11.4.4 Tensor Contravariant Component Direct Factor Relation

The Tensor Contravariant Component Direct Factor Relation shows how components transform via direct factorization in tensor algebra.

Tensor Contravariant Component Direct Factor Relation is the precise statement that a contravariant tensor component transforms under a change of basis by contraction with the direct Jacobian factor, establishing the exact algebraic dependence between the transformed component, the original component, and the derivative of the new coordinate with respect to the old coordinate.

Statement of the Relation

Single-Index Form

For a contravariant vector, the relation states that the transformed component equals the direct Jacobian factor contracted with the original component, where the direct factor is the partial derivative of the new coordinate with respect to the old coordinate.

V^{i^{'}} = J_{i}^{i^{'}} V^{i} where J_{i}^{i^{'}} = \frac{\partial x^{i^{'}}}{\partial x^{i}}

Multi-Index Extension

For a tensor with several contravariant indices, the relation extends by attaching one direct Jacobian factor to each contravariant slot, with all the factors multiplied together before contracting with the original components.

T^{i^{'} j^{'}} = J_{i}^{i^{'}} J_{j}^{j^{'}} T^{i j}

Derivation of the Relation

Origin in the Chain Rule Applied to a Coordinate Differential

The relation is derived by applying the chain rule to an infinitesimal displacement expressed in the new coordinates, writing the new coordinate differential as a sum over old coordinate differentials weighted precisely by the direct Jacobian factor, which establishes the relation for the prototypical contravariant object and motivates its extension to general contravariant tensors.

d x^{i^{'}} = \sum_{i} \frac{\partial x^{i^{'}}}{\partial x^{i}} d x^{i}

Distinguishing Feature Compared to the Inverse Factor Relation

Roles of Numerator and Denominator

The direct factor relation is distinguished from the covariant transformation relation by which coordinate appears in the numerator of the partial derivative: the contravariant relation places the new coordinate in the numerator and the old coordinate in the denominator, exactly opposite to the inverse factor used for covariant components.

Consequence for Which Index Position Is Required

Because this specific direct-factor dependence is tied to the superscript position of the index, the relation only applies to indices already established as contravariant; applying it to an index actually belonging to a covariant slot would produce an incorrect transformation, underscoring why index position must be assigned correctly before invoking this relation.

Consistency With the Reciprocity Identity

Compatibility With the Inverse Factor

The direct factor appearing in this relation is linked to the inverse Jacobian factor through the reciprocity identity, guaranteeing that applying the contravariant relation to convert from the old to the new system, and then applying the corresponding relation in reverse, restores the original components exactly.

J_{i}^{k^{'}} J_{k^{'}}^{j} = δ_{j}^{i}

Practical Application

Direct Use in Deriving Contravariant Transformation Formulas

Whenever a new contravariant object is introduced, the direct factor relation is the formula applied first to confirm that the proposed object transforms correctly, making this relation the standard computational starting point for validating and applying contravariant component behavior in any specific calculation.