7.11.1 Tensor Higher Order Multi Index

Tensor Higher Order Multi Index extends multi-index notation to higher-order tensors, enabling structured indexing of multidimensional arrays.

Tensor Higher Order Multi Index is the generalized indexing scheme used for tensors of rank greater than two, in which a component is located not by a single index or a pair of indices, but by an ordered string of several indices, each corresponding to one of the tensor's argument slots and each carrying its own variance type.

Generalizing Beyond Rank Two

From Double Index to Multi Index

Where a rank-two tensor's component is written with two indices, such as (T_{ij}), a rank-(k) tensor requires (k) indices to fully specify a single entry, one for every slot the tensor accepts as an argument.

T_{i_{1} i_{2} \dots i_{k}} = T (e_{i_{1}}, e_{i_{2}}, \dots, e_{i_{k}})

Total Number of Components

In an (n)-dimensional space, a rank-(k) tensor's multi-indexed component array contains (n^k) independent entries, since each of the (k) indices independently ranges over all (n) basis directions.

number of components = n^{k}

Mixed Variance Within a Multi Index

Combining Upper and Lower Indices

A general tensor may possess several contravariant indices and several covariant indices simultaneously, distinguished by writing the contravariant ones as superscripts and the covariant ones as subscripts within the same multi-index expression.

T_{j_{1} j_{2}}^{i^{1} i^{2}}

Rank as the Sum of Index Counts

The total rank of the tensor equals the sum of the number of contravariant indices and the number of covariant indices, and a tensor is often described by the pair of these two counts, such as a "type (2,1) tensor" meaning two upper indices and one lower index.

Transformation of a Multi Index

General Multi-Index Transformation Law

Under a change of basis governed by a transition matrix (A), each contravariant index transforms using (A^{-1}) and each covariant index transforms using (A), with the full transformation applying one such factor for every index present.

T_{j^{'}}^{i^{'}} = {(A^{- 1})}_{k}^{i^{'}} A_{j^{'}}^{l} T_{l}^{k}

Consistency of Index-by-Index Behavior

Every index within the multi index obeys its own individual transformation rule independently of the others; a mixed tensor's transformation law is simply the product of the individual factors for each index, one contributed per slot.

Reading and Slicing a Multi Index

Fixing Some Indices to Reveal a Substructure

Just as a double index can be sliced into rows and columns by fixing one index at a time, a multi index can be sliced by fixing all but one of its indices, revealing a one-dimensional list of components varying along the single remaining axis.

slice along i_{3} = (T_{i_{1} i_{2} 1}, \dots, T_{i_{1} i_{2} n})

Symmetry Properties Across Multiple Indices

A multi index may be symmetric or antisymmetric under the interchange of any pair of indices of the same variance type, extending the two-index notions of symmetry and antisymmetry to more elaborate patterns, such as total symmetry across all indices or antisymmetry restricted to a particular subset.

Diagrammatic Illustration

A rank-three tensor's multi index can be visualized as a cube of entries, in contrast to the flat grid used for rank-two tensors.

Significance in General Tensor Algebra

Foundation for Contraction and Products

Multi-index notation is what makes tensor contraction and tensor products tractable for arbitrary rank, since contraction is defined as summing over one repeated upper and lower index within a multi index, and the tensor product simply concatenates the multi indices of two tensors into a longer combined multi index.

Necessity for Physical and Geometric Applications

Fields such as elasticity theory, electromagnetism, and general relativity routinely require tensors of rank three or four, making multi-index notation indispensable for expressing quantities such as the Riemann curvature tensor or the elasticity tensor in a compact and unambiguous form.