TensorsEdit

Tensors provide a unifying language for describing relations that are independent of the particular coordinates or basis chosen. At its core, a scalar is a 0th-order tensor, a vector is a 1st-order tensor, and higher-order tensors encode more complex multilinear relationships. While tensors can be represented by components in a chosen basis, the essential content is coordinate-free and can be expressed in terms of linear maps and tensor products. This perspective makes tensors central to a wide range of disciplines, from pure mathematics to physics and engineering. See for instance scalar and vector for foundational ideas, and multilinear algebra for the broader framework in which tensors live.

In physics and geometry, the tensor formalism ensures that laws and quantities are described in a way that does not depend on the observer’s vantage point. The same mathematical object can be described using different coordinate systems, with the transformation rules guaranteeing that physical predictions stay invariant. This invariance under coordinate changes is a powerful organizing principle that underpins modern theories, from general relativity to electromagnetism and beyond. See also manifold for the setting where tensor fields live, and covariant and contravariant indices for how components transform.

Tensors also appear naturally in computational contexts, where they are treated as multi-dimensional arrays that organize data for algorithms in machine learning and scientific computing. Although the notation and conventions can vary, the underlying ideas—multilinear maps, tensor products, and contractions—provide a robust toolkit for modeling and computation.

Formalism

A tensor of type (r, s) on a finite-dimensional vector space V over a field (often the real or complex numbers) is a multilinear map that takes r vectors and s covectors (dual vectors) and returns a single scalar. Equivalently, tensors can be viewed as elements of the r+s fold tensor product of V and its dual space V*: T ∈ V⊗⋯⊗V⊗V*⊗⋯⊗V*. The order r+s is sometimes called the rank or the total order of the tensor, while r and s separately indicate the numbers of contravariant and covariant indices, respectively. See vector for the basic objects, dual space for covectors, and tensor product for the construction that combines them.

In a chosen basis, a tensor is represented by components with a transformation law that encodes how those components change under a change of coordinates. If the coordinates are transformed by a matrix A, the components T' transform via a product involving A and its inverse (or transpose, depending on the index placement). This manifests the principle that the same geometric object can be described in many ways, but its invariant content remains the same. See also Einstein summation for a compact way of writing these contracts, and basis for how basis choices relate to components.

Rank-2 tensors are especially common: they can be covariant (two lower indices), contravariant (two upper indices), or mixed. A covariant 2-tensor acts on two vectors to produce a scalar, while a contravariant 2-tensor acts on two covectors. A mixed type tensor mixes these perspectives. A familiar example is the metric tensor, which provides distances and angles in a space and serves to raise or lower indices.

Contraction is a key operation: it sums over a pair of one contravariant and one covariant index, producing a tensor of lower order. When performed on a matrix (a rank-2 tensor in a simple setting), contraction reduces to familiar traces. Symmetry properties are also important: a tensor can be symmetric under index permutations or antisymmetric (skew-symmetric), leading to distinct classes with special geometric or physical meaning. See also contraction (tensor) and symmetry (mathematics).

Tensors on manifolds, i.e., tensor fields, assign a tensor to every point of a space in a smooth way. This generalizes the linear algebraic tensors described above to a setting where the basis itself may vary from point to point, which is essential for describing curved spaces in differential geometry and general relativity. See tensor field and manifold for the geometric framework.

Types, operations, and properties

Covariant, contravariant, and mixed tensors: index placement determines how components transform and how the tensor can act on vectors or covectors. See covariant and contravariant for details.
Tensor product: a fundamental operation that combines tensors to form higher-order tensors. It underpins constructions in both pure math and applied disciplines. See tensor product.
Contraction and trace: summing over an index pair yields a tensor of reduced order; in matrix language, this corresponds to the trace operation in many cases. See contraction (tensor).
Symmetry and antisymmetry: tensors can be symmetric or antisymmetric with respect to index permutation, which leads to special objects such as differential forms (antisymmetric tensors) and their role in integration on manifolds. See differential form and antisymmetric tensor.
Basis-free and component aspects: a tensor is best understood as an invariant multilinear object, but components in a basis allow concrete calculation. See basis and component representation.

Examples and notable tensors

Metric tensor: a non-degenerate symmetric covariant 2-tensor that encodes distances and angles in a space, enabling the definition of lengths and inner products. See metric tensor.
Riemann curvature tensor: a fourth-order tensor that encodes how a space is curved, central to general relativity and differential geometry. See Riemann curvature tensor.
Stress-energy (energy-momentum) tensor: a symmetric rank-2 tensor in physics that describes the distribution and flux of energy and momentum in spacetime. See stress-energy tensor.
Electromagnetic field tensor: an antisymmetric rank-2 tensor that unifies electric and magnetic fields in relativistic formulations. See electromagnetic tensor.
Strain and rotation tensors: objects in continuum mechanics that describe deformation and orientation changes in materials. See strain tensor and rotation tensor.
Representation in coordinates: in many practical calculations, a tensor is represented by a matrix of components relative to a basis, with the abstract tensorial relations guaranteed by the transformation laws. See vector and matrix.

Applications and context

In physics, tensors organize the laws of nature in a coordinate-independent way, from the geometry of spacetime in general relativity to the description of fields in gauge theory and the continuum mechanics of solids and fluids. See also stress-energy tensor and Riemann curvature tensor.
In geometry and analysis, tensors provide a language for curvature, connections, and differential operators, enabling precise statements about smooth manifolds and their properties. See tensor field and differential geometry.
In computation and data science, tensors appear as multi-dimensional arrays used to store heterogeneous data. Algorithms for tensor decomposition, factorization, and learning leverage the same core ideas that underpin the algebraic aspects of tensors. See tensor decomposition and machine learning.

History and development

The modern tensor calculus owes much to the work of mathematicians who extended linear algebra to multilinear contexts and to the developers of differential geometry in the 19th and early 20th centuries. Notable figures include Gregorio Ricci-Curbastro and Tullio Levi-Civita, whose formulation of tensor calculus on manifolds provided the backbone of general relativity. Earlier, the concept of multilinear forms and transformations appeared in the work of mathematicians such as William Kingdon Clifford and Oliver Heaviside in efforts to generalize vector operations. The term “tensor” and its formal use were popularized in the late 19th century by contributors such as Woldemar Voigt and later refined in the context of differential geometry and physics. See also Einstein field equations for a landmark physical application.