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Longer titles found: Vector field reconstruction (view), Vector fields in cylindrical and spherical coordinates (view), Vector fields on spheres (view), Conservative vector field (view), Solenoidal vector field (view), Lie bracket of vector fields (view), Killing vector field (view), Hamiltonian vector field (view), Complex lamellar vector field (view), Conformal Killing vector field (view), Boundary vector field (view), Homothetic vector field (view), Vector Field Histogram (view), Symplectic vector field (view), Laplacian vector field (view), Fundamental vector field (view), Beltrami vector field (view), Projective vector field (view), Affine vector field (view), Time dependent vector field (view), Reeb vector field (view), Gradient-like vector field (view), Variational vector field (view), Polyvector field (view), Straightening theorem for vector fields (view), Isotropic vector field (view)

searching for Vector field 167 found (803 total)

alternate case: vector field

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1)-dimensional space. The del operator illustrates the relationship between a vector field and its associated scalar potential field. An equipotential region might

curves of the vector field. The local flow of a vector field is used to define the Lie derivative of tensor fields along the vector field. A curve ϕ :

of a Dirac field with a vector field in dimension two. The Lagrangian density is made of three terms: the free vector field A μ {\displaystyle A^{\mu

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used

modeled as a continuous vector field V. The implicit function f is found by integrating the vector field V. Since not every vector field is the gradient of

contravariant tensor field may be referred to as a k {\displaystyle k} -vector field. A tensor A that is antisymmetric on indices i {\displaystyle i} and

Differential equations of this form are notable for having a discontinuous vector field. Projected dynamical systems have evolved out of the desire to dynamically

bundles, there are distinguished holomorphic sections. A holomorphic vector field is a holomorphic section of T 1 , 0 M {\displaystyle T^{1,0}M} . A holomorphic

In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra

of MSTG, in which the skew symmetric tensor field was replaced by a vector field, is scalar–tensor–vector gravity (STVG). STVG, like Milgrom's Modified

the exterior derivative of this form Canonical vector field, the corresponding special vector field defined on the tangent bundle TM of a manifold M

key results due to Thom: Given a vector field ζ {\displaystyle \zeta } on N, there exists a controlled vector field η {\displaystyle \eta } on S that

Simulation noise is a function that creates a divergence-free vector field. This signal can be used in artistic simulations for the purpose of increasing

\partial _{v}} , the coordinate vector field dual to the covector field d v {\displaystyle dv} , is a null vector field. Indeed, geometrically speaking

Simulation noise is a function that creates a divergence-free vector field. This signal can be used in artistic simulations for the purpose of increasing

smooth vector field on the two sphere vanishing only at 0 and ∞. It has index 1 at 0 and -1 at ∞. Near 0 the vector field equals the radial vector field pointing

{\displaystyle y^{\prime }=F(y)+\alpha (x-y)} where F {\displaystyle F} is the vector field modeling the isolated chaotic dynamics and α {\displaystyle \alpha }

vector field. Assuming normalization Δ(1)=0, the odd Laplacian Δ ρ {\displaystyle {\Delta }_{\rho }} is just the Δ operator, and the modular vector field

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 ‖

as follows. Given a geodesic S1 → M with an orthogonal and parallel vector field along the geodesic (i.e. a parallel section of the normal bundle to the

payload of the three spacecraft consists of the following instruments: Vector Field Magnetometer (VFM): Linear and low-noise measurements of the Earth's

base, on top of which the vector field can be applied. That means for every particle looking backward in the vector field to find out the new value for

that any Lorentzian manifold which admits a covariantly constant null vector field k {\displaystyle k} is called a pp-wave spacetime. That is, the covariant

vector fields in a frame field, taken with respect to the timelike unit vector field in the frame field, are used to define non-inertial and non-rotating

Bumblebee models are effective field theories describing a vector field with a vacuum expectation value that spontaneously breaks Lorentz symmetry. A bumblebee

)=-\mathbf {b} \cdot (\mathbf {a\times c} )} . If F = (F1, F2, F3) is a vector field defined on some open set of R 3 {\displaystyle \mathbb {R} ^{3}} as a

that | h μ ν | ≪ 1 {\displaystyle |h_{\mu \nu }|\ll 1} . Given then a vector field ξ μ {\displaystyle \xi ^{\mu }} defined on the flat background spacetime

that | h μ ν | ≪ 1 {\displaystyle |h_{\mu \nu }|\ll 1} . Given then a vector field ξ μ {\displaystyle \xi ^{\mu }} defined on the flat background spacetime

analytic function f defined on Rn and a trajectory x(t) of the gradient vector field of f having a limit point x0 ∈ Rn, where f has an isolated critical point

3D-surfaces. They are obtained by calculating the streamlines of a derived vector field on the surface, the wall shear stress. Skin friction arises from the

_{TM})_{*}\subset TTM.} The vertical lift lets us define the canonical vector field V : T M → T T M ; V ξ := vl ξ ⁡ ξ , {\displaystyle V:TM\to TTM;\qquad

the vector field performs around the knot. Knowing how many times the vector field is twisted around the knot allows one to determine the vector field up

configuration of the body at a given time t {\displaystyle t\,\!} . It is not a vector field because it depends not only on the position x {\displaystyle \mathbf

observability Killing horizon, a null surface on which there is a Killing vector field Particle horizon, the maximum distance from which particles can have

is the vector field 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

{\displaystyle t=t_{0}} admit a Bianchi III (cf. the fourth Killing vector field). This can be rewritten as the symmetry group containing three-dimensional

sections. Intuitively this means that on a neighborhood of a point, with a vector field in that neighborhood, the local parallel transport can be extended globally

data visualization (using an OpenCL octree raycaster) Streamline-based vector field visualization Multimodal volume rendering Geometry rendering with support

1991/1992 by Yulii Ilyashenko and Jean Écalle that every polynomial vector field in the plane has only finitely many limit cycles (a 1923 article by Henri

previously been observed (in quantum electrodynamics with a charged vector field, by V.S. Vanyashin and M.V. Terent'ev in 1965; and Yang–Mills theory

dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis. Banchoff

and whose principal curvatures are constant along any parallel normal vector field. The set of isoparametric manifolds is stable under the mean curvature

dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis. Banchoff

\|\alpha ^{\prime }\|=1} , and T {\displaystyle T} is the unit tangent vector field along α {\displaystyle \alpha } , then the curve σ = T {\displaystyle

) ctg – cotangent function. (Also written as cot.) curl – curl of a vector field. (Also written as rot.) cvc – covercosine function. (Also written as

smooth manifolds, the set of integral curves defined by a nonvanishing vector field defined on the manifold Congruence (general relativity), in general relativity

vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of integral curves of this vector field.

symmetrized derivative of a vector field appears also as the Lie derivative of the metric tensor g with respect to the vector field. T i j = ( L U g ) i j

varieties. The Carrell–Liebermann theorem on the zero set of a holomorphic vector field is used in complex algebraic geometry. He is a fellow of the American

In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks

\Omega \subseteq \mathbb {R} ^{n}} and a once-weakly differentiable vector field u ∈ H 1 ( R n ) n {\displaystyle u\in H^{1}(\mathbb {R} ^{n})^{n}} which

non-zero section in the normal bundle of the knot, i.e. a (non-zero) normal vector field. Given a framed knot C, the self-linking number is defined to be the

on a body are necessary and sufficient so that there exists a unique vector field v ( X ) {\displaystyle \mathbf {v} (\mathbf {X} )} that satisfies ∇ v

a counterexample to the Seifert conjecture that every non-vanishing vector field on the 3-sphere has a closed integral curve. In 1995 he demonstrated

science Source of a morphism Source, a point where the divergence of a vector field is positive Source of a representation, in finite group theory Source

the mathematical term 'vector field'. Heightmap is the more correct description because most heightfields are not a (vector) field in mathematical terms

covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent

can be shown that the upper-convected time derivative of a spacelike vector field is just its Lie derivative by the velocity field of the continuum. The

skew-symmetric. So, ( ∇ X J ) X = 0 {\displaystyle (\nabla _{X}J)X=0} for every vector field X {\displaystyle X} on M {\displaystyle M} . In particular, a Kähler

decomposition of the Riemann tensor, with respect to a timelike unit vector field X → {\displaystyle {\vec {X}}} , not necessarily geodesic or hypersurface

equations, i.e., solutions which admit a time-translational Killing vector field everywhere, not just in a small neighborhood of the black hole. While

Initially it was based on a model from an existing research work regarding vector-field Consistency, awarded the best-paper prize in the ACM/IFIP/Usenix Middleware

{c} )} where c is an arbitrary constant vector and v is a vector field. Consider a vector field v and an arbitrary constant vector c. In index notation

an interval I, with c ′ ( s ) ≠ 0 {\displaystyle c'(s)\neq 0} and a vector field n ( s ) {\displaystyle n(s)} along c such that | | n ( t ) | | = 1 {\displaystyle

on a regular grid and then bend this grid according to the flow (the vector field). The bent grid is then sampled at the original grid locations. Thus

neither.) The pair ( f , g ) {\displaystyle (f,g)} gives us a gradient vector field. We say that ( f , g ) {\displaystyle (f,g)} is Morse–Smale if the stable

and γ′(0) = v. The Lie derivative of a vector field W μ ( x ) {\displaystyle W^{\mu }(x)} along a vector field V μ ( x ) {\displaystyle V^{\mu }(x)} is

hypersurface embedding of M into M, together with a continuous unit normal vector field, such that the induced metric is g and the second fundamental form with

_{t\to \infty }x(t)=0,\quad \lim _{t\to \infty }y(t)=c.} Consider the vector field given by x ˙ = − x + x 2 y , y ˙ = x y 2 {\displaystyle {\dot {x}}=-x+x^{2}y

resolved to a vector field by using one of the two eigenvectors to represent the tensor each point in the field and then visualized using vector field visualization

Vector Field, Springer (2024). Albert C. J. Luo, Two-dimensional Single-Variable Cubic Nonlinear Systems, Vol. I:A Self-univariate Cubic Vector Field

spacetime, a spacetime having a global, non-vanishing, timelike Killing vector field which is irrotational Statics, a branch of physics concerned with physical

fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particularly susceptible to topological

change of the components of the metric tensor along the flow of a given vector field, since ∇ μ ≡ ∇ ∂ μ {\displaystyle \nabla _{\mu }\equiv \nabla _{\partial

N is the inward normal vector field. The signed distance function is thus a differentiable extension of the normal vector field. In particular, the Hessian

fields which, therefore, is globally defined. It is a nowhere vanishing vector field on X {\displaystyle X} . Accordingly, the dual time-like covector field

as well as the layered structures that can be described by a lamellar vector field. In medical professions, especially orthopedic surgery, the term is used

symmetry are absorbed in the description of the gauge vector field, providing massive vector field modes, like the plasma mode in a superconductor, or the

unit normal vector field along f such that f *g = g and such that the second fundamental form of f, relative to the given normal vector field, is k. In

semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions

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 , {\displaystyle

dy~.} A 1-parameter group of conformal transformations gives rise to a vector field X with the property that the Lie derivative of g along X is proportional

_{t}f_{t}(z)=v(f_{t}(z)),\,\,\,f_{t}(z)=0~,} the flow equation for a vector field. Restricting to the case with 0 < λ < 1, the h(D) must be starlike so

n + 1. Given an immersion f of S into M together with a unit normal vector field along f, the second fundamental form of f can be viewed as a symmetric

H+Hh^{2}-|h|^{2}h,} where, relative to a local choice of unit normal vector field, h is the second fundamental form, H is the mean curvature, and h2 is

when it is 0. The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n = 2, this is sometimes called the "hairy

φ = φ(x) be a well defined scalar field and v = v(x) a well-defined vector field, and λ1, λ2... be parameters of the coordinates Tangent vector: If x(λ)

domain of fluid dynamics to analyze turbulences, is to decompose a random vector field u(x, t) into a set of deterministic spatial functions Φk(x) modulated

\mathbf {r} =x^{i}=x^{i}(t),\quad a\leq t\leq b\,,} then the tangent vector field T = T i {\displaystyle \mathbf {T} =T^{i}} is given by T i = d x i d

Example hyperbolic flow, illustrating stable and unstable manifolds. The vector field equation is ( x + exp ⁡ ( − y ) , − y ) {\displaystyle (x+\exp(-y),-y)}

behavior can be expressed as a dynamical system represented as a vector field. In this vector field, attractors reflect stable behavioral solutions, where as

Irradiance the moment of inertia intensity in physics, typically the vector field I Luminous intensity, typically Iv the incenter of a triangle the electric

corresponding vector field Y = r∂x + s∂y. Thus Y corresponds to the vector field ∂θ. The radial vector field on the annulus r∂r is a commuting vector field which

that is, for a k-vector field ψ(r), the derivative ∇ψ generally contains terms of grade k + 1 and k − 1. For example, a vector field (k = 1) generally

vessel but never taken over by the Navy Spray (mathematics), a type of vector field in differential geometry defined on the tangent bundle of a manifold

{\displaystyle \delta _{\omega }^{*}h=\omega ^{2}h} Let X be the vertical vector field generating the scaling action. Then the following properties are immediate:

If (M, ω) is a symplectic manifold, then a smooth vector field Y on M is a Hamiltonian vector field if the contraction ω(Y, ·) is an exact 1-form (i.e

{\displaystyle f_{i}:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}} is a vector field representing the right hand side of an ordinary difference equation,

research. For example, if X {\displaystyle X} is a recurrent non-null vector field on a pseudo-Riemannian manifold satisfying ∇ X = ω ⊗ X {\displaystyle

{F} (\mathbf {r} ,t)\right]\cdot d\mathbf {s} ,} where: F(r, t) is a vector field at the spatial position r at time t, Σ is a surface bounded by the closed

three-dimensional contact manifolds, thus establishing that the Reeb vector field on such a manifold always has a closed orbit. Expanding both on this

decomposition of the Riemann tensor, with respect to a timelike unit vector field X → {\displaystyle {\vec {X}}} , not necessarily geodesic or hypersurface

Principle, Lecture 23, Math 634", by Christopher Grant. Consider the vector field ( x ˙ , y ˙ ) = ( − y − x 3 , x 5 ) {\displaystyle ({\dot {x}},{\dot

is smooth. Let X ( t , x ) {\displaystyle X(t,x)} be a time dependent vector field on M {\displaystyle M} (in C ∞ ( R , X ( M ) ) {\displaystyle C^{\infty

Relativistic vector field

would be required by the Divergence Theorem applied to the constant vector field r ^ {\displaystyle \mathbf {\hat {r}} } ) and dividing by two: A ′ =

aid other decisions; Treat the colour image as a vector field, and use derivatives of the vector field as the colour gradient. 2.5D (visual perception)

even Lipschitz) submanifold of M. The flow of any continuous timelike vector field defines a homeomorphism S × ℝ → M. By considering the restriction of

and normal components. For each X ∈ T M {\displaystyle X\in TM} and vector field Y on M, ∇ X ′ Y = ⊤ ( ∇ X ′ Y ) + ⊥ ( ∇ X ′ Y ) . {\displaystyle \nabla

of any tensor is the subscripted nabla symbol ∇β. For the case of a vector field Aα: ∇ β A α = A α ; β . {\displaystyle \nabla _{\beta }A^{\alpha }=A^{\alpha

theorem of a signed measure Helmholtz decomposition, decomposition of a vector field Indecomposable continuum Lebesgue's decomposition theorem, decomposition

condition φ ˙ 0 = v 0 {\displaystyle {\dot {\varphi }}_{0}=v_{0}} , vector field dynamics φ ˙ t = v t ∘ φ t , t ∈ [ 0 , 1 ] {\displaystyle {\dot {\varphi

mesonic theory of nuclear forces, including the equations for the mesonic vector field that carry his name (Proca's equations), elected post-mortem as a Member

local coordinates at a point and use the Einstein notation. Consider a vector field v = v i ( x ) ∂ / ∂ x i {\displaystyle v=v^{i}(x)\partial /\partial x^{i}}

Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The

divergence and ∇ × X {\displaystyle \nabla \times \mathbf {X} } the curl of a vector field X {\displaystyle \mathbf {X} } ; ∂ B ∂ t {\displaystyle {\frac {\partial

RG transformations map actions to nearby ones, thus giving rise to a vector field on theory space. The scale dependence of an action is encoded in a "running"

group, and the presentation above is a standard exposition of a Lie vector field. The connection to the Anosov flow comes from the realization that g

maximal (minimal) with respect to perturbations of the initial normal vector field n 0 {\displaystyle n_{0}} . As earlier, we refer to repelling and attracting

{\displaystyle {\mathfrak {g}}^{\ast }} gives a translation-invariant vector field on g {\displaystyle {\mathfrak {g}}} . As a result, the algebra S ( g

written in the language of Newman–Penrose formalism. Denote the tangent vector field of an observer's worldline (in a timelike congruence) as Z a {\displaystyle

calculus, involving a scalar-valued function u and vector-valued function (vector field) V. The product rule for divergence states: ∇ ⋅ ( u V )   =   u ∇ ⋅ V

^{2}{\mathbf {v} }} where: v {\displaystyle \mathbf {v} } is the fluid velocity vector field, p {\displaystyle p} is the fluid pressure field, μ {\displaystyle \mu

vector for a strongly continuous group action; see group action Smooth vector field on a differentiable manifold; see tangent space A smooth vector in a

vector for a strongly continuous group action; see group action Smooth vector field on a differentiable manifold; see tangent space A smooth vector in a

calculus, involving a scalar-valued function u and vector-valued function (vector field) V. The product rule for divergence states: ∇ ⋅ ( u V )   =   u ∇ ⋅ V

Line, by Mohamed Amine Khamsi, S.O.S. Math, last Update 1998-6-22 "The phase line and the graph of the vector field". math.bu.edu. Retrieved 2015-04-23.

{\displaystyle \partial _{v}} (a null translation, "opposing" the wave vector field ∂ u {\displaystyle \partial _{u}} ) ∂ x , ∂ y {\displaystyle \partial

collaborated with T. H. Wise to publish the two volumes of Hydrodynamics and Vector Field Theory, in 1962 and 1963. She published a textbook on optimization in

second step is feature extraction. Out of the two- or higher-dimensional vector field received from the preprocessing algorithms, higher-dimensional data is

\nabla f\,,} with a vector field F, the divergence is written as ∇ ⋅ F , {\displaystyle \nabla \cdot F,} and with a vector field F, the curl is written

U(x,t) = UJEJ in the Eulerian description. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates

solutions. A ray is a family of geodesic light-like curves. That is tangent vector field l a {\displaystyle l^{a}} is null and geodesic: l a l a = 0 {\displaystyle

tensor of 3-dimensional space. In this coordinate system the Killing vector field ξ μ {\displaystyle \xi ^{\mu }} has the components ξ μ = ( 1 , 0 , 0

This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X )

multipliers: the lapse function, N {\displaystyle N} , and components of shift vector field, N i {\displaystyle N_{i}} . These describe how each of the "leaves"

curvature of M: Suppose that the Ricci curvature of M satisfies, for every vector field X, Ric ⁡ ( X , X ) ≥ k ( n − 1 ) | X | 2 . {\displaystyle \operatorname

namely the Rindler wedge). The integral curves of the timelike unit vector field e → 0 {\displaystyle {\vec {e}}_{0}} give a timelike congruence, consisting

through time. The fluid parcels are labelled by some (time-independent) vector field x0. (Often, x0 is chosen to be the position of the center of mass of

the method of averaging is to tell us the qualitative behavior of the vector field when we average it over a period of time. It guarantees that the solution

(Common) symbol/s Defining equation SI units Dimension Flow velocity vector field u u = u ( r , t ) {\displaystyle \mathbf {u} =\mathbf {u} \left(\mathbf

1990). "An algorithm for the visual representation of a two-dimensional vector field". Conference Record of the 1990 IEEE Industry Applications Society Annual

present and future population and job distributions. It is based on vector field calculations obtained by assuming that, in a given space, all inhabitants

Lusztig, G.; Procesi, C. (1988). "Homology of the zero-set of a nilpotent vector field on a flag manifold". Journal of the American Mathematical Society. 1:

present and future population and job distributions. It is based on vector field calculations obtained by assuming that, in a given space, all inhabitants

operators. This can only be done in spacetimes with a timelike Killing vector field. This decomposition happens to be different in Cartesian and Rindler

1990). "An algorithm for the visual representation of a two-dimensional vector field". Conference Record of the 1990 IEEE Industry Applications Society Annual

equations are then a particular singular ordinary differential equation with vector field f(x)/g(x), where both f and g may vanish at a certain point (angle and

a vertex with 0 out-degree Sink, a point where the divergence of a vector field is negative Sink, a flow network node that has more incoming flow than

F:M\to N} be an immersion. If there is a global choice of unit normal vector field, then relative to this choice, the second fundamental form is a Codazzi

of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero. So in the Abelian Higgs model, the gauge field acquires a

matrices. This characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence

the local baryon number density of the nucleon. With the pion isospin vector field oriented in the shape of a hedgehog space, the model is readily solvable

\tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}} Let A be a vector field of the form: A = A σ σ ^ + A τ τ ^ + A z z ^ {\displaystyle \mathbf {A}

The Noether current is defined up to a solenoidal (divergenceless) vector field. In the context of gravitation, Felix Klein's statement of Noether's

V_{\mu }+\partial _{\mu }A} , where V μ {\displaystyle V_{\mu }} is a vector field and A {\displaystyle A} is the gauge function. The main difficulty in

present and future population and job distributions. It is based on vector field calculations obtained by assuming that, in a given space, all inhabitants

(X_{f})\right)+f} . where X f {\displaystyle X_{f}} is the Hamiltonian vector field associated to f {\displaystyle f} . More generally, suppose ( M , ω )

the definition of the strain tensor as a function of the displacement vector field, which means that these constraints introduce no new concepts or information

in order to establish the optimum operating window. Flow velocity – Vector field which is used to mathematically describe the motion of a continuum Body

\nabla \cdot \mathbf {F} } is the divergence on the two-dimensional vector field F {\displaystyle \mathbf {F} } , and n ^ {\displaystyle \mathbf {\hat

always continuous (and even Lipschitz continuous). By using the flow of a vector field chosen to be complete, smooth, and timelike, it is elementary to prove

objects Killing horizon, a null surface on which there is a Killing vector field Radio horizon, the locus of points in telecommunication at which direct

fundamental form (relative to one of the two choices of unit normal vector field) h. Furthermore, each of these embeddings is uniquely determined up to

– measured output error. Hence, due to this high-gain feedback, the vector field of the observer has a crease in it so that observer trajectories slide

depending on what basis we use for the vector spaces, but the canonical unit vector field ∂ ∂ t {\displaystyle {\frac {\partial }{\partial t}}} on R {\displaystyle

{\displaystyle \chi _{\alpha }} and λ α {\displaystyle \lambda ^{\alpha }} A real vector field (gauge field) A μ {\displaystyle A_{\mu }} Their transformation properties