11.9.2 Tensor Mixed Law Lower Index Factor

The Tensor Mixed Law Lower Index Factor governs how indices interact in tensor algebra, defining contraction and transformation rules in multi-linear contexts.

Tensor Mixed Law Lower Index Factor is the individual component of the mixed variance transformation law responsible for updating each lower index of a mixed tensor, taking the form of the inverse Jacobian matrix of partial derivatives of the old coordinates with respect to the new coordinates, applied separately to every lower index while the upper 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 lower index factor is the specific multiplicative term contracted against an old lower index, and it can be identified and studied on its own because it does not depend on how many upper indices the tensor also carries.

\frac{\partial x^{k}}{\partial x^{l^{'}}}

Its Placement in a Full Mixed Tensor Transformation

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

Identical to the Pure Covariant Factor

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

Behavior Under Multiple Lower Indices

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

Interaction With the Upper Index Factor

Independence of the Two Factors

The lower index factor and the upper 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 a lower index and an upper index to remain invariant, since the lower index factor and a matching direct factor from another tensor's upper index cancel exactly under contraction.

Role Within Tensor Algebras

Building Block for General Transformation Rules

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

Relation to Tensor Rank and Index Counting

The number of times the lower index factor must be applied in a given transformation is determined entirely by counting the lower 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.