Editing Tensor field (section)

{{Short description|Assignment of a tensor continuously varying across a region of space}}
{{distinguish|text=the [[Tensor product of fields]]}}
{{Use American English|date=March 2019}}

In [[mathematics]] and [[physics]],  a '''tensor field''' is a [[function (mathematics)|function]] assigning a [[tensor]] to each point of a [[region (mathematics)|region]] of a [[mathematical space]] (typically a [[Euclidean space]] or [[manifold]]) or of the [[physical space]]. Tensor fields are used in [[differential geometry]], [[algebraic geometry]], [[general relativity]], in the analysis of [[stress (physics)|stress]] and [[strain tensor|strain]] in material object, and in numerous applications in the [[physical sciences]].  As a tensor is a generalization of a [[scalar (physics)|scalar]] (a pure number representing a value, for example speed) and a [[vector (physics)|vector]] (a magnitude and a direction, like velocity), a tensor field is a generalization of a ''[[scalar field]]'' and a ''[[vector field]]'' that assigns, respectively, a scalar or vector to each point of space. If a tensor {{mvar|A}} is defined on a vector fields set {{mvar|X(M)}} over a module {{mvar|M}}, we call {{mvar|A}} a tensor field on {{mvar|M}}.<ref>O'Neill, Barrett. ''Semi-Riemannian Geometry With Applications to Relativity''</ref>
A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the ''[[Riemann curvature tensor]]'' refers a tensor ''field'', as it associates a tensor to each point of a [[Riemannian manifold]], a [[topological space]].

[[File:Tensor field.png|thumb|center|600px|Compared to a scalar field which has 1 value at a given point, and a vector field which has 2 (direction and magnitude), a tensor field has more than 2 values at each point, here represented by an ellipse at each point with semi-major axis length, semi-minor axis length, and direction]]
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