tenseurs

Tensors, called tenseurs in French, are mathematical objects that generalize scalars, vectors, and matrices. They provide a coordinate-free description of linear relationships between vectors, covectors, and higher-order entities. Formally, a tensor of type (k, l) is a multilinear map that takes k vectors from a vector space V and l covectors from its dual V to a scalar. In a chosen basis, its components are written as T^{i1...ik}_{j1...jl}.

Scalars are tensors of type (0, 0); vectors are of type (1, 0) or (0, 1) depending on covariant vs contravariant conventions. Second-order tensors include matrices, which can be viewed as either (1, 1), (0, 2), or (2, 0) tensors depending on index placement. For an n-dimensional space, the space of (k, l) tensors has dimension n^{k+l}. The components transform under a change of basis according to well-defined rules, so the tensor’s geometric meaning is basis-independent; the Einstein summation convention is commonly used to simplify expressions.

Key operations include the tensor product, which builds higher-order tensors from existing ones, and contraction, which sums over a pair of matched indices to produce a tensor of lower order. Indices can be raised or lowered with a metric or bilinear form, and symmetry properties may be imposed to yield symmetric, antisymmetric, or mixed-symmetry tensors.

Common examples appear in physics and geometry. The metric tensor is a symmetric (0, 2) tensor that defines distances and angles; the Cauchy stress tensor is a symmetric (0, 2) tensor describing internal forces; the inertia tensor encodes rotational properties. Curvature tensors, such as the Riemann tensor, are higher-order tensors that describe geometric structure. Tensors provide a unified language for linear relations across coordinates and physical quantities.