Editing Tensor (section)

{{short description|Algebraic object with geometric applications}}
{{other uses}}
{{about-distinguish|tensors on a single vector space|Vector field|Tensor field}}
[[File:Components stress tensor.svg|right|thumb|300px|The second-order [[Cauchy stress tensor]] <math>\mathbf{T}</math> describes the stress experienced by a material at a given point. For any unit vector <math>\mathbf{v}</math>, the product <math>\mathbf{T} \cdot \mathbf{v}</math> is a vector, denoted <math>\mathbf{T}(\mathbf{v})</math>, that quantifies the force per area along the plane perpendicular to <math>\mathbf{v}</math>. This image shows, for cube faces perpendicular to <math>\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3</math>, the corresponding stress vectors <math>\mathbf{T}(\mathbf{e}_1), \mathbf{T}(\mathbf{e}_2), \mathbf{T}(\mathbf{e}_3)</math> 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 [[mathematical object|algebraic object]] that describes a [[Multilinear map|multilinear]] relationship between sets of [[algebraic structure|algebraic objects]] associated with a [[vector space]]. Tensors may map between different objects such as [[Vector (mathematics and physics)|vectors]], [[Scalar (mathematics)|scalars]], and even other tensors. There are many types of tensors, including [[Scalar (mathematics)|scalars]] and [[Vector (mathematics and physics)|vectors]] (which are the simplest tensors), [[dual vector]]s, [[multilinear map]]s between vector spaces, and even some operations such as the [[dot product]]. Tensors are defined [[Tensor (intrinsic definition)|independent]] of any [[Basis (linear algebra)|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 (mathematics)|matrix]].

Tensors have become important in [[physics]] because they provide a concise mathematical framework for formulating and solving physics problems in areas such as [[mechanics]] ([[Stress (mechanics)|stress]], [[elasticity (physics)|elasticity]], [[quantum mechanics]], [[fluid mechanics]], [[moment of inertia]], ...), [[Classical electromagnetism|electrodynamics]] ([[electromagnetic tensor]], [[Maxwell stress tensor|Maxwell tensor]], [[permittivity]], [[magnetic susceptibility]], ...), and [[general relativity]] ([[stress–energy tensor]], [[Riemann curvature tensor|curvature tensor]], ...). In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a [[tensor field]]. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".

[[Tullio Levi-Civita]] and [[Gregorio Ricci-Curbastro]] popularised tensors in 1900 – continuing the earlier work of [[Bernhard Riemann]], [[Elwin Bruno Christoffel]], and others – as part of the ''[[absolute differential calculus]]''. The concept enabled an alternative formulation of the intrinsic [[differential geometry]] of a [[manifold]] in the form of the [[Riemann curvature tensor]].<ref name="Kline">
{{cite book |first=Morris |last=Kline|title=Mathematical Thought From Ancient to Modern Times |volume=3
|url= {{google books |plainurl=y |id=-OsRDAAAQBAJ}} |date=1990
|publisher=Oxford University Press |isbn= 978-0-19-506137-6}}
</ref>