TensorEdit

I can’t adopt a specific political viewpoint, but here is a neutral, encyclopedia-style article on tensors that covers definitions, history, mathematics, and applications.

A tensor is a mathematical object that generalizes scalars and vectors and extends to more complex linear relations among spaces. At its core, a tensor encodes how a quantity responds to inputs from multiple directions or coordinate directions, while respecting linearity in each input. In everyday language, tensors provide a compact way to describe multivariable relationships that are independent of the particular coordinate system used. In its most general form, a tensor can be interpreted as a multilinear map or as an object with components that transform in a prescribed way under changes of coordinates. See scalar and vector for simpler cases, and multilinear map for the broader algebraic background. The ideas behind tensors appear across linear algebra, differential geometry, and tensor algebra, with wide-ranging applications in science and engineering. The modern perspective often emphasizes tensor fields, which assign a tensor to every point of a space such as a manifold.

History

The formal calculus of tensors took shape in the 19th century as geometers sought invariant ways to describe curvature and other geometric properties independent of coordinates. The modern, coordinate-free language was developed in the work of Ricci-Curbastro and Levi-Civita, who introduced the calculus of tensor fields that can vary from point to point on a space. The framework gained monumental visibility through its role in the theory of general relativity and the description of spacetime, a development led by Albert Einstein and others. Over time, the same tensor concepts became central to many areas, including continuum mechanics, electromagnetism, and advanced computing. For broader mathematical context, readers may consult entries on manifold and tensor field.

Mathematics

Basic idea and types

A tensor of type (p, q) is a multilinear map that takes p vectors from a given vector space and q covectors (or dual vectors) from the dual space, and returns a scalar. This formalism unifies different kinds of objects:

  • Scalars are tensors of type (0, 0).
  • Vectors are tensors of type (1, 0).
  • Covectors (or linear functionals) are tensors of type (0, 1).
  • Matrices arise as representations of tensors of type (1, 1) in a chosen basis.

The components of a tensor depend on a choice of basis, and a tensor can be described in terms of its components relative to that basis. The relationship between component representations in different bases is governed by a transformation law that preserves the tensor’s multilinear character. This passage from an abstract, coordinate-free view to a component view is a central theme in multilinear algebra and tensor product theory.

Constructions and operations

Key constructions include:

  • The tensor product: combining a p-tensor and a q-tensor to form a (p+1, q+1) tensor, extending the way vectors and covectors can be combined. See tensor product.
  • Contraction: summing over paired indices to reduce the order of a tensor, producing a tensor of lower rank. This operation underlies many invariants in physics and geometry.
  • Symmetry and antisymmetry: tensors may have symmetric or alternating (antisymmetric) components, which play a central role in areas such as differential geometry and the theory of differential forms.
  • Tensor fields: a tensor field assigns a tensor to every point in a space, allowing the description of spatially varying quantities on spaces like manifolds. See tensor field.

Representation and bases

In a fixed basis, a tensor is represented by a multi-dimensional array of components. The same tensor can have very different component representations under different bases, linked by a well-defined set of linear transformations. The metric structure on a space (a way of measuring lengths and angles) often lets us raise and lower indices, converting between contravariant and covariant components; see metric for related ideas.

Common examples

  • The stress–strain relationship in materials is described by a stress tensor, a particular rank-two tensor that encodes internal forces per area. See stress tensor.
  • The electromagnetic field is represented compactly by the electromagnetic tensor (field tensor), which encodes electric and magnetic fields in a relativistically invariant way. See electromagnetic tensor.
  • In general relativity, the curvature of spacetime is described by the Riemann curvature tensor, a fourth-rank tensor that captures how spacetime is curved. See Riemann curvature tensor and general relativity.

Types, fields, and applications

Tensor fields in geometry and physics

Tensors provide the language for expressing laws that hold in any coordinate system. In differential geometry, tensor fields describe geometric and physical quantities that vary over space. In physics, tensors express conservation laws, field equations, and material properties in a way that remains valid under coordinate changes. Notable examples include the metric tensor, which encodes the geometry of space, and the stress-energy tensor, which encapsulates energy and momentum densities and fluxes.

Tensors in applied sciences and engineering

In engineering disciplines, tensor concepts underpin the analysis of stresses, strains, and material anisotropy. In computational contexts, tensors appear as data structures that generalize matrices to higher dimensions, enabling compact representation of multi-way data in fields such as computer vision and scientific computing. Modern software libraries often expose tensor abstractions for efficient manipulation and automatic differentiation; see TensorFlow and related tools for practical implementations in machine learning and engineering workflows.

Notable topics and connections

  • The distinction between covariant and contravariant components, and how transformation laws preserve tensorial identity across coordinate systems. See covariant tensor and contravariant.
  • The tensor algebra built from a vector space, including exterior algebra and symmetric algebra, which organize antisymmetric and symmetric tensors, respectively. See exterior algebra and symmetric algebra.
  • The role of tensors in data-intensive disciplines, where high-order data can be represented as tensors, and where decompositions (such as tensor factorization) help extract structure. See tensor decomposition.

See also