Relation between Volume Forms of Conformally Related Metrics on Oriented Riemannian Manifold

Theorem
Let $M$ be an $n$-dimensional oriented Riemannian manifold.

Let $g_1$ and $g_2$ be conformally related Riemannian metrics.

That is, suppose:


 * $g_2 = f g_1$

where $f \in \map {C^\infty} M$ is a positive smooth real function.

Let $\rd V_{g_1}$ and $\rd V_{g_2}$ be Riemannian volume forms associated with $g_1$ and $g_2$.

Then:


 * $\ds \rd V_{g_2} = f^{\frac n 2} \rd V_{g_1}$