11.18.1 Tensor Variance Type Covariant Count

Tensor Variance Type Covariant Count measures how tensor components change under coordinate transformations, preserving geometric relationships in multi-dimensional spaces.

Tensor Variance Type Covariant Count is the integer component of a tensor's type notation that records how many lower, covariant indices the tensor carries, conventionally written as the second entry $q$ in the ordered pair $(p, q)$ , and used to determine how many factors of the direct basis-change matrix must appear in the tensor's transformation law.

Foundational Setting

Locating the Count Within the Type Pair

When a tensor's variance type is written as $(p, q)$ , the first entry $p$ counts contravariant, upper indices, while the second entry $q$ is the covariant count, tallying exactly how many lower indices appear on the tensor symbol.

Reading the Count from Notation

For a tensor written as $T_{j k l}^{i}$ , the covariant count is determined simply by counting the subscript positions, giving $q = 3$ for the three lower indices $j$ , $k$ , and $l$ .

Role in the Transformation Law

One Direct-Matrix Factor per Covariant Index

The covariant count directly determines how many times the basis-change matrix $A$ appears, applied directly rather than inverted, in the tensor's transformation law. A tensor with covariant count $q$ and covariant indices $i_{1}$ through $i_{q}$ transforms as:

{\tilde{T}}_{i_{1} \dots i_{q}} = \sum_{j_{1}, \dots, j_{q}} A_{i_{1}}^{j_{1}} \dots A_{i_{q}}^{j_{q}} T_{j_{1} \dots j_{q}}

Special Values of the Count

When $q = 0$ , the tensor has no covariant indices at all and is purely contravariant or, if $p$ is also zero, an invariant scalar. When $q = 1$ , the tensor carries exactly one lower index, the simplest nontrivial covariant case, exemplified by the components of a gradient.

Visual Overview

Counting Lower Indices

Behavior of the Count Under Tensor Operations

Addition Under Tensor Products

When two tensors are combined by tensor product, their covariant counts add together, since the lower indices of both factors are simply placed alongside one another in the resulting object:

q (S \otimes T) = q (S) + q (T)

Reduction Under Contraction

Contracting one upper index against one lower index of the same tensor, or across two tensors joined in a product, decreases the covariant count by exactly one, alongside a matching decrease of one in the contravariant count:

q (contracted) = q (original) - 1

Contribution to Overall Rank

Summing with the Contravariant Count

The covariant count contributes additively to the tensor's total rank alongside the contravariant count $p$ , giving rank equal to $p + q$ . Two tensors sharing the same rank can nonetheless have entirely different covariant counts, reflecting different splits between upper and lower indices.

Distinguishing Tensors of Equal Rank

A rank-two tensor, for instance, may have covariant count $q = 2$ and contravariant count $p = 0$ , as with a metric tensor, or covariant count $q = 1$ and contravariant count $p = 1$ , as with a mixed linear operator, despite both having the same overall rank.

Summary of Key Traits

Defining Characteristics

The covariant count is the number of lower indices a tensor carries, recorded as the second entry in its type pair.
It determines exactly how many direct basis-change matrix factors appear in the tensor's transformation law.
Tensor products add covariant counts; contraction of an upper-lower pair subtracts one from the count.
Together with the contravariant count, it determines the tensor's total rank while still distinguishing tensors of equal rank but different index splits.