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...

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...

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...

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 →...

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...

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

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

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...

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

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...

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...

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...

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

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...

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...

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...

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...

the volume as a whole is to locate and describe the concrete forms which grow out of the movements of capital as a whole. Thus, the various forms of capital...

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...

two-dimension recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also be expressed in terms...

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