tensornotation

Tensor notation, or tensornotation, is a system for representing tensors and multilinear maps in a way that reveals their algebraic structure and coordinate dependence. The most widely used framework is index notation with the Einstein summation convention. A tensor of type (k,l) has k contravariant indices (superscripts) and l covariant indices (subscripts); its components in a chosen basis are written T^{i1...ik}_{j1...jl}. When an index is repeated once up and once down, it is summed over, producing a contraction. Tensors act linearly on vectors and covectors, and their components transform under a change of basis according to a tensor transformation law, so that the whole object is basis-independent.

Second-order tensors include matrices: A^i_j represents a linear map, with a matrix interpretation when a basis is fixed. A covariant rank-(0,2) tensor like a metric g_{ij} defines a bilinear form; a mixed tensor (1,1) corresponds to a linear map with components A^i_j.

Raising and lowering indices is accomplished with a metric: contractions with g or its inverse g^{ij} convert covariant components to contravariant ones and vice versa. Tensor operations include contraction (summing over an index), tensor product, and symmetrization or antisymmetrization of indices.

Tensor notation also extends to tensor fields on manifolds, where each point carries a tensor, i.e., a smoothly varying multilinear object. In physics, tensornotation is essential in relativity and field theory; in mathematics, it provides a language for differential geometry and continuum mechanics. There are also index-free approaches that describe tensors as multilinear maps without reference to coordinates.