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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a volume form or top-dimensional form is a
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of
sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
. A volume form provides a means to define the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form'' or ''pseudo-volume form''. It also defines a measure, but exists on any differentiable manifold, orientable or not.
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
s, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the nth
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented pseudo-Riemannian manifolds have an associated canonical volume form.


Orientation

The following will only be about orientability of ''differentiable'' manifolds (it's a more general notion defined on any topological manifold). A manifold is orientable if it has a coordinate atlas all of whose transition functions have positive Jacobian determinants. A selection of a maximal such atlas is an orientation on M. A volume form \omega on M gives rise to an orientation in a natural way as the atlas of coordinate charts on M that send \omega to a positive multiple of the Euclidean volume form dx^1 \wedge \cdots \wedge dx^n. A volume form also allows for the specification of a preferred class of frames on M. Call a basis of tangent vectors (X_1, \ldots, X_n) right-handed if \omega\left(X_1, X_2, \ldots, X_n\right) > 0. The collection of all right-handed frames is acted upon by the group \mathrm^+(n) of general linear mappings in n dimensions with positive determinant. They form a principal \mathrm^+(n) sub-bundle of the linear frame bundle of M, and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of M to a sub-bundle with structure group \mathrm^+(n). That is to say that a volume form gives rise to \mathrm^+(n)-structure on M. More reduction is clearly possible by considering frames that have Thus a volume form gives rise to an \mathrm(n)-structure as well. Conversely, given an \mathrm(n)-structure, one can recover a volume form by imposing () for the special linear frames and then solving for the required n-form \omega by requiring homogeneity in its arguments. A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed, \mathrm(n) \to \mathrm^+(n) is a deformation retract since \mathrm^+ = \mathrm \times \R^+, where the positive reals are embedded as scalar matrices. Thus every \mathrm^+(n)-structure is reducible to an \mathrm(n)-structure, and \mathrm^+(n)-structures coincide with orientations on M. More concretely, triviality of the determinant bundle \Omega^n(M) is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus, the existence of a volume form is equivalent to orientability.


Relation to measures

Given a volume form \omega on an oriented manifold, the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, \omega, is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds. Any volume pseudo-form \omega (and therefore also any volume form) defines a measure on the
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s by \mu_\omega(U) = \int_U\omega . The difference is that while a measure can be integrated over a (Borel) ''subset'', a volume form can only be integrated over an ''oriented'' cell. In single variable
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, writing \int_b^a f\,dx = -\int_a^b f\,dx considers dx as a volume form, not simply a measure, and \int_b^a indicates "integrate over the cell ,b/math> with the opposite orientation, sometimes denoted \overline". Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form need not be absolutely continuous.


Divergence

Given a volume form \omega on M, one can define the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of a vector field X as the unique scalar-valued function, denoted by \operatorname X, satisfying (\operatorname X)\omega = L_X\omega = d(X \mathbin \omega) , where L_X denotes the Lie derivative along X and X \mathbin \omega denotes the interior product or the left
contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
of \omega along X. If X is a compactly supported vector field and M is a manifold with boundary, then
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
implies \int_M (\operatorname X)\omega = \int_ X \mathbin \omega, which is a generalization of the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
. The solenoidal vector fields are those with \operatorname X = 0. It follows from the definition of the Lie derivative that the volume form is preserved under the
flow Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psyc ...
of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, a natural volume form may be defined by translation. That is, if \omega_e is an element of ^n T_e^*G, then a left-invariant form may be defined by \omega_g = L_^*\omega_e, where L_g 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 M is a 2 n-dimensional manifold with symplectic form \omega, then \omega^n 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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
pseudo-Riemannian (including Riemannian)
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
has a natural volume form. In local coordinates, it can be expressed as \omega = \sqrt dx^1\wedge \dots \wedge dx^n where the dx^i are
1-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
s that form a positively oriented basis for the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
of the manifold. Here, , g, is the absolute value of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix representation of the metric tensor on the manifold. The volume form is denoted variously by \omega = \mathrm_n = \varepsilon = (1). Here, the is the
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of ...
, thus the last form, (1), emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita ''tensor'' \varepsilon. Although the Greek letter \omega is frequently used to denote the volume form, this notation is not universal; the symbol \omega often carries many other meanings in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
(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 f on M, and a volume form \omega, f\omega is a volume form on M. Conversely, given two volume forms \omega, \omega', 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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative of \omega' with respect to \omega. On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon–Nikodym theorem.


No local structure

A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space . That is, for every point p in M, there is an open neighborhood U of p and a diffeomorphism \varphi of U onto an open set in \R^n such that the volume form on U is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
of dx^1\wedge\cdots\wedge dx^n along \varphi. As a corollary, if M and N are two manifolds, each with volume forms \omega_M, \omega_N, then for any points m \in M, n \in N, there are open neighborhoods U of m and V of n and a map f : U \to V such that the volume form on N restricted to the neighborhood V pulls back to volume form on M restricted to the neighborhood U: f^*\omega_N\vert_V = \omega_M\vert_U. In one dimension, one can prove it thus: given a volume form \omega on \R, define f(x) := \int_0^x \omega. Then the standard
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
dx pulls back to \omega under f: \omega = f^*dx. Concretely, \omega = f'\,dx. In higher dimensions, given any point m \in M, it has a neighborhood locally homeomorphic to \R\times\R^, and one can apply the same procedure.


Global structure: volume

A volume form on a connected manifold M has a single global invariant, namely the (overall) volume, denoted \mu(M), which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on \R^n. On a disconnected manifold, the volume of each connected component is the invariant. In symbols, if f : M \to N is a homeomorphism of manifolds that pulls back \omega_N to \omega_M, then \mu(N) = \int_N \omega_N = \int_ \omega_N = \int_M f^*\omega_N = \int_M \omega_M = \mu(M)\, and the manifolds have the same volume. Volume forms can also be pulled back under covering maps, in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In the case of an infinite sheeted cover (such as \R \to S^1), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.


See also

* * * Poincaré metric provides a review of the volume form on the complex plane *


References

* . * . {{Tensors Determinants Differential forms Differential geometry Integration on manifolds Riemannian geometry Riemannian manifolds