11.3.4 Tensor Covariant Component Inverse Factor Relation

The Tensor Covariant Component Inverse Factor Relation links components through metric tensor inversion, essential for coordinate transformations in tensor algebra.

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

Statement of the Relation

Single-Index Form

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

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

Multi-Index Extension

For a tensor with several covariant indices, the relation extends by attaching one inverse Jacobian factor to each covariant 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 Scalar Gradient

The relation is derived by applying the chain rule to the partial derivative of a scalar function with respect to a new coordinate, expressing that derivative as a sum over old coordinates weighted precisely by the inverse Jacobian factor, which establishes the relation for the prototypical covariant object and motivates its extension to general covariant tensors.

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

Distinguishing Feature Compared to the Direct Factor Relation

Reversed Roles of Numerator and Denominator

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

Consequence for Which Index Position Is Required

Because this specific inverse-factor dependence is tied to the subscript position of the index, the relation only applies to indices already established as covariant; applying it to an index actually belonging to a contravariant 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 Direct Factor

The inverse factor appearing in this relation is linked to the direct Jacobian factor through the reciprocity identity, guaranteeing that applying the covariant 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^{'}}^{i} J_{i}^{k^{'}} = δ_{k^{'}}^{i^{'}}

Practical Application

Direct Use in Deriving Covariant Transformation Formulas

Whenever a new covariant object is introduced, the inverse 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 covariant component behavior in any specific calculation.