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 of dimension , a volume form is an -form. It is an element of the space of sections of the line bundle , denoted as . A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.
A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold, orientable or not.
Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented pseudo-Riemannian manifolds have an associated canonical volume form.
In mathematics, a volumeform 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...
obtained from the 3-dimensional volumeform, a (0,3)-tensor, by raising an index. In detail, the 3-dimensional volumeform defines a product V × V × V →...
standard volumeform given by dx1 ∧ ⋯ ∧ dxn. Given a volumeform on M, the collection of all charts U → Rn for which the standard volumeform pulls back...
bilinear form on an n-dimensional real vector space V, ω ∈ Λ2(V). Then ω is non-degenerate if and only if n is even and ωn/2 = ω ∧ ... ∧ ω is a volumeform. A...
that affine differential geometry studies manifolds equipped with a volumeform rather than a metric. Here we consider the simplest case, i.e. manifolds...
the metric space structure of a Riemannian manifold, together with its volumeform. This established a deep link between Ricci curvature and Wasserstein...
{\displaystyle {\sqrt {-1}}\,\omega \wedge {\bar {\omega }}} is a positive volumeform, from which Lefschetz was able to rederive Riemann's inequalities. In...
to define a natural volumeform from the metric tensor. In a positively oriented coordinate system (x1, ..., xn) the volumeform is represented as ω =...
the volume of parallelepipeds, cones, pyramids, cylinders, and spheres. The formula were determined by prior mathematicians by using a primitive form of...
notation for Form A is SPST-NO. Form B contacts ("break contacts") are normally closed contacts. Its operation is logically inverted from Form A. An alternate...
French term is volume, which will sometimes be used by the British: "1-volumeform". Two-box form A categorization based on overall form design using rough...
technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example...
sometimes called a volumeform (or orientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact...
components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have...
integrating a Riemannian volumeform on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface. Consider...
{\displaystyle (x_{1},\ldots ,x_{k})\in U\subset \mathbb {R} ^{k}} , the volumeform ω {\displaystyle \omega } on M {\displaystyle M} induced by the embedding...
the proportional volume of the breast in relation to the size of the nipple-areola complex, and thus created a breast of natural form and appearance; greater...
notion of left-handed and right-handed. These structures give rise to a volumeform, and also the cross product, which is used pervasively in vector calculus...
Sonata form (also sonata-allegro form or first movement form) is a musical structure generally consisting of three main sections: an exposition, a development...
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