tensor

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

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A tensor is a mathematical object that takes in one or more vectors and produces an output, allowing it to describe complex relationships like how forces are distributed across different directions in a material. Tensors matter because they provide a precise way to represent physical quantities—such as stress in materials—that depend on direction, making them essential for engineering and physics calculations.

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Commons category: Tensors

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Article

38 sections

Contents

Definition
As multidimensional arrays
As multilinear maps
Using tensor products
Tensors in infinite dimensions
Tensor fields
History
Examples
Properties
Notation
Ricci calculus
Einstein summation convention
Penrose graphical notation
Abstract index notation
Component-free notation
Operations
Tensor product
Contraction
Raising or lowering an index
Applications
Continuum mechanics
Other examples from physics
Computer vision and optics
Machine learning
Generalizations
Tensor products of vector spaces
Tensors in infinite dimensions
Tensor densities
Geometric objects
Spinors
See also
Foundational
Applications
Explanatory notes
References
Specific
General
External links

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 (which are the simplest tensors), 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.