11.9 Tensor Mixed Variance Transformation Law

Tensor Mixed Variance Transformation Law is the rule governing how the components of a tensor carrying both upper and lower indices change under a coordinate transformation, obtained by applying the direct Jacobian matrix factor independently to each upper index and the inverse Jacobian matrix factor independently to each lower index, with all factors combined by ordinary multiplication and summation over the corresponding old indices.

Definition and General Form

Combining Both Transformation Rules

A tensor of mixed type carries some indices that behave contravariantly and others that behave covariantly, and its transformation law is formed by taking the tensor product of one direct Jacobian factor for every upper index together with one inverse Jacobian factor for every lower index.

Generalization to Arbitrary Rank

For a tensor with any number of upper indices and any number of lower indices, the same pattern extends directly: each upper index contributes its own direct Jacobian factor and each lower index contributes its own inverse Jacobian factor, with the old indices belonging to each factor summed over independently.

Justification for the Combined Rule

Consistency With Single-Type Tensors

The mixed variance law reduces correctly to the pure contravariant law when no lower indices are present and to the pure covariant law when no upper indices are present, confirming that it is a genuine generalization rather than an independent rule invented separately for mixed tensors.

Preservation of Contraction Invariance

Because each upper index transforms with the direct factor and each lower index transforms with the inverse factor, contracting any upper index of a mixed tensor with a lower index of another tensor still produces the cancellation of Jacobian factors that yields a coordinate-independent result.

Index-by-Index Independence

Separate Summation Variables

Each index of a mixed tensor, whether upper or lower, is associated with its own independent summation variable when contracting with the appropriate Jacobian factor, so the update of one index never depends on the value or transformation of any other index of the same tensor.

Order of Application Does Not Matter

Because the update associated with each index is a separate linear contraction, the indices of a mixed tensor can be updated in any order, or all at once, without affecting the final transformed component, mirroring the same independence property found in the pure upper-index and pure lower-index cases.

Role Within Tensor Algebras

Defining Criterion for General Tensors

The mixed variance transformation law is the defining criterion by which an array of numbers indexed by both upper and lower labels is recognized as a genuine tensor, since only arrays obeying exactly this combined rule represent objects with a coordinate-independent meaning.

Relationship to Raising and Lowering Indices

The mixed variance law governs how a mixed tensor transforms once it already carries a fixed pattern of upper and lower indices, and this pattern is itself established, prior to any coordinate change, through the metric-based operations of raising a lower index to an upper one or lowering an upper index to a lower one.