11.8.2 Tensor Contravariant Law Upper Index Update

Tensor Contravariant Law Upper Index Update describes how upper indices transform under coordinate changes, reflecting contravariant behavior in tensor algebra.

Tensor Contravariant Law Upper Index Update is the specific operation, performed as part of the contravariant transformation law, by which an upper-placed index of a tensor's components is recalculated in the new coordinate system through contraction with the direct Jacobian matrix factor, replacing the old value attached to that index with a new value that reflects the new coordinate labeling.

Definition and Mechanics

What the Update Operation Does

The upper index update takes a component labeled by an old upper index, multiplies it by the corresponding entry of the direct Jacobian matrix, and sums over all values of the old index to produce the new component labeled by the new upper index, replacing the old label entirely in the resulting expression.

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

Single-Index Versus Multi-Index Update

For a tensor with a single upper index, the update is performed exactly once using the full Jacobian contraction; for a tensor with several upper indices, the update is applied independently and simultaneously to each upper index, with a separate summation variable used for each one.

Order and Independence of Updates

Independence From Update Order

Because the contraction defining the upper index update is linear and each upper index has its own summation variable, updating several upper indices of the same tensor can be performed in any order, or simultaneously, without changing the final result.

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

Non-Interference With Lower Indices

When a mixed tensor carries both upper and lower indices, the upper index update proceeds using only the direct Jacobian factor and does not interact with the separate update applied to lower indices, which instead uses the inverse Jacobian factor, so the two kinds of updates can be computed independently before being combined into the full new component.

Practical Considerations

Dependence on the Point of Evaluation

Because the Jacobian matrix used in the upper index update is generally a function of position rather than a constant matrix, the update must be recomputed at each point of the space under consideration, using the value of the Jacobian evaluated at that specific point.

Verification Through Repeated Application

Applying the upper index update forward to a new coordinate system and then applying the corresponding update backward to return to the old coordinate system reproduces the original component exactly, since the direct and inverse Jacobian factors are matched inverses of one another.

Role Within Tensor Algebras

Relation to the General Transformation Law

The upper index update is the elementary operation from which the full contravariant transformation law for tensors of arbitrary rank is built, since a tensor with several upper indices is transformed by applying this same update operation once for every upper index it possesses.

Consistency With Contraction Operations

When an upper index is later contracted with a lower index from another tensor, the upper index update ensures that the value entering the contraction is always expressed consistently in the current coordinate system, which is essential for the resulting contracted scalar to remain invariant.