11.8 Tensor Contravariant Transformation Law

Tensor Contravariant Transformation Law is the rule describing how the components of a contravariant tensor change when the coordinate system used to describe them is replaced by another, expressed through the forward Jacobian matrix of partial derivatives of the new coordinates with respect to the old coordinates. This law ensures that the underlying geometric object, such as a displacement or a velocity, remains a single invariant entity even though its numerical components differ from one coordinate description to another.

Definition and Formal Statement

The Transformation Rule

A tensor is called contravariant in a given index when that index transforms using the Jacobian of the new coordinates with respect to the old ones, applied once for each contravariant index the tensor possesses.

Vectors as the Prototypical Case

The archetype of a contravariant object is the differential displacement vector, whose components transform exactly by the chain rule applied to the coordinate functions, which is the origin of the name contravariant transformation law.

Origin of the Term Contravariant

Opposition to Basis Vector Behavior

The term contravariant reflects the fact that the components change in the opposite sense to the basis vectors themselves: when the basis vectors are stretched by a forward coordinate change, the components describing a fixed geometric vector must shrink to compensate, so that the vector as a whole does not change.

Contrast With Covariant Behavior

Contravariant transformation stands in direct contrast to covariant transformation, where components use the inverse of the Jacobian rather than the Jacobian itself, a distinction that becomes essential once tensors with mixed index types are considered.

Consistency Properties

Composition of Coordinate Changes

If a coordinate transformation is followed by a second coordinate transformation, the contravariant components transform according to the Jacobian of the composed map, which by the chain rule equals the product of the two individual Jacobian matrices taken in the natural order of application.

Identity Transformation Check

When the new coordinate system coincides with the old one, the Jacobian matrix reduces to the identity matrix, and the contravariant transformation law correctly leaves every component unchanged, serving as a basic consistency check on the law.

Role Within Tensor Algebras

Higher-Rank Contravariant Tensors

For a tensor with several contravariant indices, the law is applied independently to each index, so that the full transformation is built from the tensor product of one Jacobian factor per contravariant slot, leaving covariant slots, if present, to transform separately by the inverse factor.

Foundational Status

The contravariant transformation law, together with its covariant counterpart, forms the defining criterion by which an array of numbers attached to a coordinate system is recognized as representing a genuine tensor rather than a coordinate-dependent collection of numbers with no invariant meaning.