= Coordinate invariance of the Riemannian volume form
{title2=$\sqrt{\det g_x}\,dx^1\wedge\cdots\wedge dx^n=\sqrt{\det g_y}\,dy^1\wedge\cdots\wedge dy^n$}
For positive coordinate charts put $J=\partial y/\partial x$. The <metric tensor> transforms by $g_x=J^Tg_yJ$, giving $\det g_x=(\det J)^2\det g_y$. The top <wedge product of differential forms> transforms by $dy^1\wedge\cdots\wedge dy^n=(\det J)dx^1\wedge\cdots\wedge dx^n$. Since $\det J>0$, the two factors match. Thus the local expressions define one global <Riemannian volume form>.
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