Riemannian Volume Form of Orientable Hypersurface

Theorem
Let $\struct {\tilde M, \tilde g}$ be an oriented Riemannian manifold.

Let $\struct {M, g}$ be an orientable hypersurface with an induced metric.

Let $N$ be a unit global normal vector for $M$.

Let $\rd V_g$ be the Riemannian volume form.

Then Riemannian volume form is given by:


 * $\rd V_g = \valueat {\paren {N \lrcorner \rd V_{\tilde g} } } M$

where $\lrcorner$ denotes the interior multiplicaiton.