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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
Mathematical notation for tensors and spinors
with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate
Abstract_index_notation
On products of sums of series products
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that ( ∑ i = 1 n a i c i ) ( ∑ j = 1
Binet–Cauchy_identity
Antisymmetric permutation object acting on tensors
vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation. The Levi-Civita symbol is most often used in three and four dimensions
Levi-Civita_symbol
Branch of mathematics
forms Component-free treatment of tensors Cramer's rule Dual space Einstein notation Exterior algebra Inner product Outer product Kronecker delta Levi-Civita
Multilinear_algebra
Operation in mathematics
between, disappears; this is the essence of tensor contraction. In Einstein notation, this would be a i j × b j k = c i k {\textstyle a_{i}{}^{j}\times
Tensor_contraction
Vector behavior under coordinate changes
(as opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Topics referred to by the same term
Summation notation may refer to: Capital-sigma notation, mathematical symbol for summation Einstein notation, summation over like-subscripted indices This
Summation_notation
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
German-born theoretical physicist (1879–1955)
written in April 1953. Bern Historical Museum – Einstein Museum Einstein notation – Shorthand notation for tensor operations Frist Campus Center at Princeton
Albert_Einstein
System of symbolic representation
physicist Albert Einstein's formula E = m c 2 {\displaystyle E=mc^{2}} is the quantitative representation in mathematical notation of mass–energy equivalence
Mathematical_notation
Tensor index notation for tensor-based calculations
For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal
Ricci_calculus
Second-order differential operator
. Note that the μ and ν summation indices range from 0 to 3: see Einstein notation. (Some authors alternatively use the negative metric signature of
D'Alembert_operator
Vector operator in vector calculus
that article adopts the second of the two definitions above. Using Einstein notation we can consider the divergence in general coordinates, which we write
Divergence
Multivariate derivative (mathematics)
the vector nature of the result. gradf ∂if and f,i : Written with Einstein notation, where repeated indices (i) are summed over. The gradient (or gradient
Gradient
Theory of gravitation as curved spacetime
theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in May 1916 and is the accepted
General_relativity
Tensor used in general relativity
The Einstein tensor G {\displaystyle {\boldsymbol {G}}} is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is
Einstein_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)
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
Mathematical notation
used for the definition of weak derivatives of distributions. Einstein notation Index notation Ricci calculus Reed, M.; Simon, B. (1980). Methods of Modern
Multi-index_notation
Tensor that describes the 4D geometry of spacetime
constant G {\displaystyle G} will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Algebraic object with geometric applications
the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. The components
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
Addition of several numbers or other values
for non-negative real b > 1, c, d Capital-pi notation Einstein notation Iverson bracket Iterated binary operation Kahan summation algorithm
Summation
also 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
Property of a mathematical space
entropy). The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats
Dimension
Theoretical framework in physics
etc. The summation over the index μ has been omitted following the Einstein notation. If the parameter λ is sufficiently small, then the interacting theory
Quantum_field_theory
Tensor equal to the negative of any of its transpositions
\delta _{ab\dots }^{cd\dots }} is the generalized Kronecker delta, and the Einstein summation convention is in use. More generally, irrespective of the number
Antisymmetric_tensor
principle Einstein frame Einstein's mass–energy relation Einstein gravitational constant Einstein's radius of the universe Einstein (unit) Einstein notation Einstein
List of things named after Albert Einstein
List_of_things_named_after_Albert_Einstein
contrast, a dyad is specifically a dyadic tensor of rank one. Einstein notation This notation is based on the understanding that whenever a multidimensional
Glossary_of_tensor_theory
Mathematical Concept
associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas
Voigt_notation
Isomorphism between the tangent and cotangent bundles of a manifold
the basis as v = v i e i {\displaystyle v=v^{i}e_{i}} using Einstein summation notation, i.e., v {\displaystyle v} has components v i {\displaystyle
Musical_isomorphism
Method for specifying point positions
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Coordinate_system
Topological space that locally resembles Euclidean space
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Manifold
Tensor in differential geometry
general relativity, the Ricci curvature tensor enters the Einstein field equations through the Einstein tensor, formed from the Ricci tensor, the scalar curvature
Ricci_curvature
Array of numbers describing a metric connection
inverse of the matrix (gjk), defined as (using the Kronecker delta, and Einstein notation for summation) g j i g i k = δ j k {\displaystyle g^{ji}g_{ik}=\delta
Christoffel_symbols
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign
List of formulas in Riemannian geometry
List_of_formulas_in_Riemannian_geometry
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
Algebra associated to any vector space
{\displaystyle t\in A^{r}(V)\subset T^{r}(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 i 2 ⊗ ⋯
Exterior_algebra
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools
Mathematics of general relativity
Mathematics_of_general_relativity
Turbulence modeling approach
incompressible Newtonian fluid, these equations can be written in Einstein notation in Cartesian coordinates as: ρ u ¯ j ∂ u ¯ i ∂ x j = ρ f ¯ i + ∂ ∂
Reynolds-averaged Navier–Stokes equations
Reynolds-averaged_Navier–Stokes_equations
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
Ways of writing certain laws of physics
Lectures on Physics Vol. II Ch. 25: Electrodynamics in Relativistic Notation Einstein, A. (1961). Relativity: The Special and General Theory. New York:
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
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
Measure of the curvature of a pseudo-Riemannian manifold
part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through
Weyl_tensor
Mathematical function, in linear algebra
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Linear_map
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
Straight path on a curved surface or a Riemannian manifold
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Geodesic
Branch of mathematics
Most prominently the language of differential geometry was used by Albert Einstein in his general theory of relativity, and subsequently by physicists in
Differential_geometry
Set of vectors used to define coordinates
jyj, for i = 1, ..., n. This formula may be concisely written in matrix notation. Let A be the matrix of the ai,j, and X=[x1⋮xn]andY=[y1⋮yn] be the column
Basis_(linear_algebra)
Origin and evolution of the symbols used to write equations and formulas
standardized matrices notation, and parenthetical matrix and box matrix notation, respectively. Albert Einstein, in 1916, introduced Einstein notation, which summed
History of mathematical notation
History_of_mathematical_notation
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
Expression that may be integrated over a region
linearly dependent is zero. A common notation for the wedge product of elementary k-forms is so called multi-index notation: in an n-dimensional context, for
Differential_form
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
Continuous surjection satisfying a local triviality condition
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Fiber_bundle
Pair of values which express a thermodynamic system's internal energy
stress-induced infinitesimal strain εij is: δw=V∑ijσijdεij or, using Einstein notation for the tensors, in which repeated indices are assumed to be summed:
Conjugate variables (thermodynamics)
Conjugate_variables_(thermodynamics)
Mathematical identities
measure of how much nearby vectors tend in a circular direction. In Einstein notation, the vector field 𝐅=(F1, F2, F3) has curl given by: ∇×𝐅=εijk𝐞i∂Fk∂xj
Vector_calculus_identities
Field-equations in general relativity
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution
Einstein_field_equations
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
Classification of irreducible representations of the Poincaré group
C 1 = P μ P μ , {\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,} (Einstein notation) where P is the 4-momentum operator, and C 2 = W α W α , {\displaystyle
Wigner's_classification
Tensor having both covariant and contravariant indices
which will also be mixed. Covariance and contravariance of vectors Einstein notation Ricci calculus Tensor (intrinsic definition) Two-point tensor D.C
Mixed_tensor
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
purpose of the Kronecker delta function is for filtering terms from an Einstein summation convention. The discrete unit sample function is more simply
Kronecker_delta
Differential form of degree one or section of a cotangent bundle
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
One-form
Mapping from p forms to p-1 forms
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Interior_product
Theory of interwoven space and time by Albert Einstein
scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, "On the Electrodynamics of Moving Bodies", the theory is presented
Special_relativity
Chess game attributed to Albert Einstein and Robert Oppenheimer
in the United States. This section uses algebraic notation to describe chess moves. White: Einstein Black: Oppenheimer 1.e4 e5 2.Nf3 Nc6 3.Bb5 a6
Einstein_versus_Oppenheimer
Generators of the Clifford algebra for relativistic quantum mechanics
Covariant gamma matrices are defined by γμ=ημνγν={γ0,−γ1,−γ2,−γ3}, and Einstein notation is assumed. Note that the other sign convention for the metric, (−
Gamma_matrices
Electromagnetism in general relativity
G_{ab}~{\stackrel {\text{def}}{=}}~R_{ab}-{\frac {1}{2}}Rg_{ab}} is the Einstein tensor, κ is the Einstein gravitational constant, gab is the metric tensor, and R is
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Linear perturbations to solutions of nonlinear Einstein field equations
solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally
Linearized_gravity
Geometric structure
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Spinor_bundle
Coordinate-free definition of a tensor
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Tensor_(intrinsic_definition)
Historical critique of quantum mechanics
The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen, which argues
Einstein–Podolsky–Rosen paradox
Einstein–Podolsky–Rosen_paradox
Formalism of general relativity
( 4 ) g μ ν {\displaystyle {^{(4)}}g_{\mu \nu }} . The text uses Einstein notation, where repeated indices indicate summation. For example, if X ∈ T
BSSN_formalism
Measure of material deformation perpendicular to loading
εij=1E[σij(1+ν)−νδij∑kσkk] where δij is the Kronecker delta. The Einstein notation is usually adopted: σkk≡∑lδklσkl to write the equation simply as:
Poisson's_ratio
Hamiltonian formulation of general relativity
) g μ ν {\displaystyle {^{(4)}}g_{\mu \nu }} . The text here uses Einstein notation in which summation over repeated indices is assumed. Two types of
ADM_formalism
Canonical connection on a pseudo-Riemannian manifold
X(f)=X^{i}{\frac {\partial }{\partial x^{i}}}f=X^{i}\partial _{i}f} where Einstein's summation convention is used. An affine connection ∇ {\displaystyle \nabla
Levi-Civita_connection
Function that is invariant under all permutations of its variables
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Symmetric_function
Notation for quantum states
Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual
Bra–ket_notation
Decomposition in multilinear algebra
higher-order tensors in FI1⊗FI2⊗⋯⊗FIM with M>2 and all Im≥2. For simplicity in notation, assume without loss of generality that the factors are ordered such that
Tensor_rank_decomposition
Assignment of a tensor continuously varying across a region of space
transforms by the inverse Jacobian. The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent
Tensor_field
Mathematical object that describes the electromagnetic field in spacetime
]}\end{aligned}}} Darrigol, O. (2005). The genesis of the theory of relativity. In Einstein, 1905–2005: Poincaré Seminar 2005 (pp. 1-31). Basel: Birkhäuser Basel J
Electromagnetic_tensor
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
Basis for the SU(3) Lie algebra
g j ) {\displaystyle \mathrm {exp} (i\theta ^{j}g_{j})} using the Einstein notation, where the eight θ j {\displaystyle \theta ^{j}} are real numbers
Gell-Mann_matrices
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
Branch of physics which studies the behavior of materials modeled as continuous media
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Continuum_mechanics
Branch of acoustics
field. The Lighthill equation is an inhomogenous wave equation. Using Einstein notation, Lighthill’s equation can be written as ∂ 2 ρ ∂ t 2 − c 0 2 ∇ 2 ρ
Aeroacoustics
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
Affine_connection
Tensor invariant under permutations of vectors it acts on
Some include, the metric tensor, g μ ν {\displaystyle g_{\mu \nu }} , the Einstein tensor, G μ ν {\displaystyle G_{\mu \nu }} and the Ricci tensor, R μ ν
Symmetric_tensor
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
the Kronecker delta. The use of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent. Although Euler
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
System of moving vectors in differential geometry
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Parallel_transport
Measure of a substance's ability to resist or conduct electric current
Jz). Equivalently, resistivity can be given in the more compact Einstein notation: E i = ρ i j J j . {\displaystyle \mathbf {E} _{i}={\boldsymbol
Electrical resistivity and conductivity
Electrical_resistivity_and_conductivity
Specialized notation for multivariable calculus
attempting to use the same layout in all situations. The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except
Matrix_calculus
Operation on differential forms
dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}} (using the Einstein summation convention). The definition of the exterior derivative is extended
Exterior_derivative
Type of physical quantity
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Pseudotensor
Theorem in calculus
the fundamental theorem of calculus, part 2. Writing the theorem in Einstein notation: ∭ V ∂ F i ∂ x i d V = {\displaystyle \iiint _{V}{\dfrac {\partial
Divergence_theorem
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
{\displaystyle j} indicate the directional components in 3 dimensions in Einstein notation. Another assumption of linear elasticity is that the stress σ {\displaystyle
Eigenstrain
Mathematical description of spacetime used in relativity
the components of a vector v are written (v0, v1, v2, v3) where the Einstein notation is used to write v = vμ eμ. The component v0 is called the timelike
Minkowski_spacetime
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
Configurations of a system that do or do not satisfy classical equations of motion
mass shell is also often written in terms of the four-momentum; in Einstein notation with metric signature (+,−,−,−) and units where the speed of light
On_shell_and_off_shell
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EINSTEIN NOTATION
EINSTEIN NOTATION
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