If $M$ is an $n$ dimensional manifold, then a differential $n$ form that is never zero is called a volume element or a 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 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 Euclidean volume element or 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 area element or 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 surface area element or surface area form.

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

Bibliography

1

Michael Spivak. Calculus on Manifolds, W.A. Benjamin, Inc., 1965.

2

William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.