Riemannian volume
= Riemannian volume
{c}
{title2=$\operatorname{Vol}_g(M)=\int_MdV_g$}
The total Riemannian volume is the integral of the <Riemannian volume form>, or its density on a nonorientable manifold. In local coordinates the density is $\sqrt{\det g}\,dx^1\cdots dx^d$. It is preserved by <Riemannian isometries>. On a closed manifold it is the leading coefficient among the integrated <heat invariants>, and on a surface it is the <surface area>.