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Concept in differential topology
a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form
Vector-valued differential form
Vector-valued_differential_form
Expression that may be integrated over a region
exact differential forms Complex differential form Vector-valued differential form Equivariant differential form Calculus on Manifolds Multilinear form Polynomial
Differential_form
Type of derivative in differential geometry
connection and vector-valued differential forms. A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be
Lie_derivative
Calculus of vector-valued functions
multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used
Vector_calculus
In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the
Lie algebra–valued differential form
Lie_algebra–valued_differential_form
Function valued in a vector space; typically a real or complex one
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional
Vector-valued_function
Quadratic form related to curvatures of surfaces
and d ν {\displaystyle d\nu } the differential of ν {\displaystyle \nu } regarded as a vector-valued differential form, and the brackets denote the metric
Second_fundamental_form
Math/physics concept
derivative. A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because
Connection_form
Broad concept generalizing scalars in mathematics and physics
Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the
Frölicher–Nijenhuis_bracket
Mathematics of smooth surfaces
unit normal vector field n to f(V), one defines the following objects as real-valued or matrix-valued functions on V. The first fundamental form depends only
Differential geometry of surfaces
Differential_geometry_of_surfaces
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued
Gradient
Concepts from linear algebra
algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear
Eigenvalues_and_eigenvectors
Assignment of a vector to each point in a subset of Euclidean space
elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field
Vector_field
Matrix of partial derivatives of a vector-valued function
natural generalization of the derivative and the differential of a usual function to vector valued functions of several variables. This generalization
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Operator in differential topology
mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an
Lie_bracket_of_vector_fields
Mathematical notion of infinitesimal difference
with a differential form on the target manifold. Covariant derivatives or differentials provide a general notion for differentiating of vector fields
Differential_(mathematics)
Concept in differential geometry
any differential k-form ω and any vector-valued form s. This may also be viewed as a direct inductive definition. For instance, for any vector-valued differential
Exterior_covariant_derivative
Circulation density in a vector field
vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume
Curl_(mathematics)
Geometric object that has length and direction
length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including
Euclidean_vector
Type of derivative in mathematics
{\displaystyle df} amalgamates these forms into a single object and is therefore an instance of a vector-valued differential form. If f : M → N {\displaystyle
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Mathematical manifold theory
Ωk(M) be the real vector space of smooth differential forms of degree k on M. The de Rham complex is the sequence of differential operators 0 → Ω 0 (
Hodge_theory
Differential equation containing derivatives with respect to only one variable
linear. A number of coupled differential equations form a system of equations. If y {\displaystyle \mathbf {y} } is a vector whose elements are functions;
Ordinary differential equation
Ordinary_differential_equation
its Levi-Civita connection gives its tangent vector bundle a flat structure. Vector-valued differential forms Local system, the more general notion of a
Flat_vector_bundle
Elliptic differential operators in geometry mathematics
In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Vector behavior under coordinate changes
Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of basis, and a covariant vector is a list of numbers that
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Differential equations involving stochastic processes
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution
Stochastic differential equation
Stochastic_differential_equation
Integration over a non-flat region in 3D space
position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it
Surface_integral
Differential form on a manifold which is permitted to have complex coefficients
complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have
Complex_differential_form
Algebraic structure in linear algebra
\mathbf {0} =(0,0)} is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example, f ′ ′
Vector_space
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Type of generalization of periodic functions in Euclidean space
The values of j may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The
Automorphic_form
Type of differential equation
classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components,
Partial_differential_equation
Polynomial with all terms of degree two
(orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of manifolds, especially
Quadratic_form
Method of mathematical optimization
Differential evolution (DE) is an evolutionary algorithm to optimize a problem by iteratively trying to improve a candidate solution with regard to a given
Differential_evolution
Statement about integration on manifolds
about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular,
Generalized_Stokes_theorem
Algebraic object with geometric applications
differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space
Tensor
Branch of mathematics
natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms. Beside Lie algebroids, also Courant algebroids start
Differential_geometry
Mathematical concept
In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information
Maurer–Cartan_form
Type of mathematical equation
derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions
Matrix_differential_equation
Mathematical function often applied to matrices
mathematics, the logarithmic norm is a real-valued functional on operators, constructed from either a vector norm or an inner product, or directly from
Logarithmic_norm
holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a
Quadratic_differential
Differential equation that is linear with respect to the unknown function
of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a vector space
Linear_differential_equation
on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. In an invariant differential operator D {\displaystyle
Invariant differential operator
Invariant_differential_operator
Algebra associated to any vector space
differential geometry, where it is used to define differential forms. Differential forms are mathematical objects that evaluate the length of vectors
Exterior_algebra
Surface specified with parameters
{r} =\mathbf {r} (u,v),} where r {\displaystyle \mathbf {r} } is a vector-valued function of the parameters (u, v) and the parameters vary within a certain
Parametric_surface
Exterior algebraic map taking tensors from p forms to n-p forms
decomposition of differential forms on a closed Riemannian manifold. Let V be an n-dimensional oriented vector space with a symmetric bilinear form ⟨ ⋅ , ⋅ ⟩
Hodge_star_operator
Notion in calculus
derivative. Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small
Differential_of_a_function
System of equations in mathematics
derivatives. The vector of dependent variables may then be written as pair ( x , y ) {\displaystyle (x,y)} and the system of differential equations of the
Differential-algebraic system of equations
Differential-algebraic_system_of_equations
Term in differential geometry
{\displaystyle {\mathfrak {g}}} -valued one-form on P). Then the curvature form is the g {\displaystyle {\mathfrak {g}}} -valued 2-form on P defined by Ω = d ω
Curvature_form
Specific mathematical differential form
of differential form. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator
Inexact_differential
Study of curves from a differential point of view
determines the curve. A parametric Cr-curve or a Cr-parametrization is a vector-valued function γ : I → R n {\displaystyle \gamma :I\to \mathbb {R} ^{n}} that
Differentiable_curve
Theorem in mathematics
subset of a Banach space. There is no exact analog of the mean value theorem for vector-valued functions (see below). However, there is an inequality which
Mean_value_theorem
Study of rates of change
real analysis, vector calculus, and multivariable calculus. The central idea of differential calculus is the derivative. For a real-valued function of one
Differential_calculus
Branch of mathematics
integration, and basic optimization. Vector analysis, also called vector calculus, is part of calculus that deals with vector-valued functions. Real analysis (traditionally
Mathematical_analysis
Fundamental construction of differential calculus
of differential equations defined by a vector valued function Rn to Rm, the Fréchet derivative A is a linear operator on R considered as a vector space
Generalizations of the derivative
Generalizations_of_the_derivative
on U {\displaystyle U} . Dual to these complex-valued one-forms are the complex-valued vector fields (that is, sections of the complexified tangent
Holomorphic_tangent_bundle
Equations describing classical electromagnetism
\mathbf {S} ,} where dS denotes the differential vector element of surface area S, normal to surface Σ. (Vector area is sometimes denoted by A rather
Maxwell's_equations
Vector differential operator
is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol)
Del
Differential operator in mathematics
{\displaystyle f} is a twice-differentiable real-valued function, then the Laplacian of f {\displaystyle f} is the real-valued function defined by: where the latter
Laplace_operator
integral curves (flow lines) of vector fields. In contrast, a flow process is defined with respect to a second-order differential operator, and thus, generalises
Stochastic analysis on manifolds
Stochastic_analysis_on_manifolds
Type of infinitesimal in calculus
calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is
Exact_differential
Calculus of functions of several variables
For example, for a real-valued function f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } with two real-valued parameters, f ( x , y ) {\displaystyle
Multivariable_calculus
Algebraic operation on coordinate vectors
numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their
Dot_product
Class of ordinary differential equations
Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x ) y {\displaystyle
Sturm–Liouville_theory
Methods of safely sharing general data
"Uber's differential privacy .. probably isn't". GitHub. Lyu, Min; Su, Dong; Li, Ninghui (1 February 2017). "Understanding the sparse vector technique
Differential_privacy
component of the tangent vectors to find a global vector field. A differential equation then can be read off the global vector field. There are various
Vector_field_reconstruction
Differential form
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
Volume_form
Typically linear operator defined in terms of differentiation of functions
coordinate-independent description of differential operators between two vector bundles. Let E and F be two vector bundles over a differentiable manifold
Differential_operator
Physics concept
covariantly (like basis vectors) and the ones that transform contravariantly (like components of a vector and differential forms) are "almost the same"
Covariant_transformation
Mathematical operation
. More generally, any covariant tensor field – in particular any differential form – on N {\displaystyle N} may be pulled back to M {\displaystyle M}
Pullback (differential geometry)
Pullback_(differential_geometry)
Topics referred to by the same term
Sesquilinear form, a generalisation of bilinear forms, especially on complex vector spaces (Hermitian forms) Argument form, a.k.a. Logical form or Test form - replacing
Form
French mathematician (born 1947)
(usually) multi-valued functions, but these multi-valued functions have merely isolated singularities without singularities that form cuts with dimension
Jean_Écalle
Matrix consisting of linearly independent solutions to a linear differential equation
ordinary differential equations x ˙ ( t ) = A ( t ) x ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)} is a matrix-valued function Ψ
Fundamental matrix (linear differential equation)
Fundamental_matrix_(linear_differential_equation)
Manifold upon which it is possible to perform calculus
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
Differentiable_manifold
Mathematical theorem
ISBN 9783642614972 Hubbard, John; Hubbard, Barbara (2015). Vector Calculus, Linear Algebra and Differential Forms (5th ed.). Matrix Editions. ISBN 9780971576681.
Symmetry of second derivatives
Symmetry_of_second_derivatives
Assignment of a tensor continuously varying across a region of space
generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field
Tensor_field
Mathematical parametrization of vector spaces by another space
associate (or "attach") a vector space V ( x ) {\displaystyle V(x)} in such a way that these vector spaces fit together to form another space of the same
Vector_bundle
System of moving vectors in differential geometry
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If
Parallel_transport
Measure of directional electromagnetic energy flux
the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition. The Poynting vector is used throughout
Poynting_vector
Operation in calculus
real-valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space
Integral
Specialized notation for multivariable calculus
when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is
Matrix_calculus
set, the value 1 corresponds to true and the value 0 to false. A two-valued vector logic requires a correspondence between the truth-values true (t) and
Vector_logic
Mathematical function
Depending on the context different exact definitions of this idea are in use. A vector field f : Rn → Rn is called coercive if f ( x ) ⋅ x ‖ x ‖ → + ∞ as ‖ x
Coercive_function
Operation on differential forms
theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k {\displaystyle k} -form is thought of as measuring the flux through an
Exterior_derivative
Algebraic variety that is a moduli space for principally polarized abelian varieties
modular varieties cannot be anabelian. Siegel modular forms arise as vector-valued differential forms on Siegel modular varieties. Siegel modular varieties
Siegel_modular_variety
Definite integral of a scalar or vector field along a path
curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes
Line_integral
Construct allowing differentiation of tangent vector fields of manifolds
so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are
Affine_connection
Coordinate-free definition of a tensor
vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor. A tensor on the vector space
Tensor_(intrinsic_definition)
Tangent vector Tangent space Tangent bundle Cotangent space Cotangent bundle Tensor Tensor bundle Vector field Tensor field Differential form Exterior
List of differential geometry topics
List_of_differential_geometry_topics
Mathematical function that outputs real values
each member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables
Real-valued_function
Instantaneous rate of change (mathematics)
independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. A function of a real variable
Derivative
Antisymmetric permutation object acting on tensors
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a
Levi-Civita_symbol
Mathematical condition
condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball
Poincaré_lemma
Type of vector space in math
spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space, and it has the additional structure of an inner product that allows
Hilbert_space
Manifold with supersymmetry structure
-dual of the module graded vector fields ∂ A ( Z ) {\displaystyle \partial A(Z)} is called the module of graded exterior one-forms O 1 ( Z ) {\displaystyle
Graded_manifold
Extension of the scalar spherical harmonics for use with vector fields
fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors. Several conventions have been used to define
Vector_spherical_harmonics
Mathematical operation on vector spaces
{\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated
Tensor_product
Vector field that is the gradient of some function
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property
Conservative_vector_field
VECTOR VALUED-DIFFERENTIAL-FORM
VECTOR VALUED-DIFFERENTIAL-FORM
Boy/Male
Spanish
Victor.
Male
Arthurian
, sir Hector de Maris; (defender).
Male
English
 Anglicized form of Scottish Gaelic Eachann, HECTOR means "brown horse." Compare with another form of Hector.
Surname or Lastname
Scottish
Scottish : Anglicized form of the Gaelic personal name Eachann (earlier Eachdonn, already confused with Norse Haakon), composed of the elements each ‘horse’ + donn ‘brown’.English : found in Yorkshire and Scotland, where it may derive directly from the medieval personal name. According to medieval legend, Britain derived its name from being founded by Brutus, a Trojan exile, and Hector was occasionally chosen as a personal name, as it was the name of the Trojan king’s eldest son. The classical Greek name, HektÅr, is probably an agent derivative of Greek ekhein ‘to hold back’, ‘hold in check’, hence ‘protector of the city’.German, French, and Dutch : from the personal name (see 2 above). In medieval Germany, this was a fairly popular personal name among the nobility, derived from classical literature. It is a comparatively rare surname in France.
Male
English
Roman Latin name VICTOR means "conqueror."Â
Male
English
Variant spelling of Middle English Alvred, ALURED means "elf counsel."
Boy/Male
American, British, Christian, Danish, Dutch, English, Finnish, French, German, Greek, Hindu, Indian, Irish, Jamaican, Latin, Romanian, Slovenia, Spanish, Swedish, Swiss, Tamil, Ukrainian
Victorious; Conqueror; Winner; Champion; One who Conquers; Victory
Male
Scandinavian
Scandinavian form of German Walther, VALTER means "ruler of the army."
Male
Scandinavian
 Scandinavian form of Roman Latin Victor, VIKTOR means "conqueror." Compare with another form of Viktor.
Boy/Male
English American
Doctor; teacher.
Male
English
Short form of English Sylvester, VESTER means "from the forest."
Male
Russian
(Cyrillic Виктор): Slavic form of Roman Latin Victor, VIKTOR means "conqueror." In use by the Bulgarians, Russians and Serbians. Compare with another form of Viktor.
Surname or Lastname
English
English : topographic name for someone who lived in a valley, Middle English valeye.
Male
Portuguese
Galician-Portuguese form of Roman Latin Victor, VITOR means "conqueror."
Female
Spanish
Spanish name SALUD means "health."
Boy/Male
Anglo, British, English, Finnish, Swedish
Valley; Usually with a Stream; From the Glen
Boy/Male
Australian, Basque, Czech, Czechoslovakian, Danish, Finnish, French, German, Hungarian, Latin, Polish, Slovenia, Swedish, Swiss, Ukrainian
The Conqueror; Victory; Victorious; Conquer
Male
Portuguese
Portuguese form of Latin Hector, HEITOR means "defend; hold fast."
Male
Spanish
Spanish form of Roman Latin Victor, VÃCTOR means "conqueror."
Male
Greek
(á¼ÎºÏ„ωÏ) Variant spelling of Greek Hektor, EKTOR means "defend; hold fast."
VECTOR VALUED-DIFFERENTIAL-FORM
VECTOR VALUED-DIFFERENTIAL-FORM
Girl/Female
Hindu, Indian
Creeper of Love
Boy/Male
Czech Czechoslovakian
Red earth.
Boy/Male
Hindu, Indian
Prince of Earth
Boy/Male
Australian, Hebrew
Jehovah Knows; A Name; Hand of God
Girl/Female
Indian
Protection
Girl/Female
Arabic, Muslim
Proper Name; Old Arabic Name
Boy/Male
Tamil
Creator of the universe, Creater of the Maya
Girl/Female
Tamil
Tungabhadra | தà¯à®‚கபதà¯à®°à®¾
Name of a river
Male
Egyptian
, the grandson of Tetet.
Boy/Male
German American
Mighty.
VECTOR VALUED-DIFFERENTIAL-FORM
VECTOR VALUED-DIFFERENTIAL-FORM
VECTOR VALUED-DIFFERENTIAL-FORM
VECTOR VALUED-DIFFERENTIAL-FORM
VECTOR VALUED-DIFFERENTIAL-FORM
n.
An African weaver bird (Textor alector).
v. t.
To confer a doctorate upon; to make a doctor.
v. t.
To tamper with and arrange for one's own purposes; to falsify; to adulterate; as, to doctor election returns; to doctor whisky.
n.
A directed quantity, as a straight line, a force, or a velocity. Vectors are said to be equal when their directions are the same their magnitudes equal. Cf. Scalar.
imp. & p. p.
of Value
n.
The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.
a.
Highly regarded; esteemed; prized; as, a valued contributor; a valued friend.
v. t.
To obtain the differential, or differential coefficient, of; as, to differentiate an algebraic expression, or an equation.
v. t.
To distinguish or mark by a specific difference; to effect a difference in, as regards classification; to develop differential characteristics in; to specialize; to desynonymize.
n.
An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.
n.
A woman who wins a victory; a female victor.
n.
The turning factor of a quaternion.
n.
Same as Radius vector.
a.
Of or pertaining to a differential, or to differentials.
n.
A belly, or protuberant part; a broad surface; as, the venter of a muscle; the venter, or anterior surface, of the scapula.
pl.
of Differentia
a.
Pertaining to a rector or a rectory; rectoral.
n.
A term made up of the two parts / + /1 /-1, where / and /1 are vectors.
a.
Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate.
n.
A small difference in rates which competing railroad lines, in establishing a common tariff, allow one of their number to make, in order to get a fair share of the business. The lower rate is called a differential rate. Differentials are also sometimes granted to cities.