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searching for Unit vector 316 found (580 total)

alternate case: unit vector

Matrix unit (384 words) [view diff] exact match in snippet view article find links to article

E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}} A vector unit is a standard unit vector. A single-entry matrix generalizes the matrix unit for matrices with

^{T}MZM^{T}\mathbf {x} \right)\;dS^{n-1}} where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration

{\displaystyle \mathbf {T} ^{(\mathbf {n} )}\,\!} on a plane with normal unit vector n {\displaystyle \mathbf {n} \,\!} passing through a given point P {\displaystyle

the latitude) and z ^ {\displaystyle {\hat {\boldsymbol {z}}}} is the unit vector in the vertical upward direction (opposing the Earth's gravity). Since

spacetime. The timelike unit vector field is often denoted by e → 0 {\displaystyle {\vec {e}}_{0}} and the three spacelike unit vector fields by e → 1 , e

gravitational lensing. The theory is based on the following ingredients: A unit vector field; A dynamical scalar field; A nondynamical scalar field; A matter

are the three quartiles (25th, 50th, 75th percentile) of the feature. Unit vector normalization regards each individual data point as a vector, and divide

{\displaystyle \theta } ) of 90 degrees past the periapsis. The third unit vector w ^ {\displaystyle \mathbf {\hat {w}} } is the angular momentum vector

_{2}\delta (\mathbf {r} ),} where μ0 is the magnetic constant, r̂ is a unit vector parallel to the line joining the centers of the two dipoles, and |r|

A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}\end{aligned}}} The unit vector of a coordinate parameter u is defined in such a way that a small positive

{\displaystyle s} ( a ≤ s ≤ b {\displaystyle a\leq s\leq b} ), and a unit vector U ( s ) {\displaystyle U(s)} perpendicular to ∂ X ∂ s ( s ) {\displaystyle

⁠f/h⁠) be the unit vector associated with q, where h = √x2 + y2 + f2. If we consider a straight line in space S with the unit vector ns ≡ (nx, ny, nz)

j}u_{ij}(\mathrm {vec} T_{ij})^{T}} Where: u i j {\displaystyle u_{ij}} is a unit vector of order 1 2 n ( n + 1 ) {\displaystyle {\frac {1}{2}}n(n+1)} having

and the incident angle is β = 180° – θ. The so-called LOS-vector is a unit vector u → = ( u , v , w ) t {\displaystyle {\vec {u}}=(u,v,w)^{t}} (in the

{\displaystyle d(u,Y)\leq 1} when ‖ u ‖ = 1 {\displaystyle \|u\|=1} is a unit vector. Using this new notation, the conclusion of Riesz's lemma may be restated

the unit vector in the direction of the shear stress acting on the body surface dS, n ^ {\displaystyle {\hat {\mathbf {n} }}} is the unit vector in the

first order derivative can be obtained by taking the dot product of a unit vector oriented in a specific direction with the gradient. The basis filters

{\displaystyle {\vec {n}}_{AB}=({\vec {x}}_{A}-{\vec {x}}_{B})/r_{AB}} is the unit vector pointing from body B to body A m A {\displaystyle m_{A}} is the mass

(complex) Hilbert space H. If ε > 0 and K is a compact subset of G, then a unit vector ξ in H is called an (ε, K)-invariant vector if ∀ g ∈ K   :   ‖ π ( g

required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle

ı ^ {\displaystyle \mathbf {\hat {\imath }} } Cartesian x-axis basis unit vector unitless J {\displaystyle \mathbf {J} } current density ampere per square

{\displaystyle O(ckN)} . A unit vector can be orthogonally projected to a random subspace. Let u {\displaystyle u} be the original unit vector, and let v {\displaystyle

perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vvT suffices. A Householder reflection is typically used

,\end{aligned}}} where k ^ {\displaystyle \mathbf {\hat {k}} } is a unit vector perpendicular to the plane of the pendulum. (The second to last step

}f(x_{0}+t\theta )dt} where x0 is an initial point on the line and θ is a unit vector in Rn giving the direction of the line L. The latter integral is not

vectors to the curves. These are respectively timelike or spacelike unit vector fields. In general relativity, a timelike congruence in a four-dimensional

x ‖ ≥ ‖ T x ‖ 2 {\displaystyle \|T^{2}x\|\geq \|Tx\|^{2}} for every unit vector x in H. The class of paranormal operators was introduced by V. Istratescu

longitude, and together they generate a plane tangent to the sphere. The unit vector β ^ ( λ , φ ) = ( sin ⁡ β ) λ ^ + ( cos ⁡ β ) φ ^ {\displaystyle \mathbf

about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector. Stoker, J. J. (2011)

eft\|F\right\|=\exp(|a|^{2}/2)\left\|F\right\|.} One may interpret a unit vector in the Segal–Bargmann space as the wave function for a quantum particle

Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like

the origin of longitude is taken to be the positive x-axis. k is the unit vector (0, 0, 1), which is the normal vector to the xy reference plane. For

Vector3 r_unit_vector = { r_vector.e[0] / r_mag, r_vector.e[1] / r_mag, r_vector.e[2] / r_mag }; a_g.e[0] += acceleration * r_unit_vector.e[0]; a_g.e[1]

inertial frame A as in Figure 3, but the coordinate axes in frame B turn so unit vector u1 always points toward the centre of rotation. This example might apply

of the Wightman functions. And with increasing distance, creating a unit vector localized outside the region requires operators of ever increasing operator

\mathbf {k} ={\frac {\partial }{\partial z}}} is the z-direction of the unit vector) to the plasma mathematical model to compute calculations. Bauer, Frances;

the two vectors. So, if n ^ {\displaystyle \mathbf {\hat {n}} } is the unit vector perpendicular to the plane determined by vectors a {\displaystyle \mathbf

the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction. v R = v ⋅ r | r | = v ⋅ r ^ {\displaystyle v_{R}={\frac

\mathbf {\hat {n}} } where n ^ {\displaystyle \mathbf {\hat {n}} } is a unit vector and the scalar function ψ {\displaystyle \psi } satisfies the Helmholtz

bundle projection by noting that the fixed unit vector along the z axis, (0,0,1), rotates to another unit vector, ( 2 ( x z + w y ) , 2 ( y z − w x ) , 1

\;{\vec {e}}_{3}} Here, the first is a timelike unit vector field and the others are spacelike unit vector fields, and e → 0 {\displaystyle {\vec {e}}_{0}}

Kumar Raychaudhuri and the Soviet physicist Lev Landau. Given a timelike unit vector field X → {\displaystyle {\vec {X}}} (which can be interpreted as a family

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

{\displaystyle =-m\omega ^{2}R\ \mathbf {u} _{R}\ ,} where uR is a unit vector pointing from the axis of rotation to one of the spheres, and Ω is a

)^{2}]\right\}} where x {\displaystyle \mathbf {x} \,} is a three-dimensional unit vector, ( ⋅ ) T {\displaystyle (\cdot )^{T}} denotes the transpose of ( ⋅ )

{e}}_{2},\;{\vec {e}}_{3}} (the first is a timelike unit vector field, the last three are spacelike unit vector fields) can always be found in which the Einstein

spacelike 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

where R is a vector with origin in the system of charges and R̂ is the unit vector in the direction of R. That is to say, R ^ i {\displaystyle {\hat {R}}_{i}}

{e}}_{0}}{\vec {e}}_{0}=0} which means that the integral curves of the timelike unit vector field e0 are timelike geodesics, and also ∇ e → 0 e → 1 = ∇ e → 0 e →

that in the specific case where u {\displaystyle \mathbf {u} } is a unit vector, and thus ‖ u ‖ 2 = 1 {\displaystyle \left\Vert \mathbf {u} \right\Vert

} 1 ≤i ≤p, where e i {\displaystyle e_{i}} is the unit vector with a 1 in the i t h {\displaystyle i^{th}} place, and c n {\displaystyle

_{ij})\right]d^{3}r} where n {\displaystyle \mathbf {n} } is a unit vector in the direction of the observer, r n ≡ r ⋅ n {\displaystyle r_{n}\equiv

momentum of particle i {\displaystyle i} , and n {\displaystyle n} is a unit vector that maximizes T {\displaystyle T} and defines the thrust axis. The sum

object and the test particle, and r ^ {\displaystyle {\hat {r}}} is the unit vector in the direction going from the massive object to the test mass. Newton's

of Σ t {\displaystyle \Sigma _{t}} and ν {\displaystyle \nu } is the unit vector opposite of the mean curvature direction, then d d t m H ( Σ t ) ≥ 0

{\mathbf {n} }}} is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and ∂ ∂ n ^ {\displaystyle

Astronomical Almanac, then it gives the x-, y- and z-components of the unit vector pointing toward the Sun, through vector analysis rather than spherical

(diffuse) lighting of a surface, the unit vector from the shading point to the light source is dotted with the unit vector normal to that surface, and the

subunits: the vector register unit, the vector add unit, vector multiply unit, vector mask unit, vector address unit and the vector control unit. The vector

University of Bristol in 2006; her dissertation, Liquid crystals and tangent unit-vector fields in polyhedral geometries, was jointly supervised by Jonathan Robbins

required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle

{\displaystyle P} . Here n ^ {\displaystyle {\hat {\mathbf {n} }}} is the unit vector perpendicular to the test surface, i.e., n ^ d s = − R d r = [ d y −

the sphere, calculate d ^ {\displaystyle {\hat {d}}} , that being the unit vector from P {\displaystyle P} to the sphere's origin. Assuming that the sphere's

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

non-orientable 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

depends only on the angle (directed) between them, the same a as in the unit vector–angle representation of a versor explained above. That's why it may be

corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a1σ1 + a2σ2 + a3σ3. As a special case, it is easy to see that, if

acceleration of gravity, and k ^ {\textstyle \mathbf {\hat {k}} } is a unit vector defining the vertical direction. Choose a reference point R in the volume

-- is linearly referenced to the unit-vector-length edges of the regular tetrahedron, each of whose six unit vector edges occur in the isotropic vector

Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like

{\displaystyle {\vec {r}}} from the heliocentre to the observer, the unit vector n ^ {\displaystyle {\hat {n}}} from the observer toward the object or

{\displaystyle t} . Then the formulas for the signed curvature k(t), the normal unit vector N(t), the radius of curvature R(t), and the center Q(t) of the osculating

compact on the real line, therefore f attains a maximum on S, at some unit vector y. By Lagrange's multiplier theorem, y satisfies ∇ f = ∇ y ∗ T y = λ

{\displaystyle {\vec {e}}_{0}} is a timelike unit vector field while the others are spacelike unit vector fields; at each event, all four are mutually

position s has one tangent unit vector s that invariably points in the direction of growing s's, as well as one unit vector n, perpendicular to s, that

\mathbf {\hat {r}} ,} where r ^ {\displaystyle \mathbf {\hat {r}} } is the unit vector pointing along the viewing direction toward the viewer, d A {\displaystyle

subscripts. For current density, t ^ {\displaystyle \mathbf {\hat {t}} } is a unit vector in the direction of flow, i.e. tangent to a flowline. Notice the dot

{\displaystyle d\mathbf {S} ={\hat {\mathbf {n} }}\ dS} where ^n is the local unit vector perpendicular to dS. Integrating gives the vector area for the surface

simple appearance. Here, the first vector is understood to be a timelike unit vector field; this is everywhere tangent to the world lines of the corresponding

probability current in the wave incident upon the barrier with normal unit vector n ^ {\displaystyle {\hat {n}}} and J → t r a n s {\displaystyle {\vec

the time-average; S is the Poynting vector; α is the angle between a unit vector normal to the surface and S. For a propagating sinusoidal linearly polarized

(i.e. the marginal distribution) on to a line in the direction of the unit vector η ^ {\displaystyle {\hat {\boldsymbol {\eta }}}} equals η ^ ′ Σ η ^ =

the position of the ith source of uniform intensity Ii, and â is the unit vector normal to the illuminated elemental opaque area dA being measured, and

notation, one can write the equivalent expression using instead the ith unit vector ( e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} ) to select the ith

{e}}_{2},\;{\vec {e}}_{3}} (the first is a timelike unit vector field, the last three are spacelike unit vector fields) can always be found in which the Einstein

in 1978. In normal mapping, the unit vector from the shading point to the light source is dotted with the unit vector normal to that surface, and the

being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing

{\displaystyle \mathbf {R} ^{n}} ( n ≥ 2 {\displaystyle n\geq 2} ). For a unit vector ξ , | ξ | = 1 {\displaystyle \xi ,|\xi |=1} and a point x ∈ R n {\displaystyle

total surface area of the cell and n {\displaystyle {\mathbf {n} }} is a unit vector normal to the surface and pointing outward. So, finally, we are able

C} if and only if it is interior to C . {\displaystyle C.} Weak order unit Vector lattice – Partially ordered vector space, ordered as a latticePages displaying

{1}{2}}K_{3}(\mathbf {\hat {n}} \times \nabla \times \mathbf {\hat {n}} )^{2}.} The unit vector n ^ {\displaystyle \mathbf {\hat {n}} } is the normalized director of

{\displaystyle \mathbf {r} } as the product of its magnitude times the unit vector in its direction ( r = | r | r ^ {\displaystyle \mathbf {r} =|\mathbf

\nu \!} , n ^ {\displaystyle \mathbf {\hat {n}} } denotes a variable unit vector with origin at the point x {\displaystyle \mathbf {x} } , d ω ( n ^ )

Poisson's ratio, E is Young's modulus, n is a unit vector directed along the direction of extension, m is a unit vector directed perpendicular to the direction

ϕ {\displaystyle \phi } is the angle of rotation about the constant unit vector S. In this case, the work of the torque becomes, W = ∫ t 1 t 2 T ⋅ ω

N b a b {\displaystyle a_{\mathrm {norm} }=N^{b}a_{b}} where Nb is a unit vector normal to the surface. In a Schwarzschild coordinate system, for example

=Ap\,\mathbf {u} \cdot \mathbf {v} =Apv} where f is the sound force of unit vector u; v is the particle velocity of projection v along u; A is the area;

\end{aligned}}} where r ^ {\displaystyle {\hat {\mathbf {r} }}} denotes a unit vector in the radial direction. There are many useful features of the effective

{\textstyle {\vec {v}}/\|{\vec {v}}\|} is interpreted as an arbitrary unit vector. Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions

defined here, τ is ωx ∘ πτ for a suitable unit vector x(=xτ) in Hτ. Thus τ is ωy ∘ Φ, where y is the unit vector Σρ∈S ⊕yρ in HΦ, defined by yτ=x, yρ=0(ρ≠τ)

of rotations about the x, y and z axes; Axis angle pair (n, θ) of a unit vector representing an axis, and an angle of rotation about it; A quaternion

but only float with their short axes in parallel to the director (a unit vector which defines the liquid-crystalline alignment and order). The first

_{i}^{T}\right)\mathbf {g} _{i}} where I is the identity matrix and τi is a unit vector representing the reaction path tangent at ri. By projecting out components

complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force

vectors on a straight line are connected by multiplication to the third unit vector on that straight line with signs according to the arrows (sign of the

{\displaystyle \mathbf {u} _{\mathrm {t} }={\frac {\mathbf {v} (t)}{v(t)}}\,,} a unit vector tangent to the path pointing in the direction of motion at the chosen

the form appropriate for a perfect fluid solution with the timelike unit vector e → 0 {\displaystyle {\vec {e}}_{0}} everywhere tangent to the world

{\displaystyle n} is the topological index, M {\displaystyle \mathbf {M} } is the unit vector in the direction of the local magnetization within the magnetic thin

with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is y = X β + e {\displaystyle

R at that point of the (unit) cotangent bundle is the corresponding (unit) vector parallel to N. First jet bundle On the other hand, one can build a contact

given location on the surface of the Earth. The idea is based on the unit vector with its origin fixed at a chosen point on the surface of the Earth and

hyperbolic quaternions. Knott wrote: [T]he assumption that the square of a unit vector is positive unity leads to an algebra whose characteristic quantities

therefore can be expressed as a scalar field multiplied by a constant unit vector z ^ {\displaystyle {\hat {z}}} : ω = ∇ × v = ( ∂ v y ∂ x − ∂ v x ∂ y

||a − b|| = ||a|| + ||b||. One way to do this is to take b = −a. If there exists a unit vector u in X that is not an extreme point of the closed unit ball in X, then

gravitation object, and r ^ {\displaystyle \mathbf {\hat {r}} } is a unit vector identifying the direction to the massive object. In the weak-field approximation

of a proper rotation R by angle θ around the axis u = (ux, uy, uz), a unit vector with u2 x + u2 y + u2 z = 1, is given by: R = [ u x 2 ( 1 − cos ⁡ θ )

and the location where the magnetic field is calculated, and r̂ is a unit vector in the direction of r. For example, in the case of a sufficiently long

W\subset V} with dimension i + j − 1 {\textstyle i+j-1} , there exists a unit vector w {\textstyle w} such that ⟨ w , ( A + B ) w ⟩ ≤ λ i ( A ) + λ j ( B

dt}\cdot {\hat {A}}} Which also gives a relation of rate of change of unit vector by taking A → {\displaystyle {\vec {A}}} , to be such a vector: d A ^

The Reeb vector field at the Sasakian manifold at unit radius is a unit vector field and tangential to the embedding. A Sasakian manifold M {\displaystyle

F λ   λ ^ {\displaystyle F_{\lambda }\ {\hat {\lambda }}\,} with the unit vector λ ^ {\displaystyle {\hat {\lambda }}\,} orthogonal to the radial direction

Burgers vectors. If a Burgers circuit of the unit area normal to the unit vector l j {\displaystyle l_{j}} has a Burgers vector B i {\displaystyle B_{i}}

vectors e ^ i {\displaystyle \mathbf {{\hat {e}}_{i}} } represent the unit vector in the surface co-ordinate directions, the dot product serving to identify

variables f, k, and F correspond to the Coriolis force, the vertical unit vector, and a given force. The subscripts a, w, r, s, and p correspond to the

with limit equal to the 0 {\displaystyle \mathbf {0} } vector. The unit vector basis of ℓ p {\displaystyle \ell ^{p}} for 1 < p < ∞ , {\displaystyle

c′, respectively. For any line l = s + t n̂ in vector form (n̂ is a unit vector) and a point p, the perpendicular distance from p to l is ‖ ( s − p )

{\displaystyle B=|\mathbf {B} |} and b {\displaystyle \mathbf {b} } a unit vector: ( B ⋅ ∇ ) B μ 0 = B 2 μ 0 ( b ⋅ ∇ ) b = B 2 μ 0 κ {\displaystyle {(\mathbf

lines so they all pass through the origin of coordinates. Choose one unit vector in each line. Form the matrix M of inner products. This matrix has 1

R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along the unit vector u ^ {\displaystyle {\hat {\mathbf {u}}}} at some point x 0 ∈ R n {\displaystyle

angle of inclination of the normal segment (the oriented angle from the unit vector of the x-axis to this segment), and p is the (positive) length of the

analytical kinematics by suggesting a mathematical model: When one selects a unit vector r in the hyperbolic quaternions, then r 2 = +1. The plane D r = { t +

absolute value of r: the distance from the dipole r̂ = ⁠r/r⁠ is the unit vector parallel to r; m is the (vector) dipole moment μ0 is the permeability

{\displaystyle \rho =|\psi \rangle \langle \psi |} is a projection onto a unit vector ψ {\displaystyle \psi } . Then σ = ⟨ ψ | ⋅ ψ ⟩ {\displaystyle \sigma

compute ⨂ n H n {\textstyle \bigotimes _{n}{H_{n}}} , first select a unit vector v n ∈ H n {\textstyle v_{n}\in H_{n}} in each Hilbert space, and then

{\displaystyle g(G^{t}(V),G^{t}(V))=g(V,V).\,} In particular, when V is a unit vector, γ V {\displaystyle \gamma _{V}} remains unit speed throughout, so the

combined instruction/data Vector processing unit: 4 FMAC units, 1 FDIV unit Vector processing unit registers: 128-bit wide, 32 entries Image processing

confined to the "blue plane" in the diagram and is the result of the unit vector on the x axis x ^ = [ 1 , 0 , 0 ] {\displaystyle {\hat {x}}=[1,0,0]}

tensor over the surface of the sphere, where er represents the radial unit-vector of spherical-coordinates: F = ∬ ∂ V ⊂ ⊃ σ ⋅ d S = ∫ 0 π ∫ 0 2 π σ ⋅ e

,t;\mathbf {r} _{1},\nu )} where: ν denotes frequency. r1 denotes a unit vector, with the direction and sense of the geometrical vector r in the line

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

for each pair A k , B k {\displaystyle A_{k},B_{k}} there exists some unit vector v k {\displaystyle v_{k}} and real number c k {\displaystyle c_{k}}

its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its vector part (also

_{r}F_{r}+\mathbf {e} _{\theta }F_{\theta }+\mathbf {e} _{z}F_{z},} where ea is the unit vector in direction a, the divergence is div ⁡ F = ∇ ⋅ F = 1 r ∂ ∂ r ( r F r

the barycentre to the observer, n ^ {\displaystyle {\hat {n}}} is the unit vector from the observer to the object or event, d {\displaystyle d} is the

)[1+S(\theta )\mathbf {P} \cdot \mathbf {n} ]} for polarization P and a unit vector n in the direction of the orbital angular momentum L {\displaystyle \mathbf

complex Hilbert space C2, with each quantum state corresponding to a unit vector in C2 (unique up to phase). In this case, the state space can be geometrically

vector Ω ( t ) {\displaystyle {\boldsymbol {\Omega }}(t)} then each unit vector u ^ {\displaystyle {\hat {\boldsymbol {u}}}} of the rotating coordinate

vector Ω ( t ) {\displaystyle {\boldsymbol {\Omega }}(t)} then each unit vector u ^ {\displaystyle {\hat {\boldsymbol {u}}}} of the rotating coordinate

with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is y = X β + e {\displaystyle

|\psi _{i}\rangle } can be represented in a classical computer as a unit vector in C 2 n {\displaystyle \mathbb {C} ^{2^{n}}} . Furthermore, each gate

ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a

b ] {\displaystyle t\in [0,b]} . This is the case if and only if the unit vector bases of ℓ M {\displaystyle \ell _{M}} and ℓ N {\displaystyle \ell _{N}}

P\cos v).} If R and P in the above flat torus parametrization form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and 0 < η < π/2 parameterize the

function on Rn. The first step of the proof is to show that, for any fixed unit vector v, the v-directional derivative of u exists almost everywhere. This is

calculates the x-, y-, and z-component of the solar vector, which is a unit vector with its tail fixed at the observer's location and its head kept pointing

gravitational constant and r ^ {\displaystyle {\hat {\mathbf {r} }}} is a unit vector pointing from M {\displaystyle M} toward m {\displaystyle m} . The force

^{3}}{3}}\right)\quad &\forall \epsilon \in (0,1)\end{aligned}}} Fix some unit vector v ∈ R n {\textstyle v\in \mathbb {R} ^{n}} . Define Q i := ∑ j R i j

The first three elements of each row of A are the components of a unit vector from the receiver to the indicated satellite. The last element of each

and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P) is assumed fixed, the normal

normal to the mean airflow; n is the normal unit vector pointing into the wing; k is the vertical unit vector, normal to the freestream direction. The above

information (including separating the facet direction and size into a unit vector and a real number, which may be negative, providing an additional bit

(the charge is decreasing), the last term has a positive sign because unit vector of surface R is from left to right while the direction of electric field

relative to rotation axis a ^ {\displaystyle \mathbf {\hat {a}} \,\!} = unit vector perpendicular to rotation (zenith) axis, radial from it g = − G M | r

{u}}_{i}{\hat {u}}_{j}\right)e^{i\mathbf {k\cdot x} }\ ,} where û is the unit vector of k, û = ⁠k/k⁠. For a discussion see, Compagno, G.; Passante, R.; Persico

{\displaystyle {\hat {a}}=(a_{x},a_{y},a_{z})} be the cylinder axis unit vector, cylinder radius r {\displaystyle r} , and height (or axis length) h

{\mathbf {v} (E)}{|\mathbf {v} (E)|}}={\frac {\mathbf {v} (E)}{v(E)}}} Unit vector (solid angle) in direction of motion t {\displaystyle t} Time v ( E )

inequality becomes u2 + H−1Hrv2 ≥ 1, where (u,v) corresponds to the unit vector ċ(t). This follows from the inequality Hr ≥ H, a consequence of the non-negativity

force about an axis that passes through the reference point O. If the unit vector along the axis is e {\displaystyle \mathbf {e} } , the moment of force

{r}}|^{2}}}{\boldsymbol {\hat {r}}}} where q1, q2 are two point charges; ^r being the unit vector direction of the distance r between charges and ε0 is the permittivity

run in opposite directions, so we define one of them as the tangent unit vector t {\displaystyle {\boldsymbol {t}}} and the other as − t {\displaystyle

three-dimensional rotation A has a unique axis that defines a line L. Let the unit vector along this line be S so that the translation vector d can be resolved

{\displaystyle {\hat {\boldsymbol {b}}}={\boldsymbol {B}}/B} is the unit vector in the direction of the magnetic field. This drift can be decomposed

derivatives with respect to the arc length s, and N(s) is the normal unit vector in the direction of T′(s). As planar curves have zero torsion, the second

{m}}=\left(0,{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}\right)} be the unit vector in the direction of C' and let n → = ( 3 3 , 3 3 , 3 3 ) {\displaystyle

=0} denote the two transverse polarizations of light in free space and unit vector ϵ ( k , 3 ) = k / | k | {\displaystyle {\boldsymbol {\epsilon }}(\mathbf

{v}} |}}\right),} where j ^ {\displaystyle \mathbf {\hat {j}} } is the unit vector defining the y-axis. The angle, α {\displaystyle \alpha } , defines the

-d\mathbf {e} _{l})\right]} Here, e l {\displaystyle \mathbf {e} _{l}} is a unit vector along direction l {\displaystyle l} . γ k ′ k {\displaystyle \gamma ^{k'k}}

the viscous drag coefficient and N {\displaystyle \mathbf {N} } is the unit vector of the plane perpendicular to the tangent of the filament curve. This

\partial _{t}+\partial _{\varphi }\right)} Here, we think of the timelike unit vector field e → 0 {\displaystyle {\vec {e}}_{0}} as tangent to the world lines

counterclockwise rotation through an angle α about an axis given by a unit vector v; z is the same rotation; see quaternions and spatial rotation. This

differentiable function, k ^ {\displaystyle {\hat {\mathbf {k} }}} is a unit vector in the direction of propagation, and x {\displaystyle {\mathbf {x} }}

-d\mathbf {e} _{l})\right]} Here, e l {\displaystyle \mathbf {e} _{l}} is a unit vector along direction l {\displaystyle l} . γ k ′ k {\displaystyle \gamma ^{k'k}}

the Coriolis parameter, k {\displaystyle \mathrm {k} } is the upward unit vector normal to the sea surface, τ a {\displaystyle \tau _{a}} and τ w {\displaystyle

{\displaystyle d\mathbf {X} =dX\mathbf {N} } (Figure) in the direction of the unit vector N {\displaystyle \mathbf {N} } at the material point P {\displaystyle

x_{3}}}{\hat {x}}_{3},} where x ^ i {\displaystyle {\hat {x}}_{i}} represents a unit vector in the x i {\displaystyle x_{i}} direction for 1 ≤ i ≤ 3 {\displaystyle

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direction cosines of the truss element (i.e., they are components of a unit vector aligned with the member). This form reveals how to generalize the element

{j}} +c\mathbf {\hat {k}} }{\sqrt {a^{2}+b^{2}+c^{2}}}},} that is, the unit vector. This leads to the variation of De Moivre's formula: q n = k n ( cos

{\beta }})^{5}}},} where n ^ {\displaystyle \mathbf {\hat {n}} } is a unit vector pointing from the particle towards the observer. In the case of linear

{\displaystyle y\,\!} plane passing through P {\displaystyle P\,\!} with a unit vector n {\displaystyle \mathbf {n} \,\!} making an angle of θ {\displaystyle

n y , n z ) {\displaystyle {\hat {n}}=(n_{x},n_{y},n_{z})} is a real unit vector in three dimensions, the rotation of the Bloch sphere about this axis

_{\text{E}}(-{\tilde {r}}_{i}/c\,-\,\delta t_{{\text{clock}},i})} . The line-of-sight unit vector of satellite observed at r rec,ECEF {\displaystyle \scriptstyle {\boldsymbol

\max _{(x,y)\in D}\|x\|_{2}=R} , and it is linearly separable by some unit vector w ∗ {\textstyle w^{*}} , with margin γ {\textstyle \gamma } : γ := min

d\quad d\in \mathbb {R} } where l {\displaystyle \mathbf {l} } is a unit vector in the direction of the line, l 0 {\displaystyle \mathbf {l_{0}} } is

{\displaystyle \alpha (n)} . g n ( x ) {\displaystyle g_{n}(x)} represents a unit vector that is perpendicular to the previous curve at x {\displaystyle x} .

{I}}(x)} also sometimes called scalar and vector parts. We define the unit vector u corresponding to x as u = I ( x ) ‖ I ( x ) ‖ {\displaystyle u={\frac

the Coriolis parameter, k {\displaystyle \mathbf {k} } is the vertical unit vector, and the subscript "p" on the gradient operator denotes gradient on a

\psi } , where e z {\displaystyle \mathbf {e_{z}} } is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is

form, except that w {\displaystyle \mathbf {w} } is not necessarily a unit vector. The parameter b ‖ w ‖ {\displaystyle {\tfrac {b}{\|\mathbf {w} \|}}}

that S {\displaystyle \mathbf {S} } transforms one unit vector on the boundary to another unit vector on the boundary, in other words, it is orthogonal

this technique is to choose a random hyperplane (defined by a normal unit vector r) at the outset and use the hyperplane to hash input vectors. Given

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}}\operatorname {sgn}(g)e_{g}} where e g {\displaystyle e_{g}} is the unit vector corresponding to g, and sgn ⁡ ( g ) {\displaystyle \operatorname {sgn}(g)}

structure can be regarded as the distribution of (2n−2)-planes which, at the unit vector u, is the pullback of the orthogonal complement of u in the tangent space

obtained from the Pauli spin matrices. Let u = (ux, uy, uz) be an arbitrary unit vector. Then the operator for spin in this direction is simply S u = ℏ 2 ( u

system is defined, is left to the developer. Whether, therefore, the unit vector of the system corresponds in reality to one meter or an Ångström depends

is sharp. Since 0 ∈ W ( A ) {\textstyle 0\in W(A)} , there exists a unit vector x 0 {\textstyle x_{0}} such that x 0 ∗ A x 0 = 0 {\textstyle x_{0}^{*}Ax_{0}=0}

linearly independent. Complete this to a basis {ru,rv,n}, by selecting a unit vector n normal to the surface. It is possible to express the second partial

950\\\end{bmatrix}}} The axis/angle description (with the axis as a unit vector) is related to the misorientation matrix by: cos ⁡ Θ = g 11 + g 22 +

\mathbf {m} )/2} where total magnetization (as macrospin) is along the unit vector m {\displaystyle \mathbf {m} } and the Pauli matrices properties involving

{v}}}{dt}}={\vec {F}}-mg{\hat {k}}\;.} Here k ^ {\displaystyle {\hat {k}}} is a unit vector in the vertical direction and m {\displaystyle m} is the instantaneous

Dixmier's condition. For every finite (or compact) subset F of G there is unit vector f in L2(G) such that λ(g)f − f is arbitrarily small in L2(G) for g in

The configuration of each measuring instrument is represented by a unit vector, and the quantum-mechanical prediction for the correlation between two

^{-1}{\boldsymbol {\mu }}\rVert } is the 1d slice of the sd along the unit vector μ {\displaystyle {\boldsymbol {\mu }}} through the means, i.e. the d

_{-}}{2}}} , and R ^ {\displaystyle {\hat {\mathbf {R} }}} the corresponding unit vector: R = r − r + + r − 2 , R ^ = R | R | . {\displaystyle \mathbf {R} =\mathbf

distributions, then the Dirac impulse is refinable with respect to the unit vector δ {\displaystyle \delta } , that is known as Kronecker delta. The n {\displaystyle

scattering in the medium. The variable s {\displaystyle \mathbf {s} } is a unit vector that specifies the direction of radiation and the variable s ′ {\displaystyle

{n}}\cdot {\vec {\sigma }}~(1-\cos(a))~.} Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator

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function f (r, ŝ, t), where r is the particle position and ŝ is the unit vector in the direction of its orientation. In d-dimensions, this equation is

A 1-frame in F n {\displaystyle \mathbb {F} ^{n}} is nothing but a unit vector, so the Stiefel manifold V 1 ( F n ) {\displaystyle V_{1}(\mathbb {F}

{\displaystyle \mathbf {e} _{i}} is the standard i {\displaystyle i} th unit vector in R n {\displaystyle \mathbb {R} ^{n}} . The matroid polytope P M {\displaystyle

theorem. As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about

0 {\displaystyle |{\bar {r}}|>0} and the uniquely defined rotation (unit) vector is: r ^ = r ¯ | r ¯ | {\displaystyle {\hat {r}}={\frac {\bar {r}}{|{\bar

}{2m}}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}),} with the subscripts denoting unit vector notation. As the dispersion relation is non-linear, the free Schrödinger

_{1}/\lambda _{1}} , where e 1 {\displaystyle \mathbf {e} _{1}} is the unit vector perpendicular to these two adjacent wavefronts and the wavelength λ 1

{x^{2}+y^{2}}}} . Note ρ is a Lipschitz function whose gradient is a unit vector | ∇ ρ | = 1 {\displaystyle |\nabla \rho |=1} (almost everywhere). Let

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N i = d X i d X {\displaystyle N_{i}={\frac {dX_{i}}{dX}}} , is the unit vector in the direction of d X {\displaystyle d\mathbf {X} } , and the left-hand-side

0 {\displaystyle k\cdot k=0} and normalizing the momentum to be a 3D unit vector: k ^ i = k i / k 0 {\displaystyle {\hat {k}}_{i}=k_{i}/k_{0}} to write

}}\right)^{5}}}} where n ^ {\displaystyle {\hat {\mathbf {n} }}} is a unit vector pointing from the particle towards the observer, and d Ω {\displaystyle

{\displaystyle {\vec {p}}=p{\hat {n}}} with n ^ {\displaystyle {\hat {n}}} a unit vector, and let p → 0 {\displaystyle p\rightarrow 0} . E = − m 1 + p 2 m 2 →

explained that each path has a complex amplitude that can be thought of as a unit vector that is perpendicular to the path and makes one complete rotation for

yi is a 1 × 1 prediction, {βj} is a collection of r vectors (each a unit vector of length p) which contain the unknown parameters, {fj} is a collection

components in horizontal (x axis) and vertical (z axis) direction; k is the unit vector in the Z-direction, kT is molecular diffusivity of heat, kS is molecular

} , and e ^ ∠ {\displaystyle \mathbf {\hat {e}} _{\angle }\,\!} is a unit vector in the direction of incident flux on the area element, θ is the angle

theory) of the observable A for the system in state represented by the unit vector ψ ∈ H is ⟨ ψ | A | ψ ⟩ {\displaystyle \langle \psi |A|\psi \rangle }

\mathbf {m} )\cdot \mathbf {H} _{\text{eff}}\,\mathrm {d} V} Since m is a unit vector, dm is always perpendicular to m. Then the above definition leaves unspecified

{s}}_{k},{\vec {\tau }}_{k}\right\vert }^{2}} S4 resembles S2 except the unit vector parallel to shear stress is substituted by the predicted shear stress

for 3 respondents to question "a". The "answer" is represented by the unit vector z a {\displaystyle \mathbf {z} _{a}} , which is projected onto a plane

apply: The direction vector d {\displaystyle d} can be chosen to be a unit vector the point vector p {\displaystyle p} can be chosen to be the point on

{\displaystyle N\times N} Hermitian matrix A {\displaystyle A} and a unit vector b → ∈ R N {\displaystyle {\vec {b}}\in \mathbb {R} ^{N}} , prepare the

_{0}\leq k.\end{aligned}}} Eq. 1 The first constraint specifies that v is a unit vector. In the second constraint, ‖ v ‖ 0 {\displaystyle \left\Vert v\right\Vert

one another. To describe this anisotropic structure, a dimensionless unit vector n called the director, is introduced to represent the direction of preferred

induced volume form on the boundary of M, N is the outward oriented unit vector field normal to the boundary, and Δu = div(grad u) is the Laplacian.

{\text{ with }}\quad \sigma =\omega -\mathbf {k} \cdot \mathbf {U} } unit vector in the wave propagation direction e k {\displaystyle \mathbf {e} _{k}}

boundary of the domain, where n {\displaystyle \mathbf {n} } is the unit vector of the bounding surface (and, if the domain extends to infinity, u ⋅

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

this we can read off the Bel decomposition with respect to the timelike unit vector field X → = e → 0 = 1 a ( r ) ∂ t {\displaystyle {\vec {X}}={\vec {e}}_{0}={\frac

{\displaystyle \theta } is the angle of rotation about the direction given by unit vector a → {\displaystyle {\vec {a}}} . The displacement part can be written

check the result for the unit ball in Rn, assuming f(x) is zero for some unit vector x and f(y) < 0 if |y| < 1. By Harnack's inequality applied to −f − f

with mean square displacements. For the mean square displacement along unit vector n ^ {\displaystyle {\hat {n}}} , simply take n ^ T U n ^ {\displaystyle

gained 2π in argument along γ. (By Euler's identity, eiπ represents the unit vector, which therefore has π as its log. This π is what is meant by the argument

{\mathbf {r} -\mathbf {r} _{s}}{|\mathbf {r} -\mathbf {r} _{s}|}}} is the unit vector pointing in the direction from the source and, The symbol ( ⋯ ) t r {\displaystyle

results: If G = SL(2,R) acts unitarily on a Hilbert space H and ξ is a unit vector fixed by the subgroup N of upper unitriangular matrices, then ξ is fixed

\over dx^{2}}.} This vector equation can be separated in the bending unit vector definition ( M {\displaystyle M} is oriented as e y {\displaystyle \mathbf

Lp([0, 1]), 1 ≤ p < ∞, the Rademacher sequence is equivalent to the unit vector basis in ℓ2. In particular, the closed linear span of the Rademacher

{\kappa }}=\left({\vec {b}}\cdot \nabla \right){\vec {b}}} with b the unit vector tangent to B. Consider a cylindrical column of fully ionized quasineutral

point a ∈ Rn, and whose long side is parallel to the direction of the unit vector e ∈ Sn−1. Then for a locally integrable function f, we define the Kakeya

\mu } , e ^ ϕ {\displaystyle {\hat {e}}_{\phi }} is the same azimuthal unit vector from spherical coordinates, and e ^ ν {\displaystyle {\hat {e}}_{\nu

= r / r {\displaystyle {\hat {\mathbf {r} }}=\mathbf {r} /r} is the unit vector parallel to r; m is the (vector) dipole moment μ0 is the permeability

second equation, the pendulum equation of motion must be dotted with a unit vector that runs perpendicular to the pendulum at all times and is typically

basis vector ⁠ V i {\displaystyle \mathbf {V} _{i}} ⁠ to the stretched unit vector ⁠ σ i U i . {\displaystyle \sigma _{i}\mathbf {U} _{i}.} ⁠ By the definition

boundary curve, respectively. e 2 {\displaystyle \mathbf {e} _{2}} is the unit vector perpendicular to the tangent vector of the curve and the surface normal

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acceleration due to gravity, and r ^ {\displaystyle {\hat {r}}} is a unit vector in the radially outward direction from the center of the gravitating

algebras. Let H be a Hilbert space and M a von Neumann algebra on H with a unit vector Ω such that M Ω is dense in H M ' Ω is dense in H, where M ' denotes

equivalently, taking the dot product of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case

1 , N m ] {\displaystyle [N_{k},N_{1},N_{m}]} are the instantaneous unit vector components of the mirror's surface normal, then the mirror elevation

we assume the non-degeneracy condition on the kernel K: For a fixed unit vector u0 | K ( x ) | ≥ a | x | − n {\displaystyle |K(x)|\geq a|x|^{-n}} whenever

{\displaystyle \mathbf {T} ^{(\mathbf {n} )}} , for a given plane with unit vector n {\displaystyle \mathbf {n} } , satisfy the following equations ( T

{\vec {\sigma }}\ ,}   n ^   {\displaystyle \ {\hat {n}}\ } is a real unit vector in 3 dimensions in the direction of   a →   , {\displaystyle \ {\vec

vector (starting at the origin) and the positive x {\displaystyle x} -unit vector. The same concept may also be applied to lines in an Euclidean space

n}}} denotes differentiation along the integral surface element normal unit vector n {\displaystyle n} (i.e., a normal derivative), i.e., ∂ f ∂ n = ∇ f

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where | Ψ ¯ A + B ⟩ {\displaystyle |{\bar {\Psi }}_{A+B}\rangle } is a unit vector orthogonal to | Ψ ⟩ {\displaystyle |\Psi \rangle } . The form of | Ψ

is done. e → r ′ {\displaystyle \textstyle {{\vec {e}}_{r'}}} is the unit vector directed from the observation point (the place where E → {\displaystyle

representation of so(n,C) of dimension 2m: the Clifford action of a unit vector u ∈ U is given by u ⋅ ψ = { ψ if  ψ ∈ ∧ e v e n W − ψ if  ψ ∈ ∧ o d d

classifying a complex vector space V {\displaystyle V} , together with a unit vector u {\displaystyle u} in V {\displaystyle V} ; together they classify u

rotation-mode algorithm described above can rotate any vector (not only a unit vector aligned along the x axis) by an angle between −90° and +90°. Decisions

angle, and the adjacent side lies on the line of centers. (X, Y) is the unit vector pointing from c1 to c2, while R is cos θ where θ is the angle between

variable but parallel field: g(r) = g(r)n, where n is some constant unit vector. Although a non-uniform gravitational field cannot be exactly parallel

The choice of this measurement basis can be parametrized by a real unit vector a ∈ R 3 , | a | = 1 {\displaystyle {\boldsymbol {a}}\in \mathbb {R} ^{3}

source time function), and p ^ {\displaystyle {\hat {p}}} is a real unit vector pointing in the displacement direction and contained in the plane orthogonal

local Lagrangian (and hence respects conservation laws), and employs a unit vector field, a dynamical and non-dynamical scalar field, a free function and

{THz}}\phi _{\beta }({\mathbf {k} })} between two exciton states. The unit vector k T H z {\displaystyle {\mathbf {k} }_{\rm {THz}}} is determined by the

\mathbf {n} dl.} where n {\displaystyle \mathbf {n} } is the outward unit vector normal to the contour line element d l {\displaystyle dl} . The left-hand

field w on S. (The tangency condition means that w(x) ⋅ x = 0 for every unit vector x.) Sometimes the theorem is expressed by the statement that "there is

{x}}-ct),} where N {\displaystyle {\boldsymbol {N}}} is a Lagrangian unit vector in the direction of propagation (i.e., parallel to the wave number k

_{\mathbf {k} }(\mathbf {r} )\right|^{2}=1} where ek is chosen to be a unit vector which specifies the polarization of the field mode. The condition k ·

geodesic congruence of world lines obtained by integrating the timelike unit vector field X → = e → 0 {\displaystyle {\vec {X}}={\vec {e}}_{0}} , the expansion

{r} '|^{2}}}} where r ^ ′ {\displaystyle \mathbf {\hat {r}} '} is the unit vector pointing from the current (non-retarded) position of the particle to

Riemann solvers, an inter-particle Riemann problem is constructed along a unit vector e i j = − r i j / r i j {\displaystyle \mathbf {e} _{ij}=-\mathbf {r}

angle, Δφ >> (Δφ)², and n → {\displaystyle {\vec {n}}} is an arbitrary unit vector. From the orthogonality of Q → r {\displaystyle {\vec {Q}}_{r}} to the

free surface boundary conditions; n → {\displaystyle {\vec {n}}} is the unit vector normal to S. For a free body (as we assume it), the latter term sums

{\displaystyle \mathbf {n} =[-1,0,0]=-\mathbf {e} _{x}} is a negative unit vector. Then, we calculated the stress vector by definition s = n ⋅ σ = [ −

{R} =\left(\cos {2\omega t},-\sin {2\omega t},0\right)} . This is a unit vector that begins pointing along x ^ {\displaystyle \mathbf {\hat {x}} } and

{\displaystyle X} . Has been proven[5] that, if X {\displaystyle X} is unit vector (i.e. ∥ X ∥= 1 {\displaystyle \parallel X\parallel =1} ), then T r s

θ: polar angle, Iph,ω: spectral, directional radiation intensity, s: unit vector), i.e., q = qk + qu + qr. Once states and kinetics of the energy conversion

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probability density for a given value of the impact metric As a result, for unit vector spatial patterns: The first PCA spatial pattern always corresponds to

\alpha } is positive or negative. In both cases one has that where is the unit vector in the direction from F 2 {\displaystyle F_{2}} to F 1 {\displaystyle

all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction. Properties:

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non-invariant tensor). Let λ {\displaystyle {\boldsymbol {\lambda }}} be the unit vector which defines the axis of symmetry of the flow, then we have two scalar

{\displaystyle {\hat {p}}^{i}\equiv {\frac {p^{i}}{|\mathbf {p} |}}} is the unit vector oriented in the direction of the fermion momentum. The above are related

group SO(3) acts transitively on S2. The stabilizer subgroup of the unit vector (0,0,1) can be identified with SO(2), so that S2 = SO(3)/SO(2). The geodesics

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^{2}}}\right)^{2}-c|\nabla I|^{2},} where t {\displaystyle {\bf {t}}} is the unit vector perpendicular to the gradient, and c {\displaystyle c} determines how

limit is of x approaching surface β from inside domain D, nβ is the unit vector normal to surface β, and ∇x is now the multidimensional gradient operator

}}\int _{y}^{\infty }v_{T}\,dy} and i {\displaystyle \mathbf {i} } is the unit vector. Neither u 1 {\displaystyle \mathbf {u} _{1}} or u 2 {\displaystyle \mathbf

emits light of wavelength λ e m {\displaystyle \lambda _{em}} . The unit vector ″ e e x ″ {\displaystyle ''e_{ex}''} gives the orientation of the transition

{\mathbf {s}}_{0}} is a unit vector parallel to the direction of sliding, n 0 {\displaystyle {\mathbf {n}}_{0}} is a unit vector perpendicular to the contact

_{1}{\hat {X}})/\gamma } , where z ^ {\displaystyle {\hat {z}}} is the unit vector along uniform magnetic field B 0 {\displaystyle B_{0}} and X ^ {\displaystyle

a real phase factor, and ϵ ^ {\displaystyle {\hat {\epsilon }}} is a unit vector which indicates the direction of the excitation. Thus as t ⇒ ∞ {\displaystyle

one-one correspondence between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by h ( g ) = ( σ (