11.1.3 Tensor Mixed Variance Behavior Scope

Tensor Mixed Variance Behavior Scope is the delineation of which tensors and index configurations combine contravariant and covariant transformation rules within a single object, specifying exactly how each index slot behaves independently and where the boundary lies between a genuinely mixed tensor and an object that merely appears to mix the two behaviors.

Defining Mixed Variance

Independent Behavior per Index Slot

A mixed tensor is one that carries at least one contravariant index slot and at least one covariant index slot simultaneously. Each slot transforms strictly according to its own upper or lower placement, with no interaction between the transformation rules applied to different slots within the same object.

Rank and Type Notation

A mixed tensor's variance structure is described by a pair of numbers, one counting the contravariant slots and one counting the covariant slots, together called the type of the tensor, which fully specifies how many direct and how many inverse Jacobian factors must appear in its transformation formula.

Prototypical Mixed Tensor

The Linear Map Interpretation

A tensor of type consisting of one contravariant and one covariant index is naturally interpreted as a linear map from the vector space to itself, since one index can absorb an input vector and the other index can absorb an input covector, leaving the mixed component array as the coordinate representation of a linear transformation.

The Kronecker Delta as a Basis-Independent Mixed Tensor

The Kronecker delta, viewed as a mixed tensor of one contravariant and one covariant index, has the distinguishing property that its components remain numerically identical in every coordinate system, since the direct and inverse Jacobian factors it is contracted with under a change of basis cancel exactly through the reciprocity condition.

Boundary of the Mixed Scope

Objects With Only One Variance Type Are Excluded

A tensor consisting entirely of contravariant indices, or entirely of covariant indices, does not fall under mixed variance behavior scope regardless of how high its rank is, since the defining feature of a mixed tensor is the simultaneous presence of both index types, not merely a high number of indices.

Non-Tensorial Objects With Both Index Types Are Excluded

An indexed quantity that carries both upper and lower indices but fails to satisfy the homogeneous transformation law, such as a connection coefficient, does not qualify as a mixed tensor even though its index pattern resembles one, since true mixed variance scope requires the pure multiplicative Jacobian-factor transformation without any additive correction term.

Composition and Contraction Within Mixed Tensors

Internal Contraction Reduces Rank Without Changing Type Balance

Contracting one upper index against one lower index of the same mixed tensor produces a new tensor with both counts reduced by one, preserving the difference between the number of contravariant and covariant indices while lowering the total rank, a process called the trace of the mixed tensor.

Products of Mixed Tensors Combine Their Types

Forming a tensor product of two mixed tensors produces a new object whose contravariant count is the sum of the two original contravariant counts and whose covariant count is the sum of the two original covariant counts, extending mixed variance behavior scope predictably under this operation.

Practical Use of the Mixed Scope Concept

Identifying the Correct Number of Jacobian Factors

Before writing the transformation formula for a tensor, correctly counting its contravariant and covariant index slots determines exactly how many direct and how many inverse Jacobian factors the formula must contain, making the mixed variance scope classification a necessary first step in constructing any transformation law for a tensor of rank greater than one.

Recognizing Genuine Mixed Tensors in Applied Contexts

Distinguishing a genuine mixed tensor from a superficially similar non-tensorial object with the same index pattern is essential whenever a physical or geometric quantity is introduced with both upper and lower indices, since only true mixed tensors can be relied upon to transform predictably and consistently under an arbitrary change of basis.