= Area density of a surface
{title2=$d\mu_g$}
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> $E\,du^2+2F\,du\,dv+G\,dv^2$, it is
$$
d\mu_g=\sqrt{EG-F^2}\,|du\,dv|.
$$
Under a coordinate change with <Jacobian matrix> $J$, the metric matrix becomes $J^TgJ$, 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 $R^2\sin\theta\,|d\theta\,d\varphi|$, 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.
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