tensor
Sign in to saveright|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
AI overview
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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Article
38 sectionsContents
- 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
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 input and gives one vector as output, it is a second-order tensor.
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.