In
mathematics, a volume element provides a means for
integrating a
function with respect to
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
in various coordinate systems such as
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' mea ...
and
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
. Thus a volume element is an expression of the form
:
where the
are the coordinates, so that the volume of any set
can be computed by
:
For example, in spherical coordinates
, and so
.
The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing
surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, on ...
s. Under changes of coordinates, the volume element changes by the absolute value of the
Jacobian determinant of the coordinate transformation (by the
change of variables formula). This fact allows volume elements to be defined as a kind of
measure on a
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 ...
. On an
orientable differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, a volume element typically arises from a
volume form In mathematics, a volume form or top-dimensional form is a differential form 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 of t ...
: a top degree
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 ...
. On a non-orientable manifold, the volume element is typically the
absolute value of a (locally defined) volume form: it defines a
1-density.
Volume element in Euclidean space
In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
, the volume element is given by the product of the differentials of the Cartesian coordinates
:
In different coordinate systems of the form
,
,
, the volume element
changes by the Jacobian (determinant) of the coordinate change:
:
For example, in spherical coordinates (mathematical convention)
:
the Jacobian determinant is
:
so that
:
This can be seen as a special case of the fact that differential forms transform through a pullback
as
:
Volume element of a linear subspace
Consider the
linear subspace of the ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
R
''n'' that is spanned by a collection of
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
vectors
:
To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the
is the square root 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 ...
of the
Gramian matrix of the
:
:
Any point ''p'' in the subspace can be given coordinates
such that
:
At a point ''p'', if we form a small parallelepiped with sides
, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix
:
This therefore defines the volume form in the linear subspace.
Volume element of manifolds
On an ''oriented''
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
of dimension ''n'', the volume element is a volume form equal to the
Hodge dual of the unit constant function,
:
:
Equivalently, the volume element is precisely the
Levi-Civita tensor
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
.
[Carroll, Sean. ''Spacetime and Geometry''. Addison Wesley, 2004, p. 90] In coordinates,
where
is 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 ...
of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allo ...
''g'' written in the coordinate system.
Area element of a surface
A simple example of a volume element can be explored by considering a two-dimensional surface embedded in ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. Such a volume element is sometimes called an ''area element''. Consider a subset
and a mapping function
:
thus defining a surface embedded in
. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form
:
that allows one to compute the area of a set ''B'' lying on the surface by computing the integral
:
Here we will find the volume element on the surface that defines area in the usual sense. The
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
of the mapping is
:
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
on the set ''U'', with matrix elements
:
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 ...
of the metric is given by
:
For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.
Now consider a change of coordinates on ''U'', given by a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
:
so that the coordinates
are given in terms of
by
. The Jacobian matrix of this transformation is given by
:
In the new coordinates, we have
:
and so the metric transforms as
:
where
is the pullback metric in the ''v'' coordinate system. The determinant is
:
Given the above construction, it should now be straightforward to understand how the volume element is invariant under an orientation-preserving 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 element takes the same expression: the expression of the volume element 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.
Example: Sphere
For example, consider the sphere with radius ''r'' centered at the origin in R
3. This can be parametrized using
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' mea ...
with the map
:
Then
:
and the area element is
:
See also
*
*
*
Surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, on ...
*
Volume integral
In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
References
*
{{reflist
Measure theory
Integral calculus
Multivariable calculus