10.14.4 Tensor Inverse Jacobian Index Placement Convention

The Tensor Inverse Jacobian Index Placement Convention standardizes index order in inverse Jacobian matrices for tensor algebra clarity and consistency.

Tensor Inverse Jacobian Index Placement Convention is the set of notational rules governing where the two indices of an inverse Jacobian entry are written, upper or lower, barred or unbarred, first or second, so that the entry can be combined correctly with tensor components under the Einstein summation convention without ambiguity about which coordinate system or which transformation direction it belongs to.

The Two Indices of One Entry

Upper and Lower Placement

An inverse Jacobian entry carries one upper index and one lower index, matching the general rule that a repeated index used for summation must appear once up and once down. The standard placement is:

{(J^{- 1})}_{j}^{i} = \frac{\partial x^{i}}{\partial {\bar{x}}^{j}}

The upper index, here $i$ , matches the index of the coordinate appearing in the numerator of the derivative, while the lower index, here $j$ , matches the index of the coordinate appearing in the denominator, a rule that holds for every derivative-based transformation coefficient regardless of which matrix it belongs to.

Barred and Unbarred Marking

Because the upper index of the inverse Jacobian belongs to the original, unbarred coordinate system while the lower index belongs to the new, barred coordinate system, some presentations additionally mark the indices themselves, writing the lower index as $\bar{j}$ to make explicit that it ranges over the new coordinate labels, so that no separate legend is needed to recall which index belongs to which coordinate system.

Distinguishing the Inverse from the Forward Matrix

Same Letter, Reversed Index Roles

The forward Jacobian is written with the same letter but the opposite assignment of index roles:

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

Here the upper index belongs to the new system and the lower index belongs to the original system, exactly reversed from the inverse Jacobian, so the convention of always reading "upper index equals numerator coordinate, lower index equals denominator coordinate" is what allows the two matrices to be told apart without needing a separate symbol beyond the explicit minus-one superscript.

Role of the Explicit Inverse Superscript

The superscript $(- 1)$ attached to $J$ is itself not a tensor index and does not participate in the summation convention; it is purely a label identifying the matrix as the inverse, and placement conventions for the two genuine tensor indices are applied only after this label has been fixed.

Consistency With the Summation Convention

Valid Contraction Pattern

A term is well formed only when a lower placement on the inverse Jacobian matches an upper placement on some tensor component being summed over, as in:

\sum_{i}^{n} {(J^{- 1})}_{j}^{i} W_{i}

Here the summed index $i$ appears as an upper index on the inverse Jacobian and as a lower index on the tensor component, satisfying the placement convention, while the free index $j$ remains lower throughout, correctly marking the result as a covariant component in the new coordinate system.

Invalid Placement Pattern

A term with two upper indices or two lower indices on the same summed letter, such as an inverse Jacobian upper index repeated against another upper index on a tensor component, violates the convention and signals either a transcription error or a genuine mismatch between the transformation law being applied and the variance type of the tensor component involved.

Diagram of Index Roles

Visual Key

Ordering Convention in Matrix Notation

Row-Column Reading

When the inverse Jacobian is written as an ordinary matrix rather than in explicit index notation, the placement convention fixes which index is the row and which is the column: the lower index numbers the row and the upper index numbers the column, so that matrix multiplication by a column vector of original-basis covariant components on the right reproduces the same summation pattern as the indexed formula.

Consistency Across Sources

Because different texts sometimes swap which index is written first, the placement convention is anchored to the derivative definition itself rather than to typographical order, meaning that regardless of whether an author writes the upper or lower index first on the page, the correct interpretation is recovered by checking which coordinate label sits in the numerator and which sits in the denominator of the underlying partial derivative.