11.8.1 Tensor Contravariant Law Direct Matrix Factor

The Tensor Contravariant Law Direct Matrix Factor links tensor transformations to matrix operations, defining how contravariant components behave under coordinate changes.

Tensor Contravariant Law Direct Matrix Factor is the transformation factor appearing in the contravariant transformation law of a tensor, expressed directly as the forward Jacobian matrix of partial derivatives of the new coordinates with respect to the old coordinates, applied without inversion, in contrast to the inverse matrix factor used for covariant components.

Definition and Structural Role

Position Within the Transformation Law

The contravariant transformation law states that a contravariant tensor component in the new basis is obtained by contracting the old components with partial derivatives of the new coordinates with respect to the old coordinates, and this partial derivative matrix, taken exactly as it stands without any inversion, is the direct matrix factor.

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

Distinction From the Inverse Matrix Factor

Where the covariant law relies on an inverse matrix factor built from derivatives of old coordinates with respect to new coordinates, the direct matrix factor used here is applied exactly as computed from the forward map, with no inversion step required before use in the contraction.

Computation of the Direct Matrix Factor

Construction From the Coordinate Map

The direct matrix factor is assembled by differentiating each new coordinate function with respect to each old coordinate variable, forming a square matrix whose rows correspond to the new coordinate indices and whose columns correspond to the old coordinate indices.

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

Requirement of Non-Singularity

Although the direct matrix factor is used without inversion in the contravariant law itself, the coordinate transformation from which it is built must still be locally invertible, since a non-invertible transformation would fail to define a well-behaved new coordinate system in the first place.

Behavior Under Composition

Direct Application Across Successive Maps

When a coordinate change is followed by a second coordinate change, the direct matrix factor of the combined transformation is obtained by multiplying the two individual direct matrix factors in the order matching the chain rule, with no inversion introduced at any stage of the composition.

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

Identity Case as a Sanity Check

When the new coordinates coincide with the old coordinates, every off-diagonal partial derivative vanishes and every diagonal partial derivative equals one, so the direct matrix factor reduces to the identity matrix and the contravariant law leaves the components unchanged, confirming the correctness of the factor in the trivial case.

Role Within Tensor Algebras

Application to Multiple Contravariant Indices

For a tensor carrying several contravariant indices, the direct matrix factor is applied once per index, forming the tensor product of one copy of the factor for each contravariant slot, while any covariant slots present in a mixed tensor instead receive the inverse matrix factor.

Complementary Relationship With the Inverse Factor

The direct matrix factor and the inverse matrix factor together fully describe how a general tensor transforms, and their status as mutual inverses is precisely what makes the pairing of covariant and contravariant indices produce coordinate-independent scalars under contraction.