Tensors may be considered as generalizations of vectors and matrices. A type (p,q) tensor T on a real vector space V is a multilinear map

where V* denotes the dual space. The order of the tensor is p + q, and the tensor may also be considered as a (p + q)-dimensional array once bases are chosen. If p = 0, the tensor is called covariant, and if q = 0, the tensor is called contravariant. If V = ℝⁿ, then the tensor is called Cartesian.

Let e₁,…, e_n be a basis of V and f₁,…,f_n be the dual basis. When p = 0, q = 1, T is a functional, and when p = 1, q = 0, T is a vector v = (Tf₁,…,Tf_n). When p = q = 1, then T represents a linear map T:V → V as follows: T is represented by the matrix (a_ij), where

More properly, considered as a tensor, these entries should be written $a_j^i$ to denote that one of these indices is covariant, one contravariant. See also Einstein’s notation.

Tensors appear widely throughout mathematics and physics; grad is a (0,1) tensor, an inner product is a (0,2) tensor, as is the first fundamental form, the vector product is a (1,2) tensor, stress and strain are tensors, and differential forms are tensors. The first fundamental form on a surface instead might be considered as assigning an inner product, a tensor, to each tangent space; this is an example of a tensor field, important in physics and general relativity particularly.