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Concept
Volume element
Summary
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. 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 integrals. 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. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density.
In Euclidean space, 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
Consider the linear subspace of the n-dimensional Euclidean space Rn that is spanned by a collection of linearly independent 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 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.
Official source
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