Volume element ($dV=|\det D\Phi |d{u}_1\cdots d{u}_n$)

Under a smooth locally invertible coordinate map $\Phi$ into Euclidean space, the positive volume element in a multiple integral is multiplied by the absolute Jacobian determinant. In three-dimensional spherical coordinates, $dV=r^2\sin \theta drd\theta d\phi$; in cylindrical coordinates, $dV=rdrd\theta dz$. The absolute value distinguishes positive integration measure from a signed orientation convention.