Glossary of Tensor Theory - Classical Notation

Classical Notation

Ricci calculus

The earliest foundation of tensor theory - tensor index notation.

Tensor order

A tensor written in component form is an indexed array. The order of a tensor is the number of indices required. (The rank of tensor used to mean the order, but now it means something different)

Rank

The rank of the tensor is the minimal number of rank-one tensor that you need to sum up to obtain this higher-rank tensor. Rank-one tensors are given the generalization of outer product to m-vectors where m is the order of the tensor.

Dyadic tensor

A dyadic tensor has order two, and may be represented as a square matrix. The conventions, and, do have different meanings (the position of the index determines its valence (variance), 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, 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.

Einstein notation

This notation is based on the understanding that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index. For example if is a matrix, then under this convention is its trace. The Einstein convention is generally used in physics and engineering texts, to the extent that if summation is not to be applied it is normal to note that explicitly.

Kronecker delta

Levi-Civita symbol

Covariant tensor, Contravariant tensor

The classical interpretation is by components. For example in the differential form the components are a covariant vector. That means all indices are lower; contravariant means all indices are upper.

Mixed tensor

This refers to any tensor with lower and upper indices.

Cartesian tensor

Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight.

Contraction of a tensor

Raising and lowering indices

Symmetric tensor

Antisymmetric tensor

Multiple cross products