11.16.3 Tensor Coordinate Change Mixed Response

Tensor Coordinate Change Mixed Response explains how tensor components transform under coordinate changes, blending vector and covector behavior in mixed tensor types.

Tensor Coordinate Change Mixed Response is the combined transformation behavior displayed by a tensor possessing both upper and lower indices when the coordinate system changes, in which each contravariant index responds through the direct Jacobian factor while each covariant index responds through the inverse-direction Jacobian factor simultaneously within the same expression.

Foundational Setting

Tensors with Both Index Types

A mixed tensor carries at least one upper index and at least one lower index at once, written generally as $T_{j}^{i}$ , where the upper position $i$ is contravariant and the lower position $j$ is covariant. A common example is a linear map from a vector space to itself, whose matrix representation naturally carries one upper and one lower index.

The Two Jacobian Building Blocks

Mixed response draws on both directional forms of the Jacobian matrix introduced for pure contravariant and pure covariant behavior:

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

K_{i}^{j} = \frac{\partial x^{j}}{\partial {\tilde{x}}^{i}}

The Mixed Transformation Law

Direct Statement for a Rank-Two Mixed Tensor

Under a change of coordinates, a rank-two mixed tensor transforms according to:

{\tilde{T}}_{l}^{k} = \sum_{i, j} J_{i}^{k} K_{l}^{j} T_{j}^{i}

Reading the Formula

The upper index of the new tensor is produced by contracting with one factor of $J$ against the old upper index, while the lower index of the new tensor is produced by contracting with one factor of $K$ against the old lower index. Each index type retains its own independent transformation rule even though both appear in the same tensor.

Invariance Preserved by the Mixed Response

The Trace as an Invariant

A central consequence of the mixed response is that contracting the upper and lower indices of a mixed tensor, forming its trace, produces a scalar unchanged by the coordinate transformation:

\sum_{i} T_{i}^{i} = \sum_{i} {\tilde{T}}_{i}^{i}

This invariance holds precisely because the $J$ and $K$ factors are inverses of one another and cancel upon contraction.

Action on Vectors Preserved

A mixed tensor acting as a linear operator on a vector produces a result whose transformation is consistent with the vector's own contravariant response, since the mixed tensor's covariant index absorbs the contravariant factor from the vector it acts on, leaving a purely contravariant output.

w^{i} = \sum_{j} T_{j}^{i} v^{j}

Diagrammatic Overview

Index Flow Under Transformation

Extension to Higher-Rank Mixed Tensors

General Rule for Arbitrary Rank

For a tensor with several upper indices and several lower indices, the mixed response applies one factor of $J$ for every upper index and one factor of $K$ for every lower index, all multiplied together within a single summation over the original indices:

{\tilde{T}}_{l m}^{k} = \sum_{i, j, n} J_{i}^{k} K_{l}^{j} K_{m}^{n} T_{j n}^{i}

Structural Importance

This uniform combination of both Jacobian directions within a single tensor is what allows mixed tensors, such as the Kronecker delta and linear operators generally, to serve as bridges between the contravariant and covariant descriptions of a vector space.

Summary of Key Traits

Defining Characteristics

Upper indices of a mixed tensor transform using the direct Jacobian factor.
Lower indices of the same tensor transform simultaneously using the inverse-direction Jacobian factor.
Contracting matched upper and lower indices of a mixed tensor yields an invariant scalar.
The rule generalizes uniformly to tensors carrying any number of upper and lower indices.