11.18.2 Tensor Variance Type Contravariant Count

Tensor Variance Type Contravariant Count measures how tensor components transform under coordinate changes, key in physics and math.

Tensor Variance Type Contravariant Count is the integer component of a tensor's type notation that records how many upper, contravariant indices the tensor carries, conventionally written as the first entry $p$ in the ordered pair $(p, q)$ , and used to determine how many factors of the inverse 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 $p$ is the contravariant count, tallying exactly how many upper indices appear on the tensor symbol, while the second entry $q$ counts lower, covariant indices separately.

Reading the Count from Notation

For a tensor written as $T_{l}^{i j k}$ , the contravariant count is determined simply by counting the superscript positions, giving $p = 3$ for the three upper indices $i$ , $j$ , and $k$ .

Role in the Transformation Law

One Inverse-Matrix Factor per Contravariant Index

The contravariant count directly determines how many times the inverse basis-change matrix $A^{- 1}$ appears in the tensor's transformation law. A tensor with contravariant count $p$ and contravariant indices $i^{1}$ through $i^{p}$ transforms as:

{\tilde{T}}^{i^{1} \dots i^{p}} = \sum_{j^{1}, \dots, j^{p}} {(A^{- 1})}_{j^{1}}^{i^{1}} \dots {(A^{- 1})}_{j^{p}}^{i^{p}} T^{j^{1} \dots j^{p}}

Special Values of the Count

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

Visual Overview

Counting Upper Indices

Behavior of the Count Under Tensor Operations

Addition Under Tensor Products

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

p (S \otimes T) = p (S) + p (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 contravariant count by exactly one, alongside a matching decrease of one in the covariant count:

p (contracted) = p (original) - 1

Contribution to Overall Rank

Summing with the Covariant Count

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

Distinguishing Tensors of Equal Rank

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

Summary of Key Traits

Defining Characteristics

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