11.18 Tensor Variance Type Behavior

Tensor Variance Type Behavior is the overall classification of how a tensor's components respond to a change of basis or coordinate system, sorting tensors into a small number of distinct behavioral categories, namely invariant, covariant, contravariant, and mixed, according to which combination of transformation rules governs their indices.

Foundational Setting

The Need for a Classification

A vector space paired with its dual space supports several distinct kinds of multi-indexed objects, each reacting differently to a basis change. Variance type behavior is the framework that names and organizes these reactions so that any tensor can be assigned unambiguously to exactly one type based on the structure of its indices.

The Governing Principle

Every classification rests on the same underlying fact: a change of basis is described by a matrix $$A$$, and each index of a tensor transforms either with $$A$$ directly or with its inverse $$A^{-1}$$, and the pattern of these choices across all indices determines the variance type.

The Four Behavioral Categories

Invariant Behavior

A quantity with no free indices at all, called a scalar or invariant, does not change value under any basis transformation:

Contravariant Behavior

A tensor whose indices are entirely upper, such as $$v^i$$, transforms with the inverse basis-change matrix on every index, compensating for stretches or shrinks in the basis so that the object represented stays fixed.

Covariant Behavior

A tensor whose indices are entirely lower, such as $${\omega }_i$$, transforms with the direct basis-change matrix on every index, moving in step with the basis rather than against it.

Mixed Behavior

A tensor carrying both upper and lower indices simultaneously, such as $$T_j^i$$, exhibits mixed behavior, applying the inverse matrix to each upper index and the direct matrix to each lower index at once.

Visualizing the Four Categories

Diagram of Behavioral Types

Rank and Type Together

The Type Notation

The precise variance type of a tensor is often recorded as a pair of nonnegative integers $$(p,q)$$, where $$p$$ counts the upper indices and $$q$$ counts the lower indices. Invariant behavior corresponds to type $$(0,0)$$, pure contravariant behavior to type $$(p,0)$$, pure covariant behavior to type $$(0,q)$$, and mixed behavior to any type with both $$p$$ and $$q$$ positive.

Rank as the Sum

The rank of a tensor, meaning the total number of indices it carries, is the sum $$p+q$$, so tensors of the same rank can still differ in variance type depending on how that rank is split between upper and lower positions.

Behavior Under Tensor Operations

Products Combine Types Additively

Forming the tensor product of two tensors of types $$(p_1,q_1)$$ and $$(p_2,q_2)$$ yields a tensor of type $$(p_1+p_2,q_1+q_2)$$, since the indices of both factors are simply combined.

Contraction Reduces Type

Contracting one upper index against one lower index removes one instance of each from the variance type, reducing a tensor of type $$(p,q)$$ to type $$(p-1,q-1)$$, and repeated contraction down to type $$(0,0)$$ yields an invariant scalar.

Summary of Key Traits

Defining Characteristics

• Variance type behavior classifies tensors into invariant, contravariant, covariant, and mixed categories.
• The classification is fully determined by the arrangement of upper and lower indices.
• The type pair $$(p,q)$$ records the count of each index kind, with their sum giving the tensor's rank.
• Tensor products add variance types, while contraction reduces them, eventually producing invariants.