6.13.3 Tensor Zero One Component Index

The Tensor Zero One Component Index identifies a tensor component with a single zero index, key in tensor algebra and coordinate transformations.

Tensor Zero One Component Index is the single lower index attached to the components of a type zero-one tensor, ranging over every basis direction of the underlying vector space, whose particular numerical value at any given position selects out one specific component from among the full array describing the one-form in a chosen basis. This index is the concrete, numerical counterpart to the tensor's single abstract covariant slot: where the slot is a structural position in the multilinear definition, the component index is the label attached to that position once a basis has actually been chosen and the one-form has been written out as a list of numbers.

The Range and Role of the Index

Ranging Over the Basis Directions

For a vector space of dimension n, the component index of a type zero-one tensor takes n distinct values, one for each basis vector spanning the dual space, and the component associated with each value of the index is the scalar the one-form assigns to the corresponding basis vector of the original space. Enumerating the index from its first value to its last recovers every one of the n independent numbers needed to specify the one-form completely.

ω = \sum_{a = 1}^{n} ω_{a} e^{a}

Free Versus Summed Occurrence

When the component index appears once in an expression, unaccompanied by a matching upper index of the same letter, it is a free index, meaning the expression represents an entire family of numbers, one for each value the index can take, rather than a single number. When the same letter appears once as this lower index and once as an upper index elsewhere in the same term, the summation convention applies, and the expression collapses to a single sum over every value of that shared index.

Notational Conventions for the Index

Abstract Letters Versus Numerical Values

The component index may be written using an abstract letter, standing generically for any one of its possible values, or it may be written using an explicit number, singling out one particular component for direct inspection. The abstract form is used when stating a rule or relation that holds for every value of the index simultaneously, while the explicit numerical form is used when a specific coordinate of the one-form is being read off or computed.

ω_{1}, ω_{2}, \dots, ω_{n}

The Index Position, Not the Letter, Carries Meaning

Renaming the component index from one letter to another, provided the renaming is applied consistently everywhere that index occurs within a single term, leaves the meaning of the expression completely unchanged, since it is the lower position of the index, not its particular letter, that signals the covariant character of the slot it labels. This freedom to relabel is routinely used to avoid clashes when two separate type zero-one tensors are combined within the same expression.

Basis Dependence Carried by the Index

Component Values Tied to the Chosen Basis

The actual numerical value obtained for any particular setting of the component index depends entirely on which basis was used to expand the one-form, since the same abstract one-form generally yields different numbers at the same index value once a different basis is adopted. The index itself, as a label ranging from one to the dimension of the space, remains meaningful across any basis, but the specific number it retrieves at each value changes together with the basis.

Transformation of the Indexed Components

Under a change of basis, every component labeled by the index is recomputed according to the inverse-Jacobian transformation law associated with the tensor's single covariant slot, with the new value at each index position built from a sum over the old components weighted by the appropriate entries of the inverse Jacobian matrix relating the two bases.

ω_{a}' = \frac{\partial x^{b}}{\partial x^{'} (a)} ω_{b}

The Index in Relation to Other Operations

Serving as the Target of Contraction

The component index of a type zero-one tensor is precisely the index that must be matched against an upper index of a vector or another contravariant tensor in order to perform a contraction, and it is the coincidence of this particular lower index with an upper index bearing the same letter that triggers the summation convention producing a scalar result.

Serving as the Anchor for Raising and Lowering

When the inverse metric is used to raise this index, converting the type zero-one tensor into a type one-zero tensor, the resulting upper index inherits the same range of values as the original lower index, differing only in its variance classification and in the transformation law it subsequently obeys, not in the set of values it ranges over.