tensors

A tensor is a mathematical object that generalizes scalars, vectors, and linear maps. In linear algebra, a tensor of order k on a vector space V over a field F is a multilinear map that takes k vectors (and/or covectors) to F, or equivalently an element of a tensor product of copies of V and its duals. More specifically, a tensor of type (r, s) is a multilinear map that takes r covectors and s vectors as input, producing a scalar; equivalently, an element of V*^{⊗ r} ⊗ V^{⊗ s}. The rank or order of a tensor is the total number of indices in a component representation. Scalars have rank 0, vectors rank 1, and matrices commonly represent rank-2 tensors. A linear operator can be viewed as a rank-(1,1) tensor.

Under a change of basis, tensor components transform according to a specific transformation law, which preserves the tensor’s intrinsic meaning across coordinate systems. This invariance makes tensors fundamental in physics and differential geometry. Tensor fields assign a tensor to each point of a space or manifold, varying smoothly in the case of a smooth field. Common examples include the metric tensor, which defines inner products, and curvature tensors, which measure geometric deviation.

Operations on tensors include addition, tensor product, contraction (which reduces rank by summing over paired indices), and index manipulation. Special tensors may be symmetric, where T_{ij} = T_{ji}, or antisymmetric, where T_{ij} = -T_{ji}.

Applications span physical theories such as classical mechanics and general relativity, where tensors encode physical quantities and laws, to differential geometry. In computing and data analysis, the term tensor is also used for multi-dimensional arrays that represent these objects in a coordinate form.