Volume form
volume element.
Let be a vector space of dimension
with a given orientation and an inner product. The corresponding volume form, or volume element, is the unique element
, the space of
-forms on
(cf. Exterior form), such that
for each orthonormal (with respect to the given inner product) basis of
. Recall that
is one-dimensional. If
with the standard inner product and orientation, then
for each
-tuple of vectors
(written out in the standard basis for calculating the determinant) and
is the volume of the parallelopipedon spanned by the line segments from zero to the
.
If is an oriented Riemannian manifold, then the volume form
on
is defined by requiring that
for each
is the unique volume element on
defined by inner product and orientation on each
. One often writes
for the volume form on
, even though there may not be an
-form
on
of which it is the exterior derivative.
In given local coordinates , let
be the two-form (matrix) determining the inner product on
(with respect to the basis
, cf. Tangent vector). Then in local coordinates,
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where depending on whether the orientation of
corresponds to the standard one on
or not (under the given coordinate chart).
On a Riemannian manifold a function
is integrated by intergrating the
-form
over
in the sense of integration on manifolds.
Let denote the Hodge star operator (cf. Laplace operator). The divergence of a vector field, locally given by
, is defined as the function
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One then has
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and, on integration over an -chain in
, by the Stokes formula there results the higher-dimensional divergence theorem, which specializes to the usual one for
a
-dimensional submanifold with boundary in
.
References
[a1] | M. Spivak, "Calculus on manifolds" , Benjamin (1965) |
[a2] | M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340 |
[a3] | Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds, and physics" , North-Holland (1977) (Translated from French) |
Volume form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Volume_form&oldid=14716