11.9.3 Tensor Mixed Law Factor Ordering

Tensor Mixed Law Factor Ordering is the observation that, within the mixed variance transformation law, the sequence in which the upper index factors and lower index factors are written or applied has no effect on the resulting transformed component, since ordinary scalar multiplication of the individual Jacobian and inverse Jacobian factors is commutative and each factor acts on a distinct summation index.

Definition and Statement

What Ordering Independence Means

Factor ordering independence means that regardless of whether the direct Jacobian factors for the upper indices are written before or after the inverse Jacobian factors for the lower indices, or in any interleaved sequence, the numerical value of the transformed component computed from the product of all factors remains exactly the same.

Basis in Commutativity of Scalar Multiplication

The independence follows directly from the fact that each individual entry of the direct Jacobian factor and each individual entry of the inverse Jacobian factor are ordinary real numbers at a given point, and multiplication of real numbers does not depend on the order in which the numbers are written.

Why Ordering Freedom Matters

Notational Flexibility Without Loss of Meaning

Because factor ordering has no mathematical consequence, the mixed variance transformation law can be written with the upper index factors listed first, the lower index factors listed first, or any other convenient arrangement, without altering the identity of the tensor being described or introducing any ambiguity.

Separation of Computation Steps

Factor ordering independence allows the update of upper indices and the update of lower indices to be computed as separate, independent steps in any sequence, which is useful when implementing tensor transformations as a series of smaller operations rather than one single combined calculation.

Scope and Limits of the Independence

Applies to Scalar Multiplication, Not Matrix Composition

The ordering independence described here concerns the scalar multiplication of individual Jacobian entries within a single tensor's transformation, and should be distinguished from the composition of Jacobian matrices across successive coordinate transformations, where the order of matrix multiplication does matter and must follow the chain rule precisely.

Summation Indices Remain Fixed to Their Roles

Although the multiplicative order of the factors is free, each factor must still be contracted with the summation index belonging to its own upper or lower slot; rearranging the order of writing the factors does not permit reassigning which summation index a given factor is contracted against.

Role Within Tensor Algebras

Simplification of Notation for Higher-Rank Tensors

Factor ordering independence becomes especially useful for tensors with many upper and lower indices, since it permits writing the transformation law by grouping all upper index factors together and all lower index factors together, or in any other order convenient for exposition, without changing the underlying mathematics.

Consistency With General Tensor Product Structure

The freedom in factor ordering reflects the deeper fact that the transformation of a mixed tensor is built as a tensor product of independent linear maps, one per index, and the commutativity of scalar multiplication among these independent factors is what permits this flexible presentation.