11.9.1 Tensor Mixed Law Upper Index Factor

The Tensor Mixed Law Upper Index Factor dictates transformation rules for upper indices in tensor algebra.

Tensor Mixed Law Upper Index Factor is the individual component of the mixed variance transformation law responsible for updating each upper index of a mixed tensor, taking the form of the direct Jacobian matrix of partial derivatives of the new coordinates with respect to the old coordinates, applied separately to every upper index while the lower indices of the same tensor are handled by a different factor.

Definition and Isolation Within the Mixed Law

The Factor in Isolation

Within the full mixed variance transformation law, the upper index factor is the specific multiplicative term contracted against an old upper index, and it can be identified and studied on its own because it does not depend on how many lower indices the tensor also carries.

\frac{\partial x^{i^{'}}}{\partial x^{j}}

Its Placement in a Full Mixed Tensor Transformation

When a mixed tensor with one upper index and one lower index transforms, the upper index factor appears exactly once, multiplying the old upper index, while a separate and distinct factor multiplies the old lower index.

A_{l^{'}}^{i^{'}} = \frac{\partial x^{i^{'}}}{\partial x^{j}} \cdot \frac{\partial x^{k}}{\partial x^{l^{'}}} A_{k}^{j}

Properties of the Upper Index Factor

Identical to the Pure Contravariant Factor

The upper index factor used within the mixed law is exactly the same direct Jacobian factor that appears alone in the pure contravariant transformation law, so no new mathematical object is introduced for the mixed case; the same factor is simply reused once per upper index.

Behavior Under Multiple Upper Indices

When a mixed tensor carries more than one upper index, the upper index factor is applied independently to each one, using a distinct summation variable for each upper index, so the total number of upper index factors appearing in the transformation equals the number of upper indices on the tensor.

Interaction With the Lower Index Factor

Independence of the Two Factors

The upper index factor and the lower index factor that together make up the mixed transformation law act on separate indices and are computed as separate multiplicative terms, so changing the value of one factor, for example through a different coordinate transformation, does not require recomputing the other.

Joint Contribution to Invariance

Although the two factors are computed independently, their combined effect is what allows a contraction between an upper index and a lower index to remain invariant, since the upper index factor and a matching inverse factor from another tensor's lower index cancel exactly under contraction.

Role Within Tensor Algebras

Building Block for General Transformation Rules

The upper index factor serves as a reusable building block: the same factor appears in the pure contravariant law, in the mixed variance law, and in any transformation rule for a tensor of arbitrary rank, wherever an upper index needs to be updated to a new coordinate system.

Relation to Tensor Rank and Index Counting

The number of times the upper index factor must be applied in a given transformation is determined entirely by counting the upper indices of the tensor, making the factor a systematic tool for constructing the transformation law of tensors with any combination of upper and lower indices.