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Representation of a tensor in Euclidean space
algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components
Cartesian_tensor
of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory
Glossary_of_tensor_theory
Operation in graph theory
has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The
Cartesian_product_of_graphs
Mathematical operation on vector spaces
two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense
Tensor_product
Representation of mechanical stress at every point within a deformed 3D object
Cauchy stress tensor (symbol σ {\displaystyle {\boldsymbol {\sigma }}} , named after Augustin-Louis Cauchy), also called true stress tensor or simply stress
Cauchy_stress_tensor
Algebraic object with geometric applications
(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In
Tensor
Type of category in category theory
abelian category, is not Cartesian closed. So the category of modules over a ring is not Cartesian closed. However, the functor tensor product − ⊗ M {\displaystyle
Cartesian_closed_category
Tensor describing energy momentum density in spacetime
stress-energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity
Stress–energy_tensor
Spinning motion in theoretical physics
theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general
Spin_tensor
Structure defining distance on a manifold
metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g ( v , v ) >
Metric_tensor
In image analysis, the generalized structure tensor (GST) is an extension of the Cartesian structure tensor to curvilinear coordinates. It is mainly used
Generalized_structure_tensor
Graph in graph theory
lexicographic order. It is one of 4 common graph products including Cartesian, tensor, and strong. The lexicographic product was first studied by Felix
Lexicographic product of graphs
Lexicographic_product_of_graphs
In physics, the gyration tensor is a tensor that describes the second moments of position of a collection of particles S m n = d e f 1 N ∑ i = 1 N
Gyration_tensor
Cosines of the angles between a vector and the coordinate axes
_{u}\beta _{v}+\gamma _{u}\gamma _{v}\right|\right).} Cartesian tensor Euler angles Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. pp. 18–19
Direction_cosine
Operation in graph theory
In graph theory, the tensor product G × H of graphs G and H is a graph such that the vertex set of G × H is the Cartesian product V(G) × V(H); and vertices
Tensor_product_of_graphs
Electric charge generated in certain solids due to mechanical stress
that is, Cartesian tensors of rank 1; and permittivity ε is a Cartesian tensor of rank 2. Strain and stress are, in principle, also rank-2 tensors. But conventionally
Piezoelectricity
Stress-strain relation in a linear elastic material
elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness
Elasticity_tensor
Mathematical set formed from two given sets
G. The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the tensor product
Cartesian_product
Arrangement that creates a quadrupole field of some sort
reflecting various orders of complexity. The quadrupole moment tensor Q is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally
Quadrupole
Coordinate system whose directions vary in space
example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient
Curvilinear_coordinates
Mathematical object that describes the electromagnetic field in spacetime
electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a tensor that describes
Electromagnetic_tensor
Mathematical form
infinite-dimensional vector spaces, one also has the: Tensor product of Hilbert spaces Topological tensor product. The tensor product, outer product and Kronecker product
Product_(mathematics)
Assignment of a tensor continuously varying across a region of space
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space
Tensor_field
Type of category in category theory
a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite
Cartesian_monoidal_category
Algebraic operation on coordinate vectors
(single-) dot product between a tensor of order n {\displaystyle n} and a tensor of order m {\displaystyle m} is a tensor of order n + m − 2 {\displaystyle
Dot_product
Tensor index notation for tensor-based calculations
notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern
Ricci_calculus
Coefficients in angular momentum eigenstates of quantum systems
is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators. By developing
Clebsch–Gordan_coefficients
Mathematical model for describing material deformation under stress
deformation tensors. In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor (the
Finite_strain_theory
Tensor operator generalizes the notion of operators which are scalars and vectors
graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which
Tensor_operator
Operation in mathematics
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. This example
Tensor_contraction
Spectroscopic technique
directions x, y, and z in the molecular frame are represented by the Cartesian tensor ρ and σ here. Analyzing Raman excitation patterns requires the use
Raman_spectroscopy
Coordinate-free definition of a tensor
mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear
Tensor_(intrinsic_definition)
Method of utilizing water in magnetic resonance imaging
more gradient directions, sufficient to compute the diffusion tensor. The diffusion tensor model is a rather simple model of the diffusion process, assuming
Diffusion-weighted magnetic resonance imaging
Diffusion-weighted_magnetic_resonance_imaging
Concept in multilinear algebra and representation theory
and representation theory, the principal invariants of the second rank tensor A {\displaystyle \mathbf {A} } are the coefficients of the characteristic
Invariants_of_tensors
Rigid body equations in classical mechanics
Balafoutis, Rajnikant V. Patel (1991). Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach. Springer. Chapter 5. ISBN 0-7923-9145-4.
Newton–Euler_equations
Relative deformation of a physical body
ISO 80000-4 (Mechanics), as a "tensor quantity representing the deformation of matter caused by stress. Strain tensor is symmetric and has three linear
Strain_(mechanics)
{\boldsymbol {T}}} is a tensor field of order n > 1 then the divergence of the field is a tensor of order n− 1. In a Cartesian coordinate system we have
Tensor derivative (continuum mechanics)
Tensor_derivative_(continuum_mechanics)
Method for specifying point positions
unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and
Coordinate_system
Moment of inertia of diff geometric shapes
the dots indicate tensor contraction and the Einstein summation convention is used. In the above table, n would be the unit Cartesian basis ex, ey, ez
List_of_moments_of_inertia
Vector operator in vector calculus
covariant index of a tensor is intrinsic and depends on the ordering of the terms of the Cartesian product of vector spaces on which the tensor is given as a
Divergence
Antisymmetric permutation object acting on tensors
independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms
Levi-Civita_symbol
Equations of motion for viscous fluids
Navier–Stokes equations from Euler equations) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems. A special
Navier–Stokes_equations
Physical quantity that expresses internal forces in a continuous material
the first and second Piola–Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor. Bending Compressive strength Critical plane
Stress_(mechanics)
expressed in Cartesian coordinates. In Cartesian coordinates the 2 basis vectors are represented by a 2 × 2 {\displaystyle 2\times 2} cell tensor h {\displaystyle
Fractional_coordinates
Abbreviation in the fields of special and general relativity
relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime. General four-tensors are usually written in tensor index notation
Four-tensor
Concept in mathematics
In mathematics, the tensor-hom adjunction is the statement that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle
Tensor–hom_adjunction
Phrase of the philosopher René Descartes
Charles Porterfield Krauth. Fumitaka Suzuki writes "Taking consideration of Cartesian theory of continuous creation, which theory was developed especially in
Cogito,_ergo_sum
Cartesian plane Cartesian tensor Cartesian monoid Cartesian monoidal category Cartesian closed category Cartesian oval Cartesian product Cartesian product of
List of things named after René Descartes
List_of_things_named_after_René_Descartes
Basis used to express spherical tensors
A vector A in 3D Euclidean space R3 can be expressed in the familiar Cartesian coordinate system in the standard basis ex, ey, ez, and coordinates Ax
Spherical_basis
Physical quantity that is a vector
example, a position vector in physical space may be expressed as three Cartesian coordinates with SI unit of meters. In physics and engineering, particularly
Vector_quantity
Tensor used in continuum mechanics
The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed
Viscous_stress_tensor
Mathematical model for describing material deformation under stress
tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor E {\displaystyle \mathbf {E} } , and the Eulerian finite strain tensor
Infinitesimal_strain_theory
Type of category in mathematics
non-cartesian example is the category of vector spaces, K-Vect, over a field K {\displaystyle K} . Here the monoidal product is the usual tensor product
Closed_monoidal_category
Category admitting tensor products
In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle
Monoidal_category
Differential operator in mathematics
any tensor field T {\displaystyle \mathbf {T} } ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor: ∇ 2
Laplace_operator
Mathematical operation on matrices
specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map
Kronecker_product
Line segment of infinitesimally small length
from the metric. The simplest line element is in Cartesian coordinates - in which case the metric tensor is just the Kronecker delta: g i j = δ i j {\displaystyle
Line_element
Mathematical identities
)^{\textsf {T}}} is a tensor field of order k + 1. For a tensor field T {\displaystyle \mathbf {T} } of order k > 0, the tensor field ∇ T {\displaystyle
Vector_calculus_identities
Second order tensor in vector algebra
mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There
Dyadics
British physicist and mathematician
Inference, Macmillan Publishers; 2nd edn. 1937; 3rd edn. 1973 1931: Cartesian Tensors. Cambridge University Press; 2nd edn. 1961 1934: Ocean Waves and Kindred
Harold_Jeffreys
Array of numbers describing a metric connection
corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γijk are zero
Christoffel_symbols
Isomorphism between the tangent and cotangent bundles of a manifold
X_{2}=y,\quad X_{3}=z} (where x,y,z are the usual Cartesian coordinates) and the Minkowski metric tensor with metric signature (− + + +) is defined as η
Musical_isomorphism
Generalised alphabetical order
order on an n-ary Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally
Lexicographic_order
Vector behavior under coordinate changes
consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Generalizations of '"`UNIQ--math-00000000-QINU`"' in algebraic structures
Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Among tensors of a given type, the zero tensor of that type
Zero_element
Specification of a derivative along a tangent vector of a manifold
fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). Given
Covariant_derivative
Degree to which a material becomes magnetized in an applied magnetic field
defined as a tensor: M i = H j χ i j {\displaystyle M_{i}=H_{j}\chi _{ij}} where i and j refer to the directions (e.g., of the x and y Cartesian coordinates)
Magnetic_susceptibility
Concept in mathematical category theory
category (i.e. a category in which a "tensor product" ⊗ {\displaystyle \otimes } is defined) such that the tensor product is symmetric (i.e. A ⊗ B {\displaystyle
Symmetric_monoidal_category
Proposed theories of gravity
Minkowski metric. g μ ν {\displaystyle g_{\mu \nu }\;} is a tensor, usually the metric tensor. These have signature (−,+,+,+). Partial differentiation is
Alternatives to general relativity
Alternatives_to_general_relativity
Mathematical gradient operator in certain coordinate systems
{\displaystyle \varphi } in the formulae shown in the table above. ^β Defined in Cartesian coordinates as ∂ i A ⊗ e i {\displaystyle \partial _{i}\mathbf {A} \otimes
Del in cylindrical and spherical coordinates
Del_in_cylindrical_and_spherical_coordinates
Force needed to pull a spring grows linearly with distance
the tensor s, called the compliance tensor, represents the inverse of said linear map. In a Cartesian coordinate system, the stress and strain tensors can
Hooke's_law
Mathematical model of how solid objects deform
{\sigma }}} is the Cauchy stress tensor, ε {\displaystyle {\boldsymbol {\varepsilon }}} is the infinitesimal strain tensor, u {\displaystyle \mathbf {u}
Linear_elasticity
Operator generalizing the Laplacian in differential geometry
Hessian tensor. Because the covariant derivative extends canonically to arbitrary tensors, the Laplace–Beltrami operator defined on a tensor T by Δ T
Laplace–Beltrami_operator
Tensor in general relativity
\partial x^{b}}}} where we are using the standard Cartesian chart for E3, with the Euclidean metric tensor d s 2 = d x 2 + d y 2 + d z 2 , − ∞ < x , y , z
Tidal_tensor
Vector operation
two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product
Outer_product
Vector differential operator
being a tensor. The tensor derivative of a vector field v {\displaystyle \mathbf {v} } (in three dimensions) is a 9-term second-rank tensor – that is
Del
Mathematical description of spacetime used in relativity
the metric tensor of Minkowski spacetime. It is a pseudo-Euclidean metric, or more generally, a constant pseudo-Riemannian metric in Cartesian coordinates
Minkowski_spacetime
Most general completion of a commutative square given two morphisms with same codomain
pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z
Pullback_(category_theory)
Coordinates comprising a distance and two angles
first in the Cartesian xy plane from (x, y) to (R, φ), where R is the projection of r onto the xy-plane, and the second in the Cartesian zR-plane from
Spherical_coordinate_system
Pakistani mathematical physicist
Vol:61 pp:471–476 (Journal) HEC Recognized:Yes "Linear invariants of a cartesian tensor" (2009) Quarterly Journal of Mechanics and Applied Mathematics Vol:62
Muneer_Ahmad_Rashid
transformation tensor F {\displaystyle {\boldsymbol {F}}} because an alternative form of the mapping between curvilinear and Cartesian bases is more useful
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
Set of coordinates where the coordinate hypersurfaces all meet at right angles
hypersurface on which qk is a constant. For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate
Orthogonal_coordinates
Component of mechanical stress without shear
it is one third of the first invariant of the stress tensor (i.e. the trace of the stress tensor): σ h = I i 3 = 1 3 tr ( σ ) {\displaystyle \sigma
Hydrostatic_stress
Solution of Einstein field equations
Riemann tensor can be computed into three pieces, the tidal or electrogravitic tensor (which represents tidal forces), the magnetogravitic tensor (which
Gödel_metric
Geometric object that has length and direction
contravariance of vectors). Tensors are another type of quantity that behave in this way; a vector is one type of tensor. In pure mathematics, a vector
Euclidean_vector
Topics referred to by the same term
often called product that yields the product of a sequence Direct product Cartesian product of sets Direct product of groups Semidirect product Product of
Product
Abstract coordinate system
points are sufficient to fully define a reference frame. Using rectangular Cartesian coordinates, a reference frame may be defined with a reference point at
Frame_of_reference
Coordinates comprising a distance and an angle
coordinate, polar angle, or azimuth. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context
Polar_coordinate_system
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
the metric tensor along an integral curve generated by the vector field (whose image is parallel to the x-axis). Furthermore, the metric tensor is independent
Killing_vector_field
Topics referred to by the same term
viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. Affine differential
Affine
Theorem in planar dynamics
involving the inertia tensor. Let Iij denote the inertia tensor of a body as calculated at the center of mass. Then the inertia tensor Jij as calculated relative
Parallel_axis_theorem
Category of coordinate systems
reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
American engineer and professor (1909–1972)
1956. By using Cartesian tensor notation in his last three books, Engineering Mechanics, Statics of Deformable Bodies, and Cartesian Tensors, Myklestad became
Nils_Otto_Myklestad
exactly two ways: with the usual categorical product and with the funny tensor product. Given two categories C {\displaystyle C} and D {\displaystyle D}
Premonoidal_category
Algebra associated to any vector space
alternating tensor subspace. On the other hand, the image A ( T ( V ) ) {\displaystyle {\mathcal {A}}(\mathrm {T} (V))} is always the alternating tensor graded
Exterior_algebra
Property of a mathematical space
two dimensions, and a cube describes three dimensions. (See Space and Cartesian coordinate system.) A temporal dimension, or time dimension, is a dimension
Dimension
Generalized Euclidean space in mathematics
this case. The Cartesian product of projective Hilbert spaces is not a projective space. The Segre mapping is an embedding of the Cartesian product of two
Projective_Hilbert_space
Transforming a function in such a way that it only takes a single argument
currying and uncurrying is known as tensor-hom adjunction. Here, an interesting twist arises: the Hom functor and the tensor product functor might not lift
Currying
Mathematical function
three dimensions, in Cartesian and spherical geometry, on surfaces and in volumes, on graphs, and in scalar, vector, and tensor forms. Without reference
Slepian_function
Angular momentum in special and general relativity
of point-like particles, the angular momentum tensor is expressed in terms of the stress–energy tensor of the rotating object. In special relativity alone
Relativistic_angular_momentum
CARTESIAN TENSOR
CARTESIAN TENSOR
Surname or Lastname
English
English : from the Old French personal name Hu(gh)e, introduced to Britain by the Normans. This is in origin a short form of any of the various Germanic compound names with the first element hug ‘heart’, ‘mind’, ‘spirit’. Compare, for example, Howard 1, Hubble, and Hubert. It was a popular personal name among the Normans in England, partly due to the fame of St. Hugh of Lincoln (1140–1200), who was born in Burgundy and who established the first Carthusian monastery in England.In Ireland and Scotland this name has been widely used as an equivalent of Celtic Aodh ‘fire’, the source of many Irish surnames (see for example McCoy).
CARTESIAN TENSOR
CARTESIAN TENSOR
Biblical
God that hears;hears;
Boy/Male
Arabic, Australian, Muslim, Pashtun
Best Friend of Prophet; Evident
Boy/Male
Indian, Telugu
Lord Shiva
Male
Hebrew
Short form of Hebrew Immanuw'el (English Immanuel), MAN means "God is with us."
Girl/Female
Hindu, Indian
Moon
Girl/Female
Indian
Perfect, Goddess, Flower
Boy/Male
Finnish, German
Valley; Stream
Girl/Female
Indian
Jewelry, Ornament, Finery
Girl/Female
Tamil
Anandithya | அநாநà¯à®¤à¯€à®¤à¯à®¯à®¾Â
Boy/Male
Gujarati, Hindu, Indian, Kannada, Kashmiri, Malayalam, Marathi, Oriya, Rajasthani, Sindhi, Tamil, Telugu, Traditional
Pearl
CARTESIAN TENSOR
CARTESIAN TENSOR
CARTESIAN TENSOR
CARTESIAN TENSOR
CARTESIAN TENSOR
n.
A muscle that stretches a part, or renders it tense.
n.
Sard; carnelian.
n.
An adherent of Descartes.
v. i.
To pass by degrees; to change gradually; to shade off; as, sandstone which graduates into gneiss; carnelian sometimes graduates into quartz.
n.
An instrument for clutching objects for the purpose of raising them; -- specially applied to devices for withdrawing drills, etc., from artesian and other wells that are drilled, bored, or driven.
n.
The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.
n.
A well known public school and charitable foundation in the building once used as a Carthusian monastery (Chartreuse) in London.
n.
A Carthusian monastery; esp. La Grande Chartreuse, mother house of the order, in the mountains near Grenoble, France.
n.
A precious stone, probably a carnelian, one of which was set in Aaron's breastplate.
a.
Of or pertaining to the French philosopher Rene Descartes, or his philosophy.
n.
A variety of chalcedony, of a clear, deep red, flesh red, or reddish white color. It is moderately hard, capable of a good polish, and often used for seals.
a.
Of or pertaining to Artois (anciently called Artesium), in France.
n.
A Carthusian.
n.
A bead of rough carnelian. Arangoes were formerly imported from Bombay for use in the African slave trade.
n.
A variety of carnelian, of a rich reddish yellow or brownish red color. See the Note under Chalcedony.
n.
Same as Carnelian.
n.
The system of occasional causes; -- a name given to certain theories of the Cartesian school of philosophers, as to the intervention of the First Cause, by which they account for the apparent reciprocal action of the soul and the body.
a.
Pertaining to the Carthusian.
n.
A member of an exceeding austere religious order, founded at Chartreuse in France by St. Bruno, in the year 1086.