= Density on a manifold
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 satisfies
$$
a_y(y)=a_x(x(y))\left|\det\frac{\partial x}{\partial y}\right|,\qquad \mu=a_x(x)|dx^1\cdots dx^n|.
$$
Densities 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 $\sqrt{\det(g_{ij})}|dx^1\cdots dx^n|$. 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.
Back to article page
