> [!definition] > > Let $V \iso \real^d$ be a vector space equipped with an [[Inner Product|inner product]]. Given an [[Orientation|orientation]], there exists a unique [[Alternating Tensor|alternating tensor]] $\Omega \in \Lambda^n(V)$ such that $\Omega(e_1, \cdots, e_n) = 1$ whenever $\seq{e_j}$ is an orthonormal basis, known as the **volume** of the orientation. > [!definition] > > Let $(X, g)$ be a [[Riemannian Manifold|Riemannian n-manifold]]. An **orientation** is a choice of orientation and a volume $\Omega_p$ at each $T_pX$, such that $\Omega$ is a [[Differential Form|n-form]], known as the **volume form**. If $X$ is connected, then there can only be two possible orientations on $X$. > > If $(U, \varphi)$ is a [[Atlas|chart]] with coordinates $\seq{x_j}$, and there exists $\varphi \in C^p(\wh U)$ such that > $ > \Omega(x) = \varphi(x)dx_1 \wedge \cdots \wedge dx_n > $ > and $\varphi \ge 0$, then $(U, \varphi)$ is an **oriented chart**. > [!theorem] > > Let $(U, \varphi)$ be a chart, and let $G = (g_{ij})$ be the local representation of $g$, such that > $ > \angles{\cdot, \cdot}_g = \angles{\cdot, G\cdot}_{\real^n} > $ > then the Riemannian volume form has density > $ > \Omega(x) = \sqrt{\det G} dx_1 \wedge \cdots \wedge dx_n > $ > *Proof*. Since $G$ is symmetric and positive definite, it admits a square root $G^{1/2}$ such that > $ > \angles{\cdot, \cdot}_g = \angles{\cdot, G\cdot}_{\real^n} = \angles{G^{1/2}\ > \cdot, G^{1/2}\cdot} > $ > Thus $G^{1/2}$ can be viewed as a linear operator $\real^n$, transforming $dx_1 \wedge \cdots \wedge dx_n$ into $\det G^{1/2}dx_1 \wedge \cdots \wedge dx_n$. As the determinant commutes with composition, we get the square root back.