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Mathematical parametrization of vector spaces by another space
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
Vector_bundle
Topics referred to by the same term
Smooth vector may refer to: Smooth vector for a strongly continuous group action; see group action Smooth vector field on a differentiable manifold; see
Smooth_vector
Assignment of a vector to each point in a subset of Euclidean space
then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable
Vector_field
Operator in differential topology
X {\displaystyle X} and Y {\displaystyle Y} on a smooth manifold M {\displaystyle M} a third vector field denoted [ X , Y ] {\displaystyle [X,Y]} . Conceptually
Lie_bracket_of_vector_fields
Smooth manifold with an inner product on each tangent space
Let M {\displaystyle M} be a smooth manifold. For each point p ∈ M {\displaystyle p\in M} , there is an associated vector space T p M {\displaystyle T_{p}M}
Riemannian_manifold
Degree of differentiability of a function or map
C^{k}} or C ∞ {\displaystyle C^{\infty }} smoothness by regarding the function as a map between real vector spaces. This should be distinguished from
Smoothness
Concepts in mathematics
topology, Riemannian geometry and Lie group theory. Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal
Vector_flow
Theorem in vector calculus
} With the above notation, if F {\displaystyle \mathbf {F} } is any smooth vector field on R 3 {\displaystyle \mathbb {R} ^{3}} , then ∮ ∂ Σ F ⋅ d Γ =
Stokes'_theorem
example of such a form.) Let M be a smooth manifold and E → M be a smooth vector bundle over M. We denote the space of smooth sections of a bundle E by Γ(E)
Vector-valued differential form
Vector-valued_differential_form
Certain vector fields are the sum of an irrotational and a solenoidal vector field
solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions
Helmholtz_decomposition
Relates the geometric vector bundles to algebraic projective modules
concerning smooth vector bundles on a smooth manifold (real, complex, or quaternionic). His topological variant is about continuous (real or complex) vector bundles
Serre–Swan_theorem
Tangent spaces of a manifold
parallelizable. A smooth assignment of a tangent vector to each point of a manifold is called a vector field. Specifically, a vector field on a manifold
Tangent_bundle
Linear approximation of smooth maps on tangent spaces
functor. Given a smooth map φ : M → N and a vector field X on M, it is not usually possible to identify a pushforward of X by φ with some vector field Y on
Pushforward_(differential)
sense that it sends smoothly parameterized families of vector spaces to smoothly parameterized families of vector spaces. Smooth functors may therefore
Smooth_functor
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Italian mathematician (born 1989)
Luigi Ambrosio and Alessio Figalli. Her dissertation, Flows of non-smooth vector fields and degenerate elliptic equations: With applications to the Vlasov-Poisson
Maria_Colombo_(mathematician)
{\displaystyle \nabla } is a connection on a Hermitian vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} which is compatible with
Hermitian_connection
Mathematical function defined piecewise by polynomials
knot vector t, a degree n, and a smoothness vector r for t, one can consider the set of all splines of degree ≤ n having knot vector t and smoothness vector
Spline_(mathematics)
Type of symmetry in physics
follows. A smooth vector field X on a spacetime M is said to preserve a smooth tensor T on M (or T is invariant under X) if, for each smooth local flow
Spacetime_symmetries
(Gel'fand & Fuks 1969–70), is a cohomology theory for Lie algebras of smooth vector fields. It differs from the Lie algebra cohomology of Chevalley-Eilenberg
Gelfand–Fuks_cohomology
Instrument in differential geometry
fundamental vector fields are instruments that describe the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find
Fundamental_vector_field
Manifold upon which it is possible to perform calculus
arbitrary nonzero vector in T e G , {\displaystyle T_{e}G,} one can use these identifications to give a smooth non-vanishing vector field on G . {\displaystyle
Differentiable_manifold
Calculus of vector-valued functions
subject of scalar field theory. A vector field is a smooth assignment of a vector to each point in a space. A vector field in the plane, for instance,
Vector_calculus
Mathematics concept
bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B {\displaystyle
Vertical and horizontal bundles
Vertical_and_horizontal_bundles
Defines a notion of parallel transport on a bundle
{\displaystyle \pi :E\to M} be a smooth real vector bundle over a smooth manifold M {\displaystyle M} . Denote the space of smooth sections of π : E → M {\displaystyle
Connection_(vector_bundle)
Mathematical result in differential geometry
on quasiconformal manifolds. X is a compact smooth manifold (without boundary). E and F are smooth vector bundles over X. D is an elliptic differential
Atiyah–Singer_index_theorem
Computer graphics images defined by points, lines and curves
Vector graphics are a form of computer graphics in which visual images are created directly from geometric shapes defined on a Cartesian plane, such as
Vector_graphics
Operation on differential forms
is, d f {\displaystyle df} is the unique 1-form such that for every smooth vector field X {\displaystyle X} , d f ( X ) = d X f {\displaystyle df(X)=d_{X}f}
Exterior_derivative
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
case of the hairy ball theorem, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks. Let
Poincaré–Hopf_theorem
Mathematical concept in particularly differential topology
secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle
Secondary vector bundle structure
Secondary_vector_bundle_structure
Quadratic form related to curvatures of surfaces
r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial
Second_fundamental_form
(semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle. Since (TM,πTM,M) is a vector bundle in its own right
Double_tangent_bundle
Structure defining distance on a manifold
pairs of tangent vectors to real numbers), and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric
Metric_tensor
Operation in Hamiltonian mechanics
algebra, the symplectic vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the Hamiltonian vector fields form an ideal
Poisson_bracket
Assignment of a tensor continuously varying across a region of space
{\displaystyle \otimes } now denotes the tensor product of vectors spaces, such that it constitutes a smooth map T : M → V ⊗ p ⊗ ( V ∗ ) ⊗ q {\displaystyle T:M\rightarrow
Tensor_field
Elliptic partial differential equation
every vector field is the gradient of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field
Poisson's_equation
Line or vector perpendicular to a curve or a surface
space at P . {\displaystyle P.} Normal vectors are of special interest in the case of smooth curves and smooth surfaces. The normal is often used in 3D
Normal_(geometry)
Isomorphism between the tangent and cotangent bundles of a manifold
every smooth manifold can be (non-canonically) endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a smooth manifold
Musical_isomorphism
Generalization of vector spaces from fields to rings
NM). If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X). The set of all smooth vector fields defined on
Module_(mathematics)
Representation theory of the symplectic group
that smooth vectors are also smooth for the metaplectic group. Moreover, a vector is in S {\displaystyle {\mathcal {S}}} if and only if it is a smooth vector
Oscillator_representation
Affine connection on the tangent bundle of a manifold
X, Y, Z are smooth vector fields on M, i. e. smooth sections of TM. [X, Y] is the Lie bracket of X and Y. It is again a smooth vector field. The metric
Levi-Civita_connection
Concept in differential geometry
{\displaystyle (M,g)} is called a Ricci soliton if, and only if, there exists a smooth vector field V {\displaystyle V} such that Ric ( g ) = λ g − 1 2 L V g ,
Ricci_soliton
Mathematical problem concerning limit cycles in dynamical systems
the globe. The problem asks whether in a generic family of smooth vector fields, smoothly parameterized over a compact set in finite dimensional Euclidean
Hilbert–Arnold_problem
Lie group homomorphism from the real numbers
The action of a one-parameter group on a set is known as a flow. A smooth vector field on a manifold, at a point, induces a local flow - a one parameter
One-parameter_group
System of moving vectors in differential geometry
tangent vector along the geodesic. Generally, the parallel transport of tangent vectors along a curve requires additional structure beyond the smooth structure
Parallel_transport
Property of vector fields in mathematics
Let M be a smooth manifold and A 0 , … , A n {\displaystyle A_{0},\dotsc ,A_{n}} be smooth vector fields on M. Assuming that these vector fields satisfy
Hörmander's_condition
Formulation of classical mechanics using momenta
\Omega ^{1}(M)} between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every f , g ∈ C ∞ ( M , R ) {\displaystyle
Hamiltonian_mechanics
Correspondsnce between Higgs bundles and fundamental group representations
following, fix a smooth complex vector bundle E {\displaystyle E} . Every Higgs bundle will be considered to have the underlying smooth vector bundle E {\displaystyle
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential
Complex_lamellar_vector_field
symplectic vector field is one whose flow preserves a symplectic form. That is, if ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold with smooth manifold
Symplectic_vector_field
Mappings between convenient vector spaces are smooth or C ∞ {\displaystyle C^{\infty }} if they map smooth curves to smooth curves. This leads to a Cartesian
Convenient_vector_space
Mathematics of smooth surfaces
the chain rule, that this vector does not depend on f. For smooth functions on a surface, vector fields (i.e. tangent vector fields) have an important
Differential geometry of surfaces
Differential_geometry_of_surfaces
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks
List_of_periodic_functions
Type of derivative in differential geometry
notion of transport (Lie transport). A smooth vector field defines a smooth flow on the manifold, which allows vectors to be transported between two points
Lie_derivative
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
Complex vector bundle on a complex manifold
bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety X (viewed
Holomorphic_vector_bundle
Definite integral of a scalar or vector field along a path
(t))\left|\mathbf {r} '(t)\right|dt.} For a vector field F: U ⊆ Rn → Rn, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is
Line_integral
Scalar potential used in fluid dynamics
theory. It was introduced by Joseph-Louis Lagrange in 1788. Suppose a smooth vector field u {\displaystyle \mathbf {u} } in a simple connected region represents
Velocity_potential
Generalization of a fiber bundle
category of morphisms in C. The category of smooth vector bundles is a bundle object over the category of smooth manifolds in Cat, the category of small categories
Bundle_(mathematics)
A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) M {\displaystyle M} whose flow preserves
Projective_vector_field
Mathematical operation
^{*}f)(x)=f(\phi (x))} is a smooth map from M {\displaystyle M} to A {\displaystyle A} . If E {\displaystyle E} is a vector bundle (or indeed any fiber
Pullback (differential geometry)
Pullback_(differential_geometry)
Concept in mathematics
Smooth vectors are dense in H by a classical argument of Lars Gårding, since convolution by smooth functions of compact support yields smooth vectors
Unitary_representation
Millennium Prize Problem
an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
In control theory, visible state of a system
^{p}} the output vector. f , g , h {\displaystyle f,g,h} are to be smooth vector fields. Define the observation space O s {\displaystyle {\mathcal {O}}_{s}}
Observability
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
Isomorphism of symplectic manifolds
the set of smooth vector fields on M {\displaystyle M} , and L X {\displaystyle {\mathcal {L}}_{X}} is the Lie derivative along the vector field X . {\displaystyle
Symplectomorphism
Extends the Jordan curve theorem to characterize the inner and outer regions
curve has a tubular neighbourhood, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at
Schoenflies_problem
Computer vision framework
that is produced by a process that smooths and diffuses an input vector field. It is usually used to create a vector field from images that points to object
Gradient_vector_flow
Tensor in differential geometry
map that takes smooth vector fields X {\displaystyle X} , Y {\displaystyle Y} , and Z {\displaystyle Z} , and returns the vector field R ( X ,
Ricci_curvature
Feature of a system that is preserved under some transformation
usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more
Symmetry_(physics)
Concept in mathematics
this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms. Structurally
Structural_stability
Type of differentiable manifold
{\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields { V 1 , … , V n } {\displaystyle \{V_{1},\ldots ,V_{n}\}} on
Parallelizable_manifold
Differential operator in mathematics
the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field
Laplace_operator
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in n {\displaystyle n} -dimensional
Vector_fields_on_spheres
Dual space to the tangent space in differential geometry
differential geometry, the cotangent space is a vector space associated with a point x {\displaystyle x} on a smooth (or differentiable) manifold M {\displaystyle
Cotangent_space
Singularity theorem in Yang–Mills theory
_{g}<\infty } the vector bundle η ↠ B 4 ∖ { 0 } {\displaystyle \eta \twoheadrightarrow B^{4}\setminus \{0\}} extends to a smooth vector bundle η ¯ ↠ B 4
Uhlenbeck's singularity theorem
Uhlenbeck's_singularity_theorem
Direct summand of a free module (mathematics)
smooth functions on a compact manifold is the space of smooth sections of a smooth vector bundle). Vector bundles are locally free. If there is some notion
Projective_module
Series of connected vectors in computer graphics
In graphics design, a vector path is a drawn or generated outline that represents a series of smooth straight (vector) lines instead of raster dots (or
Vector_path
Topics referred to by the same term
of algebraic topology Vect(X), the space of smooth vector fields on a manifold X, see Lie bracket of vector fields vect-, a Latin morpheme, see List of
VECT
Objects extending the notion of functions
duality for topological vector spaces. Its main rival in applied mathematics is mollifier theory, which uses sequences of smooth approximations (the 'James
Generalized_function
Group that is also a differentiable manifold with group operations that are smooth
derivation. If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is
Lie_group
Algebraic structure used in topology
can be made more explicit when E is a smooth vector bundle over a smooth manifold X, since then a general smooth section of X vanishes on a codimension-r
Cohomology
Distance along a curve
mathematically for smooth curves using vector calculus and differential geometry, or for curves that might not necessarily be smooth as a limit of lengths
Arc_length
Differential geometry construct on fiber bundles
connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber
Ehresmann_connection
Expression that may be integrated over a region
tangent vector at all. Since a vector field on N determines, by definition, a unique tangent vector at every point of N, the pushforward of a vector field
Differential_form
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
mathematics and theoretical physics, a Killing vector field or Killing field (named after Wilhelm Killing) is a vector field on a Riemannian manifold or pseudo-Riemannian
Killing_vector_field
Vector tangent to a curve or surface at a given point
tangent vector, we discuss its use in calculus and its tensor properties. Let r ( t ) {\displaystyle \mathbf {r} (t)} be a parametric smooth curve. The
Tangent_vector
{\displaystyle n} , its tangent bundle as a smooth vector bundle is a real rank 2 n {\displaystyle 2n} vector bundle T M {\displaystyle TM} on M {\displaystyle
Holomorphic_tangent_bundle
Differential map between manifolds whose differential is everywhere surjective
diffeomorphisms Riemannian submersions The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a
Submersion_(mathematics)
Algebra associated to any vector space
smooth functions in k {\displaystyle k} variables. The two-dimensional Euclidean vector space R 2 {\displaystyle \mathbf {R} ^{2}} is a real vector space
Exterior_algebra
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
Principal bundle associated to a vector bundle
E} . The above all works in the smooth category as well: if E {\displaystyle E} is a smooth vector bundle over a smooth manifold M {\displaystyle M} then
Frame_bundle
Fitting an approximating function to data
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data
Smoothing
Theorem in Optimal Transport
be a smooth vector field it can then be written in a unique way as V = w + ∇ p {\displaystyle V=w+\nabla p} where p {\displaystyle p} is a smooth real
Polar_factorization_theorem
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
Operation in differential geometry
a curve through p is a tangent vector. A tangent vector at p is a first-order differential operator acting on smooth real-valued functions at p. In local
Jet_(mathematics)
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
concepts must be clearly defined. One begins by noting that an arbitrary smooth vector field X {\displaystyle X} on a manifold M {\displaystyle M} defines
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions
Nuclear_space
pseudoscalar of the vector manifold is a (pseudoscalar-valued) function of the points on the vector manifold. If i.e. this function is smooth then one says
Universal_geometric_algebra
Algebraic object with geometric applications
of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There
Tensor
SMOOTH VECTOR
SMOOTH VECTOR
Girl/Female
Hindu, Indian
Smooth
Surname or Lastname
English
English : occupational name for a worker in metal, from Middle English smith (Old English smið, probably a derivative of smītan ‘to strike, hammer’). Metal-working was one of the earliest occupations for which specialist skills were required, and its importance ensured that this term and its equivalents were perhaps the most widespread of all occupational surnames in Europe. Medieval smiths were important not only in making horseshoes, plowshares, and other domestic articles, but above all for their skill in forging swords, other weapons, and armor. This is the most frequent of all American surnames; it has also absorbed, by assimilation and translation, cognates and equivalents from many other languages (for forms, see Hanks and Hodges 1988).
Girl/Female
Hindu, Indian
Smooth
Girl/Female
Arabic, Indian
Smooth; Soft
Surname or Lastname
English (South Yorkshire)
English (South Yorkshire) : unexplained.
Boy/Male
Latin American English Irish Norse
Smooth.
Boy/Male
Chinese
Smooth.
Girl/Female
Indian, Telugu
Smooth
Boy/Male
Hindu, Indian
Smooth; Tender
Girl/Female
German, Polish
Smooth-brow
Surname or Lastname
English
English : from Middle English south, hence a topographic name for someone who lived to the south of a settlement or a regional name for someone who had migrated from the south.
Female
Egyptian
, Child of Mouth.
Boy/Male
Tamil
Smooth
Boy/Male
Indian
Smooth
Surname or Lastname
English (south and south Midlands)
English (south and south Midlands) : variant spelling of Laing.
Boy/Male
Greek, Indian
Smooth Rock
Girl/Female
Hindu, Indian
Soft; Smooth
Boy/Male
Australian, Chinese, Danish, Latin
Smooth; Polished
Boy/Male
Hindu, Indian
Smooth
Girl/Female
Indian, Telugu
Inspiration
SMOOTH VECTOR
SMOOTH VECTOR
Boy/Male
Hindu, Indian, Punjabi, Sikh
Inner Soul; Light
Surname or Lastname
English
English : habitational name from Alvingham in Lincolnshire, named in Old English as Aluingeham ‘homestead (Old English hÄm) of the family or followers of Ælf(a)’. Reaney also mentions a lost place called Allingham in Kent as a possible source; this is perhaps the same as one of the two places in Kent called Allington.
Boy/Male
Christian & English(British/American/Australian)
Cunning
Boy/Male
Indian, Punjabi, Sikh
Liberation through Guru
Boy/Male
Australian, Danish, French, Greek, Vietnamese
Thinker
Female
English
Variant spelling of English Rosanne, ROSEANN means "rose of grace."
Girl/Female
Tamil
Dawn
Girl/Female
American, Australian
Combination of Prefix Sha with Rae
Boy/Male
Hebrew
Comfort.
Girl/Female
American, British, English
Combination of Krystal and Lynn; Sparkling K from the Greek Spelling of Krystallos
SMOOTH VECTOR
SMOOTH VECTOR
SMOOTH VECTOR
SMOOTH VECTOR
SMOOTH VECTOR
adv.
Smoothly.
adv.
In a smooth manner.
v. t.
To smooth.
imp. & p. p.
of Smooth
adv.
From the south; as, the wind blows south.
n.
That which is smooth; the smooth part of anything.
a.
Speaking smoothly; plausible; flattering; smooth-tongued.
a.
Having a smooth chin; beardless.
superl.
Evenly spread or arranged; sleek; as, smooth hair.
a.
To palliate; to gloze; as, to smooth over a fault.
superl.
Having an even surface, or a surface so even that no roughness or points can be perceived by the touch; not rough; as, smooth glass; smooth porcelain.
a.
To assuage; to mollify; to calm; to comfort; as, to soothe a crying child; to soothe one's sorrows.
n.
The act of making smooth; a stroke which smooths.
a.
To make smooth; to make even on the surface by any means; as, to smooth a board with a plane; to smooth cloth with an iron.
v. t.
To make smooth.
n.
One who, or that which, smooths.
a.
Having a smooth tongue; plausible; flattering.
a.
To give a smooth or calm appearance to.
v. i.
To put mouth to mouth; to kiss.
superl.
Gently flowing; moving equably; not ruffled or obstructed; as, a smooth stream.