In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold...

differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space. In general, a k-form is...

spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form d V = ρ ( u 1 , u 2 , u 3 ) d u 1 d u 2 d u 3 {\displaystyle...

that affine differential geometry studies manifolds equipped with a volume form rather than a metric. Here we consider the simplest case, i.e. manifolds...

denote the density, and d n s {\displaystyle \mathrm {d} ^{n}s} is the volume form of the field function, i.e., the measure of the domain of the field function...

standard volume form given by dx1 ∧ ⋯ ∧ dxn. Given a volume form on M, the collection of all charts U → Rn for which the standard volume form pulls back...

obtained from the 3-dimensional volume form, a (0,3)-tensor, by raising an index. In detail, the 3-dimensional volume form defines a product V × V × V →...

{\displaystyle {\sqrt {-1}}\,\omega \wedge {\bar {\omega }}} is a positive volume form, from which Lefschetz was able to rederive Riemann's inequalities. In...

up volume in Wiktionary, the free dictionary. Volume is the quantity of space an object occupies in a 3D space. Volume may also refer to: Volume (thermodynamics)...

the metric space structure of a Riemannian manifold, together with its volume form. This established a deep link between Ricci curvature and Wasserstein...