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searching for Local coordinates 97 found (124 total)

alternate case: local coordinates

Centripetal force (7,171 words) [view diff] exact match in snippet view article find links to article

equation. To deal directly with this issue, local coordinates are preferable, as discussed next. Local coordinates mean a set of coordinates that travel with

(see Exterior product). The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. Indeed,

(structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols. The Levi-Civita connection is named

Levi-Civita connection. Arguably, the definitions directly using local coordinates are preferable, since the "crucial property" of the Riemann tensor

that can are referred to as exact differentials. For a given set of local coordinates xk, the differentials dxk p form a basis of the cotangent space at

several ways. The most elementary method uses local coordinates. Suppose that xi are local coordinates on the base manifold M. In terms of these base

V of q in N, and local coordinates (x1, …, xm) at p and (x1, …, xn) at q such that f(U) = V, and the map f in these local coordinates is the standard projection

{\mathcal {A}}} in local coordinates xi, 1 ≤ i ≤ m, is given by ½ΔLB, where ΔLB is the Laplace–Beltrami operator given in local coordinates by

that ∇XY − ∇Y X = [X, Y] for arbitrary X and Y. Written in terms of local coordinates, this is equivalent to 0=XjYk(Γjki−Γkji),{\displaystyle 0=X^{j}Y^{k}(\Gamma

_{\alpha }:U_{\alpha }\to \mathbb {R} ^{n}} is a diffeomorphism. These local coordinates on Uα{\displaystyle U_{\alpha }} give rise to an isomorphism TxM→Rn{\displaystyle

dq^{i}} Here (q1,…,qn){\displaystyle (q^{1},\ldots ,q^{n})} are any local coordinates on Q{\displaystyle Q} and (p1,…,pn){\displaystyle (p_{1},\ldots ,p_{n})}

also a local chart for the manifold M{\displaystyle M}, admitting local coordinates x=(x1,…,xn);xi:U→R{\displaystyle \mathbf {x} =(x^{1},\dots ,x^{n});\quad

the deployment of the Mars Orbiter Laser Altimeter (MOLA) to obtain local coordinates of the topography shown in the HiRISE images, and then maps the HiRISE

of a function is the same as its ordinary differential. Choosing local coordinates {xi}{\displaystyle \left\{x^{i}\right\}} gives a local expression

this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates

are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the

satisfying linearity and a Leibniz rule is called a derivation. If in local coordinates (x1,...,xn){\displaystyle (x_{1},...,x_{n})} the vector field X{\displaystyle

smooth submanifold of a manifold X{\displaystyle X}, we can pick local coordinates (x1,…,xn){\displaystyle (x_{1},\dots ,x_{n})} around p∈Y{\displaystyle

curve through p in terms of its Taylor series in local coordinates on M. Examples in local coordinates: As indicated previously, the 1-jet of a curve through

}}{\partial \theta }}+{\frac {\partial F_{z}}{\partial z}}.} The use of local coordinates is vital for the validity of the expression. If we consider x the

In one use they are local coordinates that are adapted to a geodesic. In a second, more general one, they are local coordinates that are adapted to any

operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of

vector field given above: The Lie bracket of X and Y at p is given in local coordinates by the formula LXY(p)=[X,Y](p)=∂XY(p)−∂YX(p),{\displaystyle {\mathcal

relativity as follows: Given a pseudo-Riemannian manifold with a local coordinates xμ and equipped with a metric tensor gμν, the proper time interval

set in random locations within a given area and marked with GPS or local coordinates, and then monitored on a specific time interval. Once the samples

the gauge covariant derivative is an expression for a connection in local coordinates after choosing a frame for the fields involved, often in the form

space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point

differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means

charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by (x,θ,θ¯){\displaystyle (x,\theta ,{\bar {\theta

a small neighbourhood around any point on M there exist suitable local coordinates p1,⋯,pn, q1,⋯,qn{\displaystyle p_{1},\cdots ,p_{n},\ q_{1},\cdots

^{p,q}\rightarrow \Omega ^{p,q+1}} To describe these operators in local coordinates, let α=∑|I|=p,|J|=q fIJdzI∧dz¯J∈Ωp,q{\displaystyle \alpha =\sum _{|I|=p

PX{\displaystyle P_{X}} is called the momentum function corresponding to X. In local coordinates, the vector field X at point q may be written as Xq=∑iXi(q)∂∂qi{\displaystyle

φ{\displaystyle \varphi } at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth

Hamiltonian vector field associated to f{\displaystyle f}. Choosing local coordinates (U,xi){\displaystyle (U,x^{i})}, any Poisson bracket is given by {f

Euler–Lagrange equation for the energy functional E[γ] reads in the local coordinates (x1, ..., xn, v1, ..., vn) of TM as gik(γ(t),γ˙(t))γ¨i(t)+(∂gik∂xj(γ(t)

)=\int \limits _{0}^{1}||\gamma '(t)||_{\gamma (t)}dt}, or, choosing local coordinates x{\displaystyle x}, ℓ(γ)=∫01±∑i,j=1ngij(x(γ(t)))dxi(γ(t))dtdxj(γ(

{\mathcal {A}}} in local coordinates xi, 1 ≤ i ≤ m, is given by ½ΔLB, where ΔLB is the Laplace-Beltrami operator given in local coordinates by

_{W}}, so jrσ{\displaystyle j^{r}\sigma } really is a section. In local coordinates, jrσ{\displaystyle j^{r}\sigma } is given by (σα,∂|I|σα∂xI)1≤|I|≤r

v} on S (an assignment of a tangent vector to each point in S) in local coordinates. If P is a point of U0 ⊂ S, then a vector field may be represented

definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an antiderivation

following result. Proposition. In a natural basis, associated with local coordinates (U, xμ), i.e., in the holonomic frame ∂μ, the (local) connection coefficients

(M,g){\displaystyle (M,g)} be a Riemannian manifold equipped with local coordinates (x1,x2,…,xn){\displaystyle (x^{1},x^{2},\dots ,x^{n})}. Let A=Ai∂∂xi{\displaystyle

pick the right window and transform the screen coordinates to window local coordinates, by applying the inverse transformations. The types of transformation

components of the Levi-Civita connection with respect to a system of local coordinates. Christoffel's ideas were generalized and greatly developed by Gregorio

σP, is the function on the cotangent bundle T∗X defined in these local coordinates by σP(x,ξ)=∑|α|=kPα(x)ξα{\displaystyle \sigma _{P}(x,\xi )=\sum _{|\alpha

over X. D is an elliptic differential operator from E to F. So in local coordinates it acts as a differential operator, taking smooth sections of E to

with respect to a coordinate basis, despite the choice of a set of (local) coordinates xμ{\displaystyle x^{\mu }} being unnecessary for the specification

Einstein manifolds with k = 0 are called Ricci-flat manifolds. In local coordinates the condition that (M, g) be an Einstein manifold is simply Rab=kgab

coordinates on the Euclidean plane. These coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally

twice-differentiable (Busemann 1955). Geodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the exponential

x{\displaystyle x}, so that we may write ω{\displaystyle \omega } in local coordinates as f(z)dz{\displaystyle f(z)\;dz}. Then, the residue of ω{\displaystyle

M} around p{\displaystyle p}, one can use such functions to build local coordinates (xi,uσj){\displaystyle (x^{i},u_{\sigma }^{j})} and provide Jk(E,m){\displaystyle

example, the equations of general relativity can be rewritten in local coordinates without affecting the physical implications. Although making a property

obtain a (1,1)-tensor field in order to take the trace. In terms of local coordinates one can write, using the Einstein notation convention, that:

classes represented by differential forms which, in some choice of local coordinates z1,…,zn{\displaystyle z_{1},\ldots ,z_{n}}, can be written as a harmonic

induction that there are local coordinates u = (u1, ... un), z = ψ(u), 0 = ψ(0), such that First, assume that there exist local coordinates y = (y1, ... yn),

different open cone containing v. On a differentiable manifold M, using local coordinates x,ξ{\displaystyle x,\xi } on the cotangent bundle, the wave front

Z) = g(∇XY, Z) + g(Y, ∇X Z), "compatibility with the metric". In local coordinates the components of the form are called Christoffel symbols: because

is extremely significant that the Ricci tensor can be defined, in local coordinates, by a formula involving the first and second derivatives of the metric

given a vector field X{\displaystyle X} on a manifold, there exist local coordinates y1,…,yn{\displaystyle y_{1},\dots ,y_{n}} such that X=∂/∂y1{\displaystyle

the differential form formulation can be seen as follows. Choose local coordinates xα which gives a basis of 1-forms dxα in every point of the open set

+V(q).} In a more general setting, (v,q){\displaystyle (v,q)} are local coordinates on the tangent bundle TM{\displaystyle T{\mathcal {M}}} of a manifold

harmonic maps i.e. those that are horizontally (weakly) conformal. In local coordinates, x{\displaystyle x} on M{\displaystyle M} and y{\displaystyle y} on

List │ │ │ ├─ 0x4140 // Mapping Coordinates List │ │ │ └─ 0x4160 // Local Coordinates System │ │ ├─ 0x4600 // Light │ │ │ └─ 0x4610 // Spotlight │ │ └─

to local ENU coordinates To transform from ECEF coordinates to the local coordinates we need a local reference point. Typically, this might be the location

where x,y,z{\displaystyle x,y,z} represent the local coordinates of OP1(4),OP1(6),OP1{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{1}}(4)

F(h){\displaystyle F(h)} if necessary). The dHYM equation may also be written in local coordinates. Fix p∈X{\displaystyle p\in X} and holomorphic coordinates (z1,…,zn){\displaystyle

in Cartesian coordinates in an inertial framework and describe the local coordinates in terms of these Cartesian coordinates. In Figure 1, the arc length

proof of the Riemannian Penrose inequality. In local coordinates, this says gijkij = 0 In local coordinates, this says R - gikgjlkijkkl + (gijkij)2 ≥ 2(gpq(gijkpi;j

conditions. If an 2-form ω{\displaystyle \omega } can be written with local coordinates ω=dp1∧dq1+⋯+dpm∧dqm{\displaystyle \omega =dp_{1}\wedge dq_{1}+\cdots

transformations of a supercurve (with one even local coordinate z and N odd local coordinates θ1,...,θN) are generated by the coefficients of a super-stress–energy

     d=∑j∂jdxj{\displaystyle d=\sum _{j}\partial _{j}\,dx_{j}} in any local coordinates) is invariant with respect to all smooth transformations of the manifold

is at the origin pointing in the x-direction. This frame is called Local coordinates and is shown in the Figure on the right. The force of one magnetic

X{\displaystyle X} in PC3{\displaystyle {\textbf {P}}_{\mathbb {C} }^{3}} with local coordinates [x0:x1:x2:x3]{\displaystyle [x_{0}:x_{1}:x_{2}:x_{3}]} is said to

ξi){\displaystyle (x^{i},\xi ^{i})} be the local coordinates on TM{\displaystyle TM} associated with the local coordinates (xi{\displaystyle (x^{i}}) on M{\displaystyle

known as a P-structure and an S-structure, respectively. Let the local coordinates be called xi{\displaystyle x^{i}}. Let the derivatives ∂if{\displaystyle

is defined by a length element ds which is expressed in conformal local coordinates z as ds = ρ(z) |dz|, where ρ is a smooth positive function with isolated

parallelogram also correspond, as do the points (0, 1) in the parallelogram's local coordinates. Thus for every pair of matching features in nearby views, three point

form η, then multiplying by the holomorphic function s. If U admits local coordinates z1, ..., zn such that for some i, ∂f/∂zi ≠ 0, then this can also be

(}\iota _{\Delta f}R^{h}{\big )}{\Big )}=0.} In terms of local coordinates xi for M and local coordinates yα for N, the biharmonic map equation is written as

principle: For an explicit description of spacetime one must work with local coordinates. The freedom in choosing such coordinates generalizes the freedom

based on the properties of mercury horizon and allowed to establish local coordinates with an error about 10 m, surpassed only much later by satellite navigation

write a map U→Rn{\displaystyle U\to \mathbb {R} ^{n}} giving explicit local coordinates ϕ=(ϕ1,⋯,ϕn){\displaystyle \phi =(\phi ^{1},\cdots ,\phi ^{n})} on

The classical approach makes use of the natural frames relative to local coordinates and works with components of tensor fields, thus giving the impression

radar Mark VII, the Parallax computer Meta converted the data to local coordinates. In the remaining heavy antiaircraft radar batteries, data were used

ρ0{\displaystyle \rho _{0}} of ω0{\displaystyle \omega _{0}} is given in local coordinates by the formula ρ0=−i∂∂¯log⁡ω0n.{\displaystyle \rho _{0}=-i\partial

converse is false. Let x1,…,xn{\displaystyle x_{1},\ldots ,x_{n}} be local coordinates on M{\displaystyle M}, and let O(x1,…,xn){\displaystyle {\mathcal

diffeomorphic to the N-torus Tn{\displaystyle T^{n}}. There exist (local) coordinates on Lc{\displaystyle L_{c}} (θ1,⋯,θn,ω1,⋯,ωn){\displaystyle (\theta

_{K}\cong {\widehat {\mathbb {A} _{K}}}.} Proof. By reduction to local coordinates, it is sufficient to show each Kv{\displaystyle K_{v}} is self-dual

}g_{ij}).} Formulas for covariant derivative can be also be derived from local coordinates (x,y) without the use of isometric embeddings. Taking ∂x{\displaystyle

quadratic element is described by Figure 5 in both parent space with local coordinates ξ{\displaystyle \xi } and η,{\displaystyle \eta ,} and by the mapped

_{i}\Omega ,}} where the integral over Ui has its usual definition in local coordinates. The integral is independent of the choices here. If Ω has the local

point-by-point determinant of the shape operator, or equivalently (relative to local coordinates) as the determinant of the second fundamental form divided by the

higher genus, the failure to be a manifold is even more dramatic). Local coordinates are given by (X,ω)↦(∫γ1ω,…,∫γnω){\displaystyle (X,\omega )\mapsto

{S}}^{\vee }}. Concretely, if T is the tangent space at a point O with local coordinates xm{\displaystyle x^{m}} (m=0,…,3{\displaystyle m=0,\ldots ,3}) so

at every point except the pole. If f is not regular at a point, in local coordinates f has the expansion f(z) = a + b zm (1 + c1z + c2z2 + ⋅⋅⋅) with b

(TxM,ωx){\displaystyle (T_{x}M,\omega _{x})}, one can always find local coordinates (U~,x~1,…,x~n,y~1,…,y~n){\displaystyle ({\tilde {U}},{\tilde {x}}^{1}