Volume form, the Glossary

Table of Contents

1. 67 relations: Absolute continuity, Almost symplectic manifold, Atlas (topology), Borel set, Calculus, Complex manifold, Complex plane, Cotangent bundle, Covering space, Degenerate bilinear form, Density on a manifold, Determinant, Diffeomorphism, Differentiable manifold, Differential form, Differential geometry, Divergence, Divergence theorem, Exterior algebra, Fluid mechanics, Frame bundle, Function (mathematics), G-structure on a manifold, General linear group, Group (mathematics), Group action, Haar measure, Hodge star operator, Integral, Interior product, Jacobian matrix and determinant, Kähler manifold, Lebesgue integral, Lebesgue measure, Levi-Civita symbol, Lie derivative, Lie group, Line bundle, Local coordinates, Manifold, Mathematics, Measure (mathematics), Metric tensor, Moving frame, One-form (differential geometry), Orientability, Poincaré metric, Positive real numbers, Principal bundle, Principal homogeneous space, ... Expand index (17 more) »

2. Determinants
3. Differential forms
4. Integration on manifolds
5. Riemannian manifolds

Absolute continuity

In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

See Volume form and Absolute continuity

Almost symplectic manifold

In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form \omega on M that is everywhere non-singular.

See Volume form and Almost symplectic manifold

Atlas (topology)

In mathematics, particularly topology, an atlas is a concept used to describe a manifold.

See Volume form and Atlas (topology)

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

See Volume form and Borel set

Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

See Volume form and Calculus

Complex manifold

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holomorphic. Volume form and complex manifold are differential geometry.

See Volume form and Complex manifold

Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers.

See Volume form and Complex plane

Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

See Volume form and Cotangent bundle

Covering space

In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself.

See Volume form and Covering space

Degenerate bilinear form

In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V&hairsp) given by is not an isomorphism.

See Volume form and Degenerate bilinear form

Density on a manifold

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Volume form and density on a manifold are differential geometry.

See Volume form and Density on a manifold

Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. Volume form and determinant are determinants.

See Volume form and Determinant

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.

See Volume form and Diffeomorphism

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

See Volume form and Differentiable manifold

Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. Volume form and differential form are differential forms and differential geometry.

See Volume form and Differential form

Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

See Volume form and Differential geometry

Divergence

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point.

See Volume form and Divergence

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

See Volume form and Divergence theorem

Exterior algebra

In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v. Volume form and exterior algebra are differential forms.

See Volume form and Exterior algebra

Fluid mechanics

Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them.

See Volume form and Fluid mechanics

Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.

See Volume form and Frame bundle

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Volume form and Function (mathematics)

G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. Volume form and g-structure on a manifold are differential geometry.

See Volume form and G-structure on a manifold

General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

See Volume form and General linear group

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Volume form and Group (mathematics)

Group action

In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane.

See Volume form and Group action

Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

See Volume form and Haar measure

Hodge star operator

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Volume form and Hodge star operator are differential forms and Riemannian geometry.

See Volume form and Hodge star operator

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations.

See Volume form and Integral

Interior product

In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. Volume form and interior product are differential forms and differential geometry.

See Volume form and Interior product

Jacobian matrix and determinant

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. Volume form and Jacobian matrix and determinant are determinants.

See Volume form and Jacobian matrix and determinant

Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. Volume form and Kähler manifold are Riemannian manifolds.

See Volume form and Kähler manifold

Lebesgue integral

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis.

See Volume form and Lebesgue integral

Lebesgue measure

In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean ''n''-spaces.

See Volume form and Lebesgue measure

Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers, for some positive integer.

See Volume form and Levi-Civita symbol

Lie derivative

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. Volume form and Lie derivative are differential geometry.

See Volume form and Lie derivative

Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

See Volume form and Lie group

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space.

See Volume form and Line bundle

Local coordinates

Local coordinates are the ones used in a local coordinate system or a local coordinate space.

See Volume form and Local coordinates

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

See Volume form and Manifold

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Measure (mathematics)

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events.

See Volume form and Measure (mathematics)

Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. Volume form and metric tensor are differential geometry and Riemannian geometry.

See Volume form and Metric tensor

Moving frame

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. Volume form and moving frame are differential geometry.

See Volume form and Moving frame

One-form (differential geometry)

In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Volume form and one-form (differential geometry) are differential forms.

See Volume form and One-form (differential geometry)

Orientability

In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise".

See Volume form and Orientability

Poincaré metric

In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. Volume form and Poincaré metric are Riemannian geometry.

See Volume form and Poincaré metric

Positive real numbers

In mathematics, the set of positive real numbers, \R_.

See Volume form and Positive real numbers

Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with. Volume form and principal bundle are differential geometry.

See Volume form and Principal bundle

Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial.

See Volume form and Principal homogeneous space

Pseudo-Riemannian manifold

In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. Volume form and pseudo-Riemannian manifold are differential geometry, Riemannian geometry and Riemannian manifolds.

See Volume form and Pseudo-Riemannian manifold

Pseudotensor

In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., an improper rotation), which is a transformation that can be expressed as a proper rotation followed by reflection. Volume form and pseudotensor are differential geometry.

See Volume form and Pseudotensor

Pullback

In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product.

See Volume form and Pullback

Pullback (differential geometry)

Let \phi:M\to N be a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by \phi), and is frequently denoted by \phi^*. Volume form and pullback (differential geometry) are differential geometry.

See Volume form and Pullback (differential geometry)

Radon–Nikodym theorem

In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space.

See Volume form and Radon–Nikodym theorem

Retraction (topology)

In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace.

See Volume form and Retraction (topology)

Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Volume form and Riemannian manifold are differential geometry, Riemannian geometry and Riemannian manifolds.

See Volume form and Riemannian manifold

Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.

See Volume form and Section (fiber bundle)

Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a '''transverse vector field''') is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf.

See Volume form and Solenoidal vector field

Stokes' theorem

Stokes' theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12 (Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu.

See Volume form and Stokes' theorem

Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero.

See Volume form and Support (mathematics)

Symplectic manifold

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form \omega, called the symplectic form.

See Volume form and Symplectic manifold

Symplectic vector space

In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.

See Volume form and Symplectic vector space

Tensor contraction

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual.

See Volume form and Tensor contraction

Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n.

See Volume form and Vector field

Vector flow

In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field.

See Volume form and Vector flow

Volume element

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.

See Volume form and Volume element

See also

Determinants

• Alternant matrix
• Antisymmetrizer
• Bareiss algorithm
• Berezinian
• Capelli's identity
• Cauchy matrix
• Cauchy–Binet formula
• Cayley–Menger determinant
• Circulant matrix
• Cramer's rule
• Cross Gramian
• Determinant
• Determinantal conjecture
• Dieudonné determinant
• Discriminant
• Distance geometry
• Dodgson condensation
• Faddeev–LeVerrier algorithm
• Fischer's inequality
• Fredholm determinant
• Frobenius determinant theorem
• Functional determinant
• Gram matrix
• Grothendieck trace theorem
• Hadamard's inequality
• Hilbert matrix
• Hurwitz determinant
• Invertible matrix
• Jacobi's formula
• Jacobian matrix and determinant
• Laplace expansion
• Leibniz formula for determinants
• Maillet's determinant
• Minor (linear algebra)
• Moore matrix
• Persymmetric matrix
• Pfaffian
• Quasideterminant
• Resultant
• Rule of Sarrus
• Slater determinant
• Sylvester's determinant identity
• Totally positive matrix
• Vandermonde matrix
• Volume form
• Weinstein–Aronszajn identity
• Wronskian

Differential forms

• Berezin integral
• Closed and exact differential forms
• Complex differential form
• De Rham cohomology
• Differential form
• Differential forms on a Riemann surface
• Differential ideal
• Exterior algebra
• Exterior calculus identities
• Exterior derivative
• Generalized Stokes theorem
• Hodge star operator
• Hodge theory
• Interior product
• Lie algebra–valued differential form
• Mimetic interpolation
• One-form (differential geometry)
• Poincaré lemma
• Positive form
• Solder form
• Sum of residues formula
• Vector-valued differential form
• Volume form

Integration on manifolds

• Chain (algebraic topology)
• Generalized Stokes theorem
• Volume form

Riemannian manifolds

• Alexandrov space
• Arithmetic hyperbolic 3-manifold
• Curvature of Riemannian manifolds
• Einstein manifold
• Finsler manifold
• Flat manifold
• Frobenius manifold
• Fundamental theorem of Riemannian geometry
• Gauss's lemma (Riemannian geometry)
• Geodesic convexity
• Gibbons–Hawking space
• Gravitational instanton
• Hadamard manifold
• Hermitian manifold
• Hilbert manifold
• Hitchin–Thorpe inequality
• Hyperbolic 3-manifold
• Hyperbolic manifold
• Hyperkähler manifold
• Kähler manifold
• Kenmotsu manifold
• Milnor conjecture (Ricci curvature)
• Musical isomorphism
• Nash embedding theorems
• Otto calculus
• Pseudo-Riemannian manifold
• Ricci curvature
• Ricci flow
• Ricci-flat manifold
• Riemann curvature tensor
• Riemannian manifold
• Riemannian submanifold
• Sasaki metric
• Schur's lemma (Riemannian geometry)
• Sectional curvature
• Simons' formula
• Spin structure
• Spin(7)-manifold
• Sub-Riemannian manifold
• Volume form

References

[1] https://en.wikipedia.org/wiki/Volume_form

Also known as Area Element, Riemannian volume form, Top-dimensional form, Volume tensor.

, Pseudo-Riemannian manifold, Pseudotensor, Pullback, Pullback (differential geometry), Radon–Nikodym theorem, Retraction (topology), Riemannian manifold, Section (fiber bundle), Solenoidal vector field, Stokes' theorem, Support (mathematics), Symplectic manifold, Symplectic vector space, Tensor contraction, Vector field, Vector flow, Volume element.