Area density of a surface ($d{\mu }_g$)

The area density of a smooth surface with Riemannian metric $g$ is the positive coordinate-invariant rule for measuring surface area. In a chart $(u,v)$, with first fundamental form $Ed{u}^2+2Fdudv+Gd{v}^2$, it isUnder a coordinate change with Jacobian matrix $J$, the metric matrix becomes $J^{T}gJ$, so its determinant acquires $(\det J{)}^2$. The square root therefore transforms by $|\det J|$, exactly the density transformation needed for an unsigned integral.

For a parametrized surface $X(u,v)\subseteq {\mathbb{R}}^3$, the same coefficient is $|X_u\times X_v|$. A radius-$R$ sphere in polar coordinates has density $R^2\sin \theta |d\theta d\phi |$, giving total area $4\pi R^2$. Unlike an oriented area two-form, an area density exists on a Möbius band; reversing orientation does not negate area.