge d{x}^n$. Since $\det J>0$, the two factors match. Thus the local expressions define one global Riemannian volume form.