Integration of a Function over a Manifold

Definition

Consider a scalar function $f:\mathcal{M}\to \mathbb{R}$ and a Riemannian Manifold $(\mathcal{M},g)$ with a local Chart $\phi :\mathcal{M}\to {\mathbb{R}}^n$. The integral of $f$ over manifold $\mathcal{M}$ can be calculated in a local coordinate space. ${\int }_{\mathcal{M}}fds={\int }_U\tilde{f}\sqrt{|g|}dx$ where $\tilde{f}:=f\circ {\phi }^{-1}$, $U=\phi (\mathcal{M})$, and $|g|$ is the absolute value of the Determinant of the inner product at ${\phi }^{-1}(x)$