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...

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...

n-dimensional manifold, a top-dimensional form (n-form) is called a volume form. The differential forms form an alternating algebra. This implies that...

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 volume form. A...

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...

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...

the volume of parallelepipeds, cones, pyramids, cylinders, and spheres. The formula were determined by prior mathematicians by using a primitive form of...

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 conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which...

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

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...

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

and for the special linear group an SL(n,R)-structure is the same as a volume form. For the trivial group, an {e}-structure consists of an absolute parallelism...

sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact...

a canonical volume form which is just the Riemannian volume form determined by g. This form is given in terms of the associated (1,1)-form ω by v o l M...

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...

The Theory of Forms or Theory of Ideas, also known as Platonic idealism or Platonic realism, is a philosophical theory widely credited to the Classical...

to define a natural volume form from the metric tensor. In a positively oriented coordinate system (x1, ..., xn) the volume form is represented as ω =...

unique n {\displaystyle n} -form d V g {\displaystyle dV_{g}} called the Riemannian volume form. The Riemannian volume form is preserved by orientation-preserving...

rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to area.) More precisely, the...

a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight ⁠1/2⁠...

The three-volume novel (sometimes three-decker or triple decker) was a standard form of publishing for British fiction during the nineteenth century. It...

political economy, discussed in the first chapter of Capital, Volume 1. It refers to the social form of tradeable things as units of value, which contrast with...

Riemannian metric on S, which itself induces a differential form on S known as the Riemannian volume form ωt. The first variation of area refers to the computation...

In botanical nomenclature, a form (forma, plural formae) is one of the "secondary" taxonomic ranks, below that of variety, which in turn is below that...

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...

Gaulois from 23 September 1909 to 8 January 1910, and was released in volume form in late March 1910 by Pierre Lafitte. The novel is partly inspired by...

integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface. Consider...

non-orientable manifolds, one cannot define a volume form globally due to the non-orientability, but one can define a volume element, which is formally a density...