Tensor

Definition

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors, dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.

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1-form 2-form Absolute differential calculus Abstract algebra Abstract index notation Addison-Wesley Advances in Mathematics Affine connection Albert Einstein Albert Nijenhuis Algebraic dual Algebraic form Algebraic invariant theory Algebraic structure Algebraic topology André Joyal Angular momentum Angular momentum operator Anisotropic Antisymmetric tensor Application of tensor theory in engineering Array data structure Array data type Artificial neural networks Augustin-Louis Cauchy Banach manifold Banach space Basis (linear algebra)Basis of a vector space Basis vectors Bernhard Riemann Bijection Bilinear form Bitensor Bivector Carl Friedrich Gauss Cartan formalism (physics)Cartesian tensor Category theory Cauchy stress tensor Change of basis Change of variables formula Classical electromagnetism Classical group Coherent sheaves Commutativity Complex number Component-free treatment of tensors Computer vision Connection form Continuous dual Continuum mechanics Coordinate basis Coordinate system Covariance and contravariance of vectors Covariant derivative Covariant transformation Covector Cross product Current density Curvature Curvilinear coordinates Density on a manifold Determinant Differential form Differential geometry Diffusion MRI Diffusion tensor imaging Dimension Dimension (vector space)Donald A. Danielson Dot product Double dual Dual space Dual vector Dyadic tensor Dyadics Einstein field equations Einstein notation Einstein summation convention Einstein tensor Elasticity (physics)Elasticity tensor Electric field Electric susceptibility Electromagnetic tensor Electromagnetism Ellipsoid Elwin Bruno Christoffel Encyclopedia of Mathematics Engineering Equivariant map Eric W. Weisstein Euclidean geometry European Mathematical Society Exterior algebra Exterior calculus Exterior covariant derivative Exterior derivative Exterior form Exterior product Fiber bundle Fibre bundle Field (mathematics)Finite deformation tensors Fluid Fluid mechanics Flux Foundations of Differential Geometry Four-tensors Fréchet manifold Functor Fundamental matrix (computer vision)General linear group General relativity Geodesic Gerard Laman Glossary of tensor theory Gluon field strength tensor Gradient Gravity Gregorio Ricci-Curbastro Group action Group homomorphism Hassler Whitney Hermann Grassmann Hermann Weyl Hilbert space Hodge star operator Homological algebra Hypervolume Identity matrix Indeterminate (variable)Index notation Inner product Jacobian matrix and determinant James Munkres Jan Arnoldus Schouten Jet (mathematics)Journal of Differential Geometry Judith R. Goodstein Katsumi Nomizu Kerr effect Kronecker delta Künneth theorem Leonhard Euler Leslie Hogben Levi-Civita connection Levi-Civita symbol Lie derivative Linear elasticity Linear functional Linear map Linear operator Linear transformation Local diffeomorphism Machine learning Magnetic susceptibility Manifold Marcel Grossmann MathWorld Mathematical object Mathematician Mathematics Mathematische Annalen Matrix (mathematics)Matrix inverse Maxwell stress tensor Mechanics Metric tensor Metric tensor (general relativity)Mixed tensor Module over a ring Moment of inertia Momentum Monoidal category Multi-index notation Multidimensional array Multilinear algebra Multilinear map Multilinear subspace learning Multipole expansion Multipole moment Multivector NASA Natural bundle Naturally isomorphic Nondegenerate bilinear form Nonlinear optics Nonlinear system Nonmetricity tensor Norm (mathematics)One-form Operation (mathematics)Ordered basis Orientation (vector space)Orientation entanglement Orthonormal basis Outer product Parallelepiped Partial derivative Penrose graphical notation Permittivity Philosophical Magazine Physics PlanetMath Plate trick Pockels effect Poisson structure Polarization density Polyadic algebra Principal homogeneous space Pseudotensor Quadratic form Quadrupole moment Quantum computing Quantum mechanics Raising and lowering indices Rational representation Ray M. Bowen Real number Representation theory Ricci calculus Ricci curvature Richard Feynman Richard L. Bishop Riemann curvature tensor Riemannian geometry Ring (mathematics)Roger Penrose Ross Street SIAM J. Matrix Anal. 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