For expositions of tensor theory from different points of view see:

- Tensor
- Classical treatment of tensors
- Tensor (intrinsic definition)
- Intermediate treatment of tensors
- Application of tensor theory in engineering science

Table of contents |

2 Algebraic notation 3 Applications 4 Tensor field theory 5 Abstract algebra |

A tensor written in component form is an indexed array. The *rank* of a tensor is the number of indices required.

**Dyadic tensor**

A *dyadic* tensor has rank two, and may be represented as a square matrix. The conventions *a*_{ij}, *a*_{i}^{j}, and *a*^{ij}, do have different meanings, in that the first may represent a quadratic form, the second a linear transformation, and the distinction is important in contexts that require tensors that aren't *orthogonal* (see below). A *dyad* is a tensor such as *a*_{i}*b*^{j}, product component-by-component of rank one tensors. In this case it represents a linear transformation, of rank one in the sense of linear algebra - a clashing terminology that can cause confusion

**Covariant tensor**, **Contravariant tensor**

The classical interpretation is by components. For example in the differential form *a*_{i}*dx*^{j} the *components**a*_{i} are a covariant vector. That means all indices are lower; contravariant means all indices are upper.

This refers to any tensor with lower and upper indices.

**Orthogonal tensor**

In the presence of a tensor δ_{i}^{j}, there is no need to maintain the distinction of upper and lower indices. That is the case given a distinguished set of orthogonal co-ordinates. Orthogonal tensors are also called *cartesian tensors*

If v and w are vectors in vector spaces V and W respectively, then is a tensor in . That is, the operation is a binary operation, but it takes values in a fresh space (it is in a strong sense *external*). The operation is bilinear; but no other conditions are applied to it.

**Pure tensor**

A pure tensor of is one that is of the form .

It could be written dyadically *a*_{i}*b*_{j}, or more accurately *a*_{i}*b*_{j} e_{i}f_{j}, where the e_{i} are a basis for V and the f_{j} a basis for W. Therefore, unless V and W have the same dimension, the array of components need not be square. Such *pure* tensors are not generic: if both V and W have dimension > 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure.

In the tensor algebra T(V) of a vector space V, the operation becomes a normal (internal) binary operation. This is at the cost of T(V) being of infinite dimension.

The wedge product is the anti-symmetic form of the operation. The quotient space of T(V) on which it becomes an internal operation is the *exterior algebra* of V; it is a graded algebra, with the graded piece of weight *k* being called the *k*-th **exterior power** of V.

**Symmetric power**

**Strain tensor**

**Tensor density**

**Tor functors**