volume element
VolumeElement
2007-12-12 13:08:32
2007-12-12 18:09:55
Definition
area element
area form
area measure
Euclidean volume element
Euclidean volume form
euclidean volume measure
surface area measure
surface area element
surface area form
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If $M$ is an $n$ dimensional manifold, then a \PMlinkname{differential $n$ form}{DifferentialForms} that is never zero is called a {\em volume element}
or a {\em volume form}. Usually one volume form is associated with the manifold. The volume element is sometimes denoted
by $dV,$ $\omega$ or $\operatorname{vol}_n.$
If the manifold is a Riemannian manifold with \PMlinkescapetext{metric} $g,$ then the natural volume form is defined in local coordinates $x^1 \ldots x^n$ by
\begin{equation*}
dV := \sqrt{\lvert g \rvert} dx^1 ~ \wedge \ldots \wedge ~dx^n .
\end{equation*}
It is not hard to show that a manifold has a volume form if and only if it is orientable.
If the manifold is ${\mathbb{R}}^n,$ then
the usual volume element $dV = dx^1~ \wedge ~ dx^2 ~ \wedge \ldots \wedge ~dx^n$ is called the {\em Euclidean volume element}
or {\em Euclidean volume form}.
In this context, ${\mathbb{C}}^n$ is usually treated as ${\mathbb{R}}^{2n}$ unless stated otherwise.
When $n=2$, then the form is frequently called the {\em area element} or {\em area form} and frequently denoted
by $dA$. Furthermore, when the manifold is a submanifold of ${\mathbb{R}}^3$, then many authors will refer to
a {\em surface area element} or {\em surface area form}.
When the context is measure theoretic, this form is sometimes called a {\em volume measure}, {\em area measure},
etc...
\begin{thebibliography}{9}
\bibitem{spivak}
Michael Spivak.
{\em \PMlinkescapetext{Calculus on Manifolds}},
W.A. Benjamin, Inc., 1965.
\bibitem{boothby}
William M.\@ Boothby.
{\em \PMlinkescapetext{An Introduction to Differentiable Manifolds and
Riemannian Geometry}},
Academic Press, San Diego, California, 2003.
\end{thebibliography}