7.22.4 Tensor Component Mixed Index Notation

Tensor Component Mixed Index Notation is the specific convention within Tensor Component Notation used to write the components of a mixed tensor, one carrying both contravariant and covariant indices together, arranging the superscripts and subscripts attached to a single base letter so that the relative ordering of all the indices, both across and within variance types, is preserved and unambiguous.

The Need for a Careful Ordering Convention

Ordering Matters Even Across Variance Types

For a tensor carrying several indices of mixed variance type, it is not sufficient merely to separate the contravariant indices as superscripts from the covariant indices as subscripts, since the relative order in which the original index positions occurred must also be preserved, particularly for tensors that do not exhibit full symmetry or antisymmetry. Mixed Index Notation addresses this by maintaining a consistent left-to-right reading order across both the superscript and subscript positions simultaneously.

The Staggered Placement Convention

To preserve this ordering unambiguously, Mixed Index Notation places each index at a horizontal position corresponding to its original order among all the tensor's indices, using vertical staggering, or an explicit placeholder dot, to indicate a variance-type slot with no index present at that particular horizontal position. A rank-three tensor with a contravariant index in the first position, a covariant index in the second position, and a contravariant index in the third position might be written as:

with the placeholder dot in each tier marking the horizontal position occupied by an index of the opposite variance type.

Illustration

Each column represents one original index position of the tensor, with the index letter appearing in the upper row if that position is contravariant and in the lower row if it is covariant, while a placeholder dot occupies the unused row at each position.

Why the Order Must Be Preserved

Order Reflects the Tensor's Definition

The order of a tensor's indices is fixed by how the tensor is originally defined, typically as a multilinear map accepting its input vectors and covectors in a specific sequence. Mixed Index Notation preserves this sequence explicitly so that a reader can determine, from the notation alone, which input slot of the original multilinear map corresponds to each index appearing in the component symbol.

Consequences of Losing the Order

If the original order among the indices were not preserved, and the contravariant indices were simply grouped together separately from the covariant indices without regard to their original positions, it would become impossible to tell, from the notation alone, which contravariant input slot pairs with which covariant input slot in the tensor's defining multilinear map, particularly whenever the tensor does not exhibit a symmetry pattern that would otherwise make such a distinction immaterial.

Interaction With the Summation Convention

Applying Summation Within Mixed Index Notation

The summation convention described within the broader Tensor Component Notation applies to Mixed Index Notation exactly as it does elsewhere: an index letter appearing once as a superscript and once as a subscript, at any horizontal position, is understood to be summed over its full Tensor Component Index Range, regardless of whether other placeholder dots appear elsewhere in the same expression.

Relationship to Other Tensor Concepts

Tensor Component Mixed Index Notation extends the basic Tensor Component Indexed Symbol to handle the additional complexity introduced by tensors combining contravariant and covariant indices in a specific, meaningful order, ensuring that expressions written within the broader Tensor Component Notation remain unambiguous even for tensors that mix variance types without any accompanying symmetry pattern to simplify their description.