Tensors

algebraic operation that takes two equal-length sequences of numbers

physical quantity defined as the rate of change of angular position whose direction is (if regarded as a vector) the axis of rotation

physical quantity that expresses internal forces in a continuous material

right|thumb|300px|The second-order Cauchy stress tensor \mathbf{T} describes the stress experienced by a material at a given point. For any unit vector \mathbf{v}, the product \mathbf{T} \cdot \mathbf{v} is a vector, denoted \mathbf{T}(\mathbf{v}), that quantifies the force per area along the plane perpendicular to \mathbf{v}. This image shows, for cube faces perpendicular to \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, the corresponding stress vectors \mathbf{T}(\mathbf{e}_1), \mathbf{T}(\mathbf{e}_2), \mathbf{T}(\mathbf{e}_3) along those faces. Because the stress tensor takes one vector as inpu

transformation of a body from a reference configuration to a current configuration

shorthand notation for tensor operations

antisymmetric permutation object acting on tensors

polynomial with roots that are the eigenvalues of a given matrix

symmetric rank (0, 2) tensor field on a smooth manifold

assignment of a tensor continuously varying across a mathematical space

manner in which a geometric object varies with a change of basis

relative change of length with respect the original length

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In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

universal construction in multilinear algebra

medical imaging technique that uses water diffusion in tissue as a source of contrast

tensor equal to the negative of any of its transpositions

(1,2)-tensor field associated to an affine connection; characterizes "twist" of geodesics; if nonzero, geodesics will be helices

thumb|170px|Parallel plane segments with the same orientation and area corresponding to the same bivector .

vector bundle of all cotangent spaces at every point in a manifold

quadratic form related to curvatures of surfaces

dual space to the tangent space in differential geometry

in geometry, transferring a differential form or fiber bundle from the codomain of a continuous map to the domain

thumb|Relations between scalars, vectors, simple -vectors, -vectors, and multivectors. Depending on the authors, a "multivector" may be either homogeneous or a mixture of different values of . This graph picks the latter.

Mathematical notation for tensors and spinors

rational surface in 5-dimensional projective space

tensor invariant under permutations of vectors it acts on

theory

in mathematics and physics, an operation on tensors

notation for representing symmetric tensors

graphical notation for multilinear algebra calculations

In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., an improper rotation), which is a transformation that can be expressed as a proper rotation followed by reflection. This is a generalization of a pseudovector. To evaluate a tensor or pseudotensor sign, it has to be contracted with some vectors, as many as its rank is, belonging to the space where the rotation is made while ke

physics concept

mathematical operations relating different types of tensor

In physics, specifically for special relativity and general relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime.

is the component of the total stress tensor in a fluid

trace-free tensor of rank 2 which is conformally invariant in four dimensions

mathematical approach to the notion of tensor as an element of a tensor product

second order tensor field characterizing the gluon interaction between quarks

rank-3 tensor defined for a 3d (pseudo-)Riemannian manifold which measures the degree to which it fails to be conformally flat

generalization of tensor fields

tensor having both covariant and contravariant indices

tensor index notation for tensor-based calculations

product of two (0,2)-tensors as a (0,4)-tensor