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A Riemannian metric on a smooth surface defines an unsigned area density of a surface in every chart:where $E,F,G$ are the coefficients of the first fundamental form. For a coordinate change with Jacobian matrix $J$, the identity $\tilde{g}=J^{T}gJ$ gives $\sqrt{\det \tilde{g}}=|\det J|\sqrt{\det g}$. This is exactly the absolute Jacobian determinant factor in coordinate area substitution, so the density defines surface area globally, including on a nonorientable surface.

If an orientation of a surface is also chosen, the corresponding oriented area two-form is, in every positively oriented chart,Changing the orientation negates this differential form, while leaving the unsigned surface area unchanged. A Möbius band has an area density but no globally nonvanishing area two-form, since such a form would orient it. The notation $dA$ is often used for the density too; an integral computing unsigned surface area uses the positive density convention.

A density on a smooth $n$-dimensional manifold transforms under coordinate changes by the absolute value of the Jacobian determinant. If $x=x(y)$, its local coefficient satisfiesDensities are sections of the corresponding rank-one density bundle. A positive density has positive local coefficient; a general density can have a signed coefficient despite the absolute Jacobian transformation.

A Riemannian metric supplies the positive density $\sqrt{\det (g_{ij})}|d{x}^1\cdots d{x}^n|$. The determinant transformation of the metric matrix gives exactly the required coordinate law. This is the higher-dimensional version of an area density of a surface.

Unlike a top-degree differential form, a density can be integrated without choosing an orientation. A Möbius band therefore has a global positive area density although it has no nowhere-zero global area two-form. The distinction separates integration of unsigned volume from orientation-sensitive integration.