A volume form on an 
-dimensional smooth manifold
is a nowhere-vanishing differential
form of degree 
.
A manifold admits a volume form iff
it is orientable, and choosing a volume form
determines a manifold orientation.
On an oriented Riemannian manifold with positively oriented local
coordinates 
and metric matrix 
, the Riemannian volume form is
 |
It provides the coordinate-independent measure used to integrate scalar functions over the manifold.
