= Metric volume form
{title2=$\omega_g=\sqrt{\det g}\,dx^1\wedge\cdots\wedge dx^n$}
On an oriented <Riemannian manifold>, the metric volume form is the <volume form> taking value one on positively oriented orthonormal frames. In an oriented <manifold chart> it has the displayed expression. Under an orientation-preserving coordinate change with Jacobian $A$, $\det g$ acquires the factor $(\det A)^2$, precisely matching the transformation of the top <exterior product>. Thus the expressions agree on chart overlaps. Orientation is needed to make a globally positive form, though the volume density exists without it.
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