9.12.4 Tensor Basis Expansion Index Form

Tensor Basis Expansion Index Form is the abbreviated way of writing a tensor's basis expansion in which only the indexed component array is displayed, while the basis vectors and dual basis covectors that the components multiply are left implicit. It is the notation used once a basis has been fixed and agreed upon, so that the tensor can be manipulated purely through its indices.

Relation to the Full Expansion

Suppressing the Basis Symbols

The full basis expansion writes a tensor as a sum of components multiplying basis vectors and dual basis covectors. Index form removes the basis symbols from this expression and retains only the indexed component, since the basis is understood from context and does not need to be repeated in every formula.

The tensor T and its indexed component are treated as interchangeable once the basis is fixed, with the understanding that the component array alone carries the same information as the full expansion.

Recovering the Full Form

The index form can always be expanded back into the full basis expansion by reinserting the appropriate basis vectors and dual basis covectors, one for each free index, following the same index pattern already present in the indexed component.

Rules Governing Index Form

Free Indices and Bound Indices

An index that appears only once in a term is a free index, and it indicates a slot of the tensor that remains open. An index that appears exactly twice in a single term, once raised and once lowered, is a bound index, and under the summation convention it indicates that a sum is implied over that index's full range.

In this expression, i and j are free indices identifying the resulting component, while k is a bound index summed over, so the expression as a whole represents a matrix-like product carried out purely in index form.

Placement of Indices

Upper indices must always correspond to contravariant slots and lower indices to covariant slots. This placement is preserved through every operation performed in index form, including addition, multiplication, and contraction, so that the type of the resulting tensor can always be read directly from the pattern of upper and lower indices left over.

Consistency Across a Formula

Every term appearing in a sum written in index form must carry exactly the same set of free indices, in the same raised or lowered position. This requirement, sometimes called index balance, ensures that the terms being added represent components of the same type of tensor and can be legitimately combined.

Advantages of Index Form

Compactness

Index form eliminates the need to write basis vectors and dual basis covectors explicitly, which shortens expressions considerably once several tensors and several operations are combined in a single formula.

Direct Computability

Because index form reduces every operation to manipulations of arrays of numbers indexed by integers, it is the form most directly suited to numerical computation, where the underlying basis vectors are never represented explicitly in memory, only the component values are stored and processed.

Transparency of Symmetry

Properties such as symmetry, antisymmetry, and specific contraction patterns are often easier to recognize and verify directly on the indices than on the full expansion, since the index pattern exposes exactly how each slot of the tensor relates to the others.