volume element

If $M$ is an $n$ dimensional manifold, then a differential $n$ form (http://planetmath.org/DifferentialForms) that is never zero is called a volume elementor a volume form. Usually one volume form is associated with the manifold. The volume element is sometimes denotedby $dV,$ $\omega$ or ${\mathrm{vol}}_n.$If the manifold is a Riemannian manifold with $g,$ then the natural volume form is defined in local coordinates $x^1\mathrm{\dots }{x}^n$ by

$$dV:=\sqrt{|g|}d{x}^1\wedge \mathrm{\dots }\wedge d{x}^n.$$

It is not hard to show that a manifold has a volume form if and only if it is orientable.

If the manifold is ${ℝ}^n,$ thenthe usual volume element $dV=d{x}^1\wedge d{x}^2\wedge \mathrm{\dots }\wedge d{x}^n$ is called the Euclidean volume elementor Euclidean volume form.In this context, ${ℂ}^n$ is usually treated as ${ℝ}^{2n}$ unless stated otherwise.

When $n=2$, then the form is frequently called the area element or area form and frequently denotedby $dA$. Furthermore, when the manifold is a submanifold of ${ℝ}^3$, then many authors will refer toa surface area element or surface area form.

When the context is measure theoretic, this form is sometimes called a volume measure, area measure,etc…