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Longer titles found: Abstract index notation (view), Multi-index notation (view)

searching for Index notation 75 found (217 total)

alternate case: index notation

Miller index (2,457 words) [view diff] no match in snippet view article find links to article

Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require

W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors. The field involved is a complex 4-potential

field, depending upon the choice of gauge. This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance

Throughout, this article uses the natural units c = G = 1, and tensor index notation. The Mathisson–Papapetrou–Dixon (MPD) equations for a mass m {\displaystyle

in 1928 and by Hermann Weyl in 1929. The index notation for tetrads is explained in tetrad (index notation). Frame fields of a Lorentzian manifold always

f(x0, t0) refers to the value of the function f at the point (x0, t0). Index notation may be used instead of functional notation. That is, instead of writing

Kulkarni–Nomizu product of the Schouten tensor with the metric. In an index notation R i j k l = W i j k l + g i k P j l − g j k P i l − g i l P j k + g

{P}}=J~{\boldsymbol {\sigma }}^{T}\cdot {\boldsymbol {F}}^{-T}} In index notation, N I j = J   F I k − 1   σ k j and P i J = J   σ k i   F J k − 1 {\displaystyle

}(\mathbf {x} \times \mathbf {n} )\nabla \cdot \mathbf {n} \ dS=0,} or, in index notation, we have ∫ S x j ∇ ⋅ n   d S k = ∫ S x k ∇ ⋅ n   d S j . {\displaystyle

function. Hooke's law may be written in terms of tensor components using index notation as σ i j = 2 μ ε i j + λ δ i j ε k k , {\displaystyle \sigma _{ij}=2\mu

v_{x}}{\partial y}}\right)\,.\end{aligned}}} We may also express this in index notation as ω i = ε i j k ∂ v k ∂ x j {\displaystyle \omega _{i}=\varepsilon

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation

Bartel Leendert van der Waerden. The Infeld–Van der Waerden symbols are index notation for Clifford multiplication of covectors on left handed spinors giving

here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Two competing notational conventions split

symmetric part, one can write the conformal Killing equation in abstract index notation as ∇ a X b + ∇ b X a = 2 n g a b ∇ c X c . {\displaystyle \nabla _{a}X_{b}+\nabla

x_{i}}}~\mathbf {e} _{k}=S_{ik,i}~\mathbf {e} _{k}\end{aligned}}} where tensor index notation for partial derivatives is used in the rightmost expressions. Note that

literature), and where space and time components can be collected, tensor index notation is shown also (frequently used in the literature), in addition the Einstein

representation of probability distributions in quantum mechanics. Tensor index notation is used when formulating physics (particularly continuum mechanics,

by e ( a + b ) x . {\displaystyle e^{(a+b)x}.} : 69  With the multi-index notation for partial derivatives of functions of several variables, the Leibniz

{FM-EN}{EG-F^{2}}}\mathbf {r} _{v}} This can be expressed compactly in index notation as ∂ a n = K a   b r b {\displaystyle \partial _{a}\mathbf {n} =K_{a}^{~b}\mathbf

curvature, including in Finsler geometry. In the context of tensor index notation, it is common to use the letter R to represent three different things:

sum of a constant tensor and a traceless symmetric tensor. Thus in index notation: ε i j = ( 1 3 ε k k δ i j ) + ( ε i j − 1 3 ε k k δ i j ) {\displaystyle

{\nabla }}\left({\frac {1}{T}}\right)+{\frac {\rho ~s}{T}}.} Now, using index notation with respect to a Cartesian coordinate system e j {\displaystyle \mathbf

different compatibility conditions. We can write these conditions in index notation as e i k r   e j l s   ε i j , k l = 0 {\displaystyle e_{ikr}~e_{jls}~\varepsilon

transformation ϕ(x) is applied in electrodynamics, then one finds (using tensor index notation): A μ → A μ + ∂ μ ϕ ( x ) ψ → exp ⁡ [ + i Q ϕ ( x ) ] ψ ψ ¯ → exp ⁡

X / G {\displaystyle X_{G}=X/G} (orbits, quotient) for upper/lower index notation. The quadratic L-groups: L n ( R ) {\displaystyle L_{n}(R)} and the

variable names altogether by using a so-called "name-free" De Bruijn index notation. Explicit substitutions were sketched in the preface of Curry's book

_{j}Y^{i}} and ∂ j X i {\displaystyle \partial _{j}X^{i}} respectively using index notation) multiplying the n × 1 {\displaystyle n\times 1} column vectors X {\displaystyle

{\displaystyle {\bar {\psi }}\square \psi -\psi \square {\bar {\psi }}=0,} or in index notation, ψ ¯ ∂ μ ∂ μ ψ − ψ ∂ μ ∂ μ ψ ¯ = 0. {\displaystyle {\bar {\psi }}\partial

notation in which four dimensional tensors are written in abstract index notation, and that Greek indices are spacetime indices taking values (0, 1, 2

product symbol, denoting multiplication. In the more convenient multi-index notation this can be written f ( x ) = ∑ α ∈ N n a α ( x − c ) α . {\displaystyle

which the distorted cubes still fit together without overlapping. In index notation, the compatibility equations are expressed as ε i j , k m + ε k m ,

&\vdots \\u_{m}v_{1}&u_{m}v_{2}&\dots &u_{m}v_{n}\end{bmatrix}}} Or, in index notation: ( u ⊗ v ) i j = u i v j {\displaystyle (\mathbf {u} \otimes \mathbf

t_{2}=t_{1}^{2},\;\;t_{3}=t_{1}^{3},\;\;t_{4}=t_{1}^{4}} In tensor index notation, this can be written using the Ricci scalar as: G a a = − R {\displaystyle

approach applies for connections in any vector bundle. Traditional index notation specifies the connection by components; see Christoffel symbols. (Note:

first time in a printed text (in Book II of his Algebra), a form of index notation in which the equation x 3 = 6 x + 40 {\displaystyle x^{3}=6x+40} appeared

planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices h, k, and ℓ as directional parameters

f)(x)=f(x+1)-f(x).} It is thus a special case of finite difference. When using the index notation for sequences, the definition becomes ( Δ a ) n = a n + 1 − a n . {\displaystyle

(see α-renaming to make name resolution trivial). In the De Bruijn index notation, any two α-equivalent terms are syntactically identical. Substitution

left-handed and right-handed Weyl spinors. The undotted and dotted index notation, known as Van der Waerden notation, distinguishes left-handed and right-handed

¯ {\displaystyle {\bar {D}}} is the covariant derivative, given in index notation as D ¯ α ˙ = − ∂ ¯ α ˙ − i θ α σ α α ˙ μ ∂ μ . {\displaystyle {\bar

\beta }\phi _{j}\\\end{matrix}}} The second column expands each term in index notation (neglecting the proper normalization coefficient). i and j are the generation

and atomic plane families, can be described using a four-value Miller index notation ( hkil ) in which the third index i denotes a degenerate but convenient

{\displaystyle {\mathfrak {so}}(3)} . The above result can be written in index notation as follows. The elements of the matrix for an active rotation by an

velocity squared, instead of stress. To illustrate, Cartesian vector index notation is used. For simplicity, consider an incompressible fluid: Given the

transform is an involution, that is, T T = 1 {\displaystyle TT=1} or, using index notation, ∑ k = 0 ∞ T n k T k m = δ n m {\displaystyle \sum _{k=0}^{\infty }T_{nk}T_{km}=\delta

ingredient of the proof is the following simple property, which uses multi-index notation for monomials in the variables Xi. Lemma. The leading term of eλt (X1

{u} )]+\rho \mathbf {a} .} in index notation, the equation can be written as Navier–Stokes momentum equation (index notation) ρ ( ∂ u i ∂ t + u k ∂ u i ∂

four dimensional vectors (except for the four-gradient), and tensor index notation. A four-vector A is a vector with a "timelike" component and three "spacelike"

narrow hats are for unit vectors (including their components in tensor index notation). The summation convention on the repeated tensor indices is used, unless

4-dimensional spacetime, to be defined. In the corresponding tensor index notation, the above matrix expression is x ′ ν = Λ ν μ x μ , {\displaystyle {x'}^{\nu

Cerium(III) nitrate crystal with Miller index notation

^{4}w}{\partial x_{2}^{4}}}=0\quad {\text{where}}\quad w:=w^{0}\,.} In index notation, w , 1111 0 + 2   w , 1212 0 + w , 2222 0 = 0 . {\displaystyle w_{,1111}^{0}+2~w_{

square root of the matrix g − 1 f {\displaystyle g^{-1}f} . Written in index notation, X {\displaystyle \mathbb {X} } is defined by the relation X μ α X α

}+a_{j}b_{k}\delta _{jk}I\end{aligned}}.~} Finally, translating the index notation for the dot product and cross product results in If i is identified

such that g(g(x)) = f(x). This function g(x) can be written using the index notation as f 1/2(x) . Similarly, f 1/3(x) is the function defined such that

scatteringPages displaying short descriptions of redirect targets Miller index – Notation system for crystal lattice planes Powder diffraction – Experimental

a crack tip is frequently expressed in a more general form (and in index notation) as J i := lim ε → 0 ∫ Γ ε ( W ( Γ ) n i − n j σ j k   ∂ u k ( Γ , x

set of curvilinear coordinates ξ = (ξ1, ξ2, ξ3), the law in tensor index notation is the "Lagrangian form" F a = m ( d 2 ξ a d t 2 + Γ a b c d ξ b d t

Common notations for the type A/B distinction index notation i1 i2 e1 e2 o1 o2 Kindaichi, Miller, Tōdō i ï e ë o ö Vovin i ï e ɛ o ə modified Mathias–Miller

{1}{2}}g^{ma}g^{nb}\varepsilon _{abpq}{\sqrt {-g}}} in terms of tensor index notation with respect to a (not necessarily orthonormal) basis { ∂ ∂ x 1 , …

which relate the different strain components. These are expressed in index notation as: ε i j , k m + ε k m , i j − ε i k , j m − ε j m , i k = 0. {\displaystyle

functions are additive. Throughout, the standard conventions of tensor index notation and Feynman slash notation are used, including Greek indices which take

is a symplectic form, it is the Berry phase or geometric phase. In index notation, the metric is: h j k = h ( ∂ ∂ θ j , ∂ ∂ θ k ) = 1 4 E [ ∂ log ⁡ p

(x_{1})m{\big )}u_{{\textbf {p}}_{1}}^{s_{1}}.} Switching to Dirac index notation (with sums over repeated indices) allows for a neater expression, in

square root of the matrix g − 1 f {\displaystyle g^{-1}f} . Written in index notation, X {\displaystyle \mathbb {X} } is defined by the relation   X μ α  

(gijkij);p)(gklkqk;l - (gklkkl);q))1/2 or, in the usual "raised and lowered index" notation, this says R - kijkij + (kii)2 ≥ 2((kpi;i - (kii);p)(kpj;j - (kjj);p))1/2

where the second expression uses the Einstein summation convention for index notation of tensors. W {\displaystyle W} is the strain energy density function

{\displaystyle \delta A(t)} is the fluctuation, be written as (use index notation with summation over repeated indices) A ˙ i ( t ) = v i ( t ) + Ω i

where the thickness of the plate is 2 h {\displaystyle 2h} . In index notation, N α β , α = 0 N α β := ∫ − h h σ α β   d x 3 M α β , α β + q = 0 M

(px, py, pz), the components can be compactly abbreviated in tensor index notation L i j = x i p j − x j p i {\displaystyle L^{ij}=x^{i}p^{j}-x^{j}p^{i}}

points (method Point.plot()); here are some examples. Introduction of index notation to denote tensor contractions and tensor symmetrizations (new class

function -- it equals 1 if the two entries are equal, and 0 otherwise. The index notation is as follows: k {\displaystyle k} ranges over English-foreign sentence

}}\times \mathbf {B} \,.} In the language of four-vectors and tensor index notation, the Lagrangian takes the form L ( τ ) = 1 2 m u μ ( τ ) u μ ( τ ) +