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Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants
Voigt_notation
Matrix representing a Euclidean rotation
two-dimensional matrix using Voigt notation. When applying a rotational transform through angle θ {\displaystyle \theta } in this notation, the rotation matrix
Rotation_matrix
German mathematician and physicist (1850–1919)
now called the Voigt effect in 1898. The word tensor in its current meaning was introduced by him in 1898. Voigt profile and Voigt notation are named after
Woldemar_Voigt
Tensor equal to the negative of any of its transpositions
{\displaystyle U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.} A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example
Antisymmetric_tensor
Array of numbers
or no columns, called an empty matrix. The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written
Matrix_(mathematics)
Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory
Multi-index_notation
Origin and evolution of the symbols used to write equations and formulas
\mathbb {C} } ) for complex number sets. Around the 1930s, Voigt notation (so named to honor Voigt's 1898 work) would be developed for multilinear algebra
History of mathematical notation
History_of_mathematical_notation
Matrix operation which flips a matrix over its diagonal
another matrix, called the transpose of A and often denoted AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician
Transpose
Surname list
engineer Wolfgang Voigt, electronic music artist The Voigt profile, a peak function The Voigt pipe, a type of loudspeaker Voigt notation, a way to represent
Voigt
Topological space that locally resembles Euclidean space
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Manifold
Graphical notation for multilinear algebra calculations
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions
Penrose_graphical_notation
German aerospace engineer
grandfather was the German physicist Woldemar Voigt (1850-1919), known for Voigt notation, Voigt profile and the Voigt effect, and who introduced the term tensor
Woldemar_Voigt_(engineer)
Algebraic object with geometric applications
in the modern sense. The contemporary usage was introduced by Woldemar Voigt in 1898. Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro
Tensor
Mathematical notation for tensors and spinors
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate
Abstract_index_notation
Shorthand notation for tensor operations
differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies
Einstein_notation
Tensor index notation for tensor-based calculations
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with
Ricci_calculus
Mathematical operation on vector spaces
differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may
Tensor_product
Method for specifying point positions
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Coordinate_system
Force needed to pull a spring grows linearly with distance
to express the anisotropic form of Hooke's law in matrix notation, also called Voigt notation. To do this we take advantage of the symmetry of the stress
Hooke's_law
Representation of mechanical stress at every point within a deformed 3D object
_{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{matrix}}\right].} The Voigt notation representation of the Cauchy stress tensor takes advantage of the symmetry
Cauchy_stress_tensor
Model of viscoelastic material
A Kelvin–Voigt material, also called a Voigt material, is the most simple model viscoelastic material showing typical rubbery properties. It is purely
Kelvin–Voigt_material
Algebra associated to any vector space
Then any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in index notation with the Einstein summation convention as t = t i 1 i 2 ⋯ i r e i 1 ⊗ e
Exterior_algebra
Electric charge generated in certain solids due to mechanical stress
6-by-6 matrix instead of a rank-3 tensor. Such a relabeled notation is often called Voigt notation. Whether the shear strain components S4, S5, S6 are tensor
Piezoelectricity
Decomposition in multilinear algebra
{\displaystyle M>2} and all I m ≥ 2 {\displaystyle I_{m}\geq 2} . For simplicity in notation, assume without loss of generality that the factors are ordered such that
Tensor_rank_decomposition
Tensor that describes the 4D geometry of spacetime
{\displaystyle g_{\mu \nu }} themselves as the metric (see, however, abstract index notation). With the quantities d x μ {\displaystyle dx^{\mu }} being regarded as
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Test for predicting engineering failures
parameters. The stresses σ i {\displaystyle \sigma _{i}} are expressed in Voigt notation. If the failure surface is to be closed and convex, the interaction
Tsai–Wu_failure_criterion
_{31}\\2\varepsilon _{12}\end{bmatrix}}} An alternative representation in Voigt notation is [ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12
Orthotropic_material
Expression that may be integrated over a region
dependent is zero. A common notation for the wedge product of elementary k {\displaystyle k} -forms is so called multi-index notation: in an n {\displaystyle
Differential_form
Stress-strain relation in a linear elastic material
properties § Mechanical properties Representation theory of finite groups Voigt notation Here, upper and lower indices denote contravariant and covariant components
Elasticity_tensor
Exterior algebraic map taking tensors from p forms to n-p forms
}(dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}} These are summarized in the index notation as ⋆ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ , ⋆ ( d x
Hodge_star_operator
Algebraic operation on coordinate vectors
specified with respect to an orthonormal basis, is defined, in summation notation, as: a ⋅ b = ∑ i = 1 n a i b i = a 1 b 1 + a 2 b 2 + ⋯ + a n b n {\displaystyle
Dot_product
Differential form of degree one or section of a cotangent bundle
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
One-form
Structure defining distance on a manifold
is increased by du units, and v is increased by dv units. Using matrix notation, the first fundamental form becomes d s 2 = [ d u d v ] [ E F F G ] [ d
Metric_tensor
Tensor describing energy momentum density in spacetime
superscripted variables (not exponents; see Tensor index notation and Einstein summation notation). The four coordinates of an event of spacetime x are given
Stress–energy_tensor
Branch of mathematics
popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma
Differential_geometry
Operation in mathematics
2x2; often 3x3 or 4x4 are used, but any size is allowed. In simple index notation, this is written ∑ j = 1 2 a i j × b j k = c i k {\textstyle \sum _{j=1}^{2}a_{ij}\times
Tensor_contraction
Branch of mathematics
tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning Multivector Geometric algebra Clifford algebra
Multilinear_algebra
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
i = j ] . {\displaystyle \delta _{ij}=[i=j].} Often, a single-argument notation δ i {\displaystyle \delta _{i}} is used, which is equivalent to setting
Kronecker_delta
Affine connection on the tangent bundle of a manifold
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Levi-Civita_connection
Type of derivative in differential geometry
=f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega .} In local coordinate notation, for a type ( r , s ) {\displaystyle (r,s)} tensor field T {\displaystyle
Lie_derivative
Mathematical object that describes the electromagnetic field in spacetime
}F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0} or using the index notation with square brackets[note 1] for the antisymmetric part of the tensor:
Electromagnetic_tensor
Straight path on a curved surface or a Riemannian manifold
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Geodesic
Tensor in differential geometry
v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation, R i c a b = R c b c a = R c a c b . {\displaystyle \mathrm {Ric} _{ab}=\mathrm
Ricci_curvature
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
\end{aligned}}} It is common in rigid body mechanics to use notation that explicitly identifies the x {\displaystyle x} , y {\displaystyle y}
Moment_of_inertia
System of moving vectors in differential geometry
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Parallel_transport
Operation on differential forms
generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows: grad f ≡ ∇ f = ( d f ) ♯ div F ≡ ∇ ⋅ F = ⋆ d ⋆ ( F ♭ )
Exterior_derivative
Mathematical function, in linear algebra
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Linear_map
Set of vectors used to define coordinates
j}y_{j},} for i = 1, ..., n. This formula may be concisely written in matrix notation. Let A be the matrix of the a i , j {\displaystyle a_{i,j}} , and X = [
Basis_(linear_algebra)
Tensor field in Riemannian geometry
noncommutativity of the second covariant derivative. In abstract index notation, R d c a b Z c = ∇ a ∇ b Z d − ∇ b ∇ a Z d . {\displaystyle R^{d}{}_{cab}Z^{c}=\nabla
Riemann_curvature_tensor
Geological concept
{C}}}}~{\underline {\underline {\boldsymbol {\varepsilon }}}}} or, using Voigt notation, [ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12
Transverse_isotropy
Construct in differenital geometry
{\displaystyle A_{j}{}^{k}\ =\ \Gamma ^{k}{}_{ij}\,dx^{i}.} The point of the notation is to distinguish the indices j, k, which run over the n dimensions of
Metric_connection
Property of a mathematical space
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Dimension
Conserved physical quantity; rotational analogue of linear momentum
about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin
Angular_momentum
Isomorphism between the tangent and cotangent bundles of a manifold
the use of the musical notation symbols ♭ {\displaystyle \flat } (flat) and ♯ {\displaystyle \sharp } (sharp). In the notation of Ricci calculus and mathematical
Musical_isomorphism
Second order tensor in vector algebra
algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two
Dyadics
Tensor having both covariant and contravariant indices
covariant, the last one contravariant, and the remaining ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance
Mixed_tensor
Array of numbers describing a metric connection
reminder that these are defined to be equivalent notation for the same concept. The choice of notation is according to style and taste, and varies from
Christoffel_symbols
Specification of a derivative along a tangent vector of a manifold
language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension
Covariant_derivative
Function that is invariant under all permutations of its variables
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Symmetric_function
Assignment of a tensor continuously varying across a region of space
curvature tensors built from them are. The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent
Tensor_field
} where C i j {\displaystyle C_{ij}} refers to elastic constants in Voigt notation. Cubic materials are special orthotropic materials that are invariant
Zener_ratio
Coordinate-free definition of a tensor
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Tensor_(intrinsic_definition)
Vector behavior under coordinate changes
opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Continuous surjection satisfying a local triviality condition
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Fiber_bundle
contrast, a dyad is specifically a dyadic tensor of rank one. Einstein notation This notation is based on the understanding that whenever a multidimensional array
Glossary_of_tensor_theory
Measure of the curvature of a pseudo-Riemannian manifold
v_{3}\right)k\left(v_{1},v_{4}\right)\end{aligned}}} In tensor component notation, this can be written as C i k ℓ m = R i k ℓ m + 1 n − 2 ( R i m g k ℓ −
Weyl_tensor
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Symmetrization
Theory of interwoven space and time by Albert Einstein
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Special_relativity
Conversion of a matrix or a tensor to a vector
{T} })\operatorname {vec} (B)} . Duplication and elimination matrices Voigt notation Packed storage matrix Column-major order Matricization Macedo, H. D
Vectorization_(mathematics)
Material made from a combination of two or more unlike substances
materials are generally anisotropic, and in many cases are orthotropic. Voigt notation can be used to reduce the rank of the stress and strain tensors such
Composite_material
Branch of physics which studies the behavior of materials modeled as continuous media
of polarization. Couple stresses and body couples were first explored by Voigt and Cosserat, and later reintroduced by Mindlin in 1960 on his work for
Continuum_mechanics
Geometric structure
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Spinor_bundle
Abbreviation in the fields of special and general relativity
four-dimensional spacetime. General four-tensors are usually written in tensor index notation as A ν 1 , ν 2 , . . . , ν m μ 1 , μ 2 , . . . , μ n {\displaystyle A_{\;\nu
Four-tensor
Mapping from p forms to p-1 forms
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Interior_product
Non-tensorial representation of the spin group
form on a complex vector space is equivalent to the standard one, this notation is often used whenever dimℂ(V) = n. If n = 2k is even, then Cℓn(ℂ) is isomorphic
Spinor
Notation used for Weyl spinors
In theoretical physics, Van der Waerden notation refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard
Van_der_Waerden_notation
German artist (born 1977)
Voigt began making drawings that have been described as projection surfaces, visualized thought models, scientific experimental designs, notations, scores
Jorinde_Voigt
Type of physical quantity
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Pseudotensor
Physics concept
{x}^{i}}}} This is the explicit form of the covariant transformation rule. The notation of a normal derivative with respect to the coordinates sometimes uses a
Covariant_transformation
Antisymmetric permutation object acting on tensors
lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:
Levi-Civita_symbol
Construct allowing differentiation of tangent vector fields of manifolds
Y]=\left(X^{j}\partial _{j}Y^{i}-Y^{j}\partial _{j}X^{i}\right)\partial _{i}} in Einstein notation. This is independent of coordinate system choice and ∂ i = ( ∂ ∂ ξ i )
Affine_connection
Second-rank tensor in quantum chromodynamics
sum to be taken (e.g. "no sum"). Below the definitions (and most of the notation) follow K. Yagi, T. Hatsuda, Y. Miake and Greiner, Schäfer. The tensor
Gluon_field_strength_tensor
Concept in mathematics
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Tensor_bundle
Ways of writing certain laws of physics
equations, one for each value of β. Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Basis used to express spherical tensors
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Spherical_basis
Theory of gravitation as curved spacetime
is the stress–energy tensor. All tensors are written in abstract index notation. Matching the theory's prediction to observational results for planetary
General_relativity
second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Naghdi, Simmonds, Green and Zerna
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm. In Einstein notation, contravariant vectors and components of tensors are shown with superscripts
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Tensor invariant under permutations of vectors it acts on
the operator is omitted: T1T2 = T1 ⊙ T2. In some cases an exponential notation is used: v ⊙ k = v ⊙ v ⊙ ⋯ ⊙ v ⏟ k times = v ⊗ v ⊗ ⋯ ⊗ v ⏟ k times =
Symmetric_tensor
Universal construction in multilinear algebra
was actually one and the same thing as ∇ {\displaystyle \nabla } ; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor
Tensor_algebra
Tensor used in general relativity
tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as G = R − 1 2 g R , {\displaystyle {\boldsymbol {G}}={\boldsymbol
Einstein_tensor
Differential form
{\displaystyle \omega } is frequently used to denote the volume form, this notation is not universal; the symbol ω {\displaystyle \omega } often carries many
Volume_form
Math/physics concept
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Connection_form
Covariant derivative of the metric tensor
Y , Z {\displaystyle X,Y,Z} arbitrary vector fields. In abstract index notation, this reads Q a b c = ∇ a g b c {\displaystyle Q_{abc}=\nabla _{a}g_{bc}}
Nonmetricity_tensor
Electromagnetism in general relativity
square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is F α β ; γ =
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Operation that pairs a left and a right R-module into an abelian group
_{R}N} . It is often called a pure tensor. Strictly speaking, the correct notation would be x ⊗R y but it is conventional to drop R here. Then, immediately
Tensor_product_of_modules
Spinning motion in theoretical physics
{\mathfrak {se}}(d)} . This article uses Cartesian coordinates and tensor index notation. The Noether current for translations in space is momentum, while the current
Spin_tensor
General relativity articles using tensors will use the abstract index notation. The principle of general covariance was one of the central principles
Mathematics of general relativity
Mathematics_of_general_relativity
Study of curves from a differential point of view
Penrose graphical notation Ricci calculus Tetrad (index notation) Van der Waerden notation Voigt notation Tensor definitions Tensor (intrinsic definition) Tensor
Differentiable_curve
Tensor operator generalizes the notion of operators which are scalars and vectors
which applies for any wavefunction, ensures the above equality. In Dirac notation: ⟨ ψ ¯ | V ^ a | ψ ¯ ⟩ = ⟨ ψ | U ( R ) † V ^ a U ( R ) | ψ ⟩ = ∑ b R a
Tensor_operator
VOIGT NOTATION
VOIGT NOTATION
VOIGT NOTATION
VOIGT NOTATION
Boy/Male
Hindu, Indian, Traditional
Powerful Like God to Create Anything
Boy/Male
Hindu, Indian
Shape
Boy/Male
Hindu
Victorious, Glorious, Famous, Successful
Girl/Female
Arabic, Muslim
Rightly Guided by Allah
Surname or Lastname
English (Somerset)
English (Somerset) : unexplained.
Boy/Male
Buddhist, Indian, Sanskrit
Made of the Different Metals of Law
Girl/Female
Arabic, Muslim, Sindhi
Princess; Friend
Boy/Male
Greek Latin
Brother of Medea.
Girl/Female
Irish
Brave.
Girl/Female
Hindu, Indian
Strong
VOIGT NOTATION
VOIGT NOTATION
VOIGT NOTATION
VOIGT NOTATION
VOIGT NOTATION
n.
The act or practice of recording anything by marks, figures, or characters.
a.
Representing sounds; as, phonetic characters; -- opposed to ideographic; as, a phonetic notation.
n.
According to the French and American notation, a thousand octillions, or a unit with thirty ciphers annexed; according to the English notation, a million octillions, or a unit with fifty-four ciphers annexed. See the Note under Numeration.
n.
A method of analysis developed by Newton, and based on the conception of all magnitudes as generated by motion, and involving in their changes the notion of velocity or rate of change. Its results are the same as those of the differential and integral calculus, from which it differs little except in notation and logical method.
n.
The practice of using symbols, or the system of notation developed thereby.
n.
According to the English notation, a million involved to the tenth power, or a unit with sixty ciphers annexed; according to the French and American notation, a thousand involved to the eleventh power, or a unit with thirty-three ciphers annexed. [See the Note under Numeration.]
n.
According to the French notation, which is used upon the Continent generally and in the United States, the number expressed by a unit with twelve ciphers annexed; a million millions; according to the English notation, the number produced by involving a million to the third power, or the number represented by a unit with eighteen ciphers annexed. See the Note under Numeration.
n.
A method of notation for all spoken sounds, proposed by Mr. Sweet; -- so called because it is based on the common Roman-letter alphabet. It is like the palaeotype of Mr. Ellis in the general plan, but simpler.
n.
Literal or etymological signification.
n.
Any particular system of characters, symbols, or abbreviated expressions used in art or science, to express briefly technical facts, quantities, etc. Esp., the system of figures, letters, and signs used in arithmetic and algebra to express number, quantity, or operations.
n.
Ornamental notes or short passages, either introduced by the performer, or indicated by the composer, in which case the notation signs are called grace notes, appeggiaturas, turns, etc.
n.
The written and printed notation of a musical composition; the score.
a.
Marked or measured by crotchets; having musical notation.
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
a.
Of or pertaining to decimals; numbered or proceeding by tens; having a tenfold increase or decrease, each unit being ten times the unit next smaller; as, decimal notation; a decimal coinage.
n.
According to the French notation, which is used on the Continent and in America, the cube of a million, or a unit with eighteen ciphers annexed; according to the English notation, a number produced by involving a million to the fifth power, or a unit with thirty ciphers annexed. See the Note under Numeration.
n.
A character used in musical notation to determine the position and pitch of the scale as represented on the staff.
n.
According to the French notation, which is followed also upon the Continent and in the United States, a unit with fifteen ciphers annexed; according to the English notation, the number produced by involving a million to the fourth power, or the number represented by a unit with twenty-four ciphers annexed. See the Note under Numeration.
n.
The act of specifying or determining by a mark or limit; notation of limits.
n.
A table showing the notation, length, or duration of the several notes.