Tensor

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a tensor is an <a href="/facts/Mathematical_object/Juhyow9M">algebraic object</a> that describes a <a href="/facts/Multilinear_map/YosSfBKE">multilinear</a> relationship between sets of algebraic objects related to a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. Tensors may map between different objects such as <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vectors</a>, <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a>, and even other tensors. There are many types of tensors, including <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> and <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vectors</a> (which are the simplest tensors), <a href="/facts/Dual_vector/oGh7Asrz">dual vectors</a>, <a href="/facts/Multilinear_map/YosSfBKE">multilinear maps</a> between vector spaces, and even some operations such as the <a href="/facts/Dot_product/tNz8MLjT">dot product</a>. Tensors are defined <a href="/facts/Tensor_(intrinsic_definition)/2KuNm0MK">independent</a> of any <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>, 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 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>.
Tensors have become important in <a href="/facts/Physics/a7aH2y08">physics</a> because they provide a concise mathematical framework for formulating and solving physics problems in areas such as <a href="/facts/Mechanics/A3yHLQA5">mechanics</a> (<a href="/facts/Stress_(mechanics)/ErAXDbbd">stress</a>, <a href="/facts/Elasticity_(physics)/wcJoIFg1">elasticity</a>, <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a>, <a href="/facts/Moment_of_inertia/TpLLAVl7">moment of inertia</a>, ...), <a href="/facts/Classical_electromagnetism/vK0YI6dj">electrodynamics</a> (<a href="/facts/Electromagnetic_tensor/MWYuKLPy">electromagnetic tensor</a>, <a href="/facts/Maxwell_stress_tensor/mW0lI7h7">Maxwell tensor</a>, <a href="/facts/Permittivity/Sd3WtgT1">permittivity</a>, <a href="/facts/Magnetic_susceptibility/hWMbGTjP">magnetic susceptibility</a>, ...), and <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> (<a href="/facts/Stress%25E2%2580%2593energy_tensor/wWKrzjfI">stress–energy tensor</a>, <a href="/facts/Riemann_curvature_tensor/KyeHCWEh">curvature tensor</a>, ...). 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 <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a>. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
<a href="/facts/Tullio_Levi-Civita/LmxVG3GP">Tullio Levi-Civita</a> and <a href="/facts/Gregorio_Ricci-Curbastro/HjBVjmPB">Gregorio Ricci-Curbastro</a> popularised tensors in 1900 – continuing the earlier work of <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a>, <a href="/facts/Elwin_Bruno_Christoffel/o66hLsvE">Elwin Bruno Christoffel</a>, and others – as part of the <a href="/facts/Absolute_differential_calculus/We4yudeq">absolute differential calculus</a>. The concept enabled an alternative formulation of the intrinsic <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> of a <a href="/facts/Manifold/rXPERJtu">manifold</a> in the form of the <a href="/facts/Riemann_curvature_tensor/KyeHCWEh">Riemann curvature tensor</a>.

Tensor open-in-new

Tensor