11.16 Tensor Coordinate Change Variance Response

Tensor Coordinate Change Variance Response is the general characterization of how any tensor, regardless of its specific type, reacts to a change of coordinates, describing this reaction as a predictable, index-by-index response governed entirely by the tensor's fixed pattern of upper and lower indices and the Jacobian of the coordinate map.

Definition and General Statement

Response as a Deterministic Function of Type

The variance response of a tensor to a coordinate change is completely determined once its type, meaning its specific counts of upper and lower indices, is known, since each upper index contributes a direct Jacobian factor to the response and each lower index contributes an inverse Jacobian factor, with no other freedom in how the response unfolds.

Response Independent of the Tensor's Specific Numerical Values

The pattern of response, meaning which factor applies to which index, does not depend on the actual numerical values of the tensor's components, but only on the abstract type of the tensor, so two entirely different tensors of the same type respond to the same coordinate change in structurally identical ways.

Categorizing the Range of Possible Responses

Purely Contravariant Response

A tensor with only upper indices responds to a coordinate change using exclusively direct Jacobian factors, one for each index, exhibiting the simplest and most direct form of variance response among the possible tensor types.

Purely Covariant Response

A tensor with only lower indices responds using exclusively inverse Jacobian factors, one for each index, exhibiting the mirror-image response to the purely contravariant case.

Mixed Response

A tensor with both upper and lower indices exhibits a combined response, applying direct factors to its upper indices and inverse factors to its lower indices simultaneously, with the two halves of the response acting independently on their respective indices.

Consistency Properties of the Response

Response Consistent Across Successive Coordinate Changes

Applying the variance response for one coordinate change followed by the variance response for a second coordinate change produces exactly the same result as applying the variance response for the single combined coordinate change, following directly from the chain rule governing composition of the underlying Jacobian factors.

Response Reduces to the Identity for the Trivial Change

When the new coordinate system coincides with the old one, the variance response leaves every component of the tensor completely unchanged, regardless of the tensor's type, since every Jacobian factor involved reduces to the appropriate Kronecker delta in this trivial case.

Role Within Tensor Algebras

Synthesizing the Individual Transformation Laws

Coordinate change variance response serves as the synthesizing concept that gathers the covariant transformation law, the contravariant transformation law, and the mixed variance transformation law into a single, unified description of how tensors of any type react to a change of coordinates.

Defining Property Distinguishing Tensors From Arbitrary Arrays

The predictable, type-determined nature of coordinate change variance response is ultimately what distinguishes a genuine tensor from an arbitrary array of numbers attached to a coordinate system, since only objects exhibiting this precise, structured response qualify as tensors in the formal sense used throughout tensor algebra.