where the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid. Special cases Lie groups For any Lie group, a natural volume form may be defined by translation. That is, if is an element of then a left-invariant form may be defined by where is left-translation. As a corollary, every Lie group is orientable. This volume form is unique up to a scalar, and the corresponding measure is known as the Haar measure. Symplectic manifolds Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If is a -dimensional manifold with symplectic form then is nowhere zero as a consequence of the nondegeneracy of the symplectic form. As a corollary, any symplectic manifold is orientable (indeed, oriented). If the manifold is both symplectic and Riemannian, then the two volume forms agree if the manifold is Kähler. Riemannian volume form Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as where the are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold. The volume form is denoted variously by Here, the is the Hodge star, thus the last form, emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita tensor Although the Greek letter is frequently used to denote the volume form, this notation is not universal; the symbol often carries many other meanings in differential geometry (such as a symplectic form). Invariants of a volume form Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function on and a volume form is a volume form on Conversely, given two volume forms their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations). In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative of with respect to On an oriented manifold, the proportionality of any two volume forms can be thought of as