In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. In local coordinates, it can be expressed as Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Volume (also called capacity) is a quantification of how much space an object occupies. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ...

where the manifold is an n dimensional manifold. Here, | g | is the absolute value of the determinant of the metric tensor on the manifold. The dxⁱ are the 1-forms providing a basis for the cotangent bundle of the manifold. In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

A number of different notations are in use for the volume form. These include

ω = vol_n = ε = * (1)

Here, the * is the Hodge dual, thus the last form, *(1), emphasizes that the volume form is the Hodge dual of the trivial constant map on the manifold. In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ...

Although the Greek letter ω is frequently used to denote the volume form, this notation is hardly universal; the symbol ω often carries many other meanings in differential geometry; thus, the appearance of ω in a formula does not necessarily mean that it is the volume form. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

Example: Volume form of a surface

A simple example of a volume form can be explored by considering a two-dimensional surface embedded in n-dimensional Euclidean space. Consider a subset and a mapping function In mathematics, a surface is a two-dimensional manifold. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

thus defining a surface embedded in . The Jacobian matrix of the mapping is In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

with index i running from 1 to n, and j running from 1 to 2. The Euclidean metric in the n-dimensional space induces a metric g = λ^Tλ on the set U, with matrix elements In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...

The determinant of the metric is given by In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

where is the wedge product. For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

Now consider a change of coordinates on U, given by a diffeomorphism In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

so that the coordinates (u₁,u₂) are given in terms of (v₁,v₂) by (u₁,u₂) = f(v₁,v₂). The Jacobian matrix of this transformation is given by

In the new coordinates, we have

and so the metric transforms as

where is the metric in the v coordinate system. The determinant is

.

Given the above construction, it should now be straightforward to understand how the volume form is invariant under a change of coordinates. In two dimensions, the volume is just the area. The area of a subset is given by the integral

Thus, in either coordinate system, the volume form takes the same expression: the expression of the volume form is invariant under a change of coordinates.

Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.

See also

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

In mathematics, the Poincaré metric is the natural metric tensor for Poincaré half-plane model of hyperbolic geometry. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

References

• Michael Spivak, Calculus on Manifolds, (1965) W.A. Benjamin, Inc. Reading, Massachusetts ISBN 0-8053-9021-9 (Provides an elementary introduction to the modern notation of differential geometry, assuming only a general calculus background)