10.14.1 Tensor Inverse Jacobian Coordinate Derivative Entry

The Tensor Inverse Jacobian Coordinate Derivative Entry explains how to invert Jacobian matrices in tensor calculus for coordinate transformations.

Tensor Inverse Jacobian Coordinate Derivative Entry is a single scalar component of the inverse Jacobian matrix, obtained as the partial derivative of one original coordinate with respect to one new coordinate, and serving as the atomic building block from which the entire inverse Jacobian matrix, and every tensor transformation that depends on it, is assembled.

Definition as a Partial Derivative

The Single Entry

Given an original coordinate system with coordinates $x^{i}$ and a new coordinate system with coordinates ${\bar{x}}^{j}$ , a single inverse Jacobian entry is written as:

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

This single entry answers a narrowly defined question: how much does the original coordinate $x^{i}$ change per unit change of the new coordinate ${\bar{x}}^{j}$ , holding every other new coordinate fixed. It is a local quantity, generally different at every point of the coordinate domain unless the change of basis is linear.

Row and Column Position

Within the inverse Jacobian matrix, the lower index labels which row the entry occupies and the upper index labels which column it occupies, so that the full matrix is the array obtained by letting both indices range over all coordinate directions:

J^{- 1} = (\begin{matrix} \frac{\partial x^{1}}{\partial {\bar{x}}^{1}} & \dots & \frac{\partial x^{1}}{\partial {\bar{x}}^{n}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial x^{n}}{\partial {\bar{x}}^{1}} & \dots & \frac{\partial x^{n}}{\partial {\bar{x}}^{n}} \end{matrix})

Role in a Single Transformation Term

Contribution to a Sum

Because tensor transformation laws involve a sum over one repeated index, a single derivative entry never acts alone: it is multiplied by one tensor component and added to similar products formed with the other entries in its row or column. For a covariant component:

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

Each entry ${(J^{- 1})}_{j}^{i}$ appears exactly once in this sum for a fixed value of the free index $j$ , weighting the contribution of the original component $W_{i}$ to the new component.

Isolating One Entry

If all original components except one are set to zero, the sum collapses and the single entry can be read off directly, which is the operational meaning of "one entry governs one directional contribution": it measures the sensitivity of a single new-basis component to a single original-basis component, independent of the others.

Computing an Entry in Practice

From an Explicit Forward Map

When the forward coordinate functions ${\bar{x}}^{j} = {\bar{x}}^{j} (x^{1}, \dots, x^{n},)$ are given explicitly, the inverse Jacobian entries are usually not obtained by differentiating an explicit inverse map, but by inverting the full forward Jacobian matrix as a linear algebra step, since an explicit closed form for the inverse coordinate functions may not exist even though the inverse map itself does, locally, by the inverse function theorem.

Matrix Inversion Formula

For a square Jacobian matrix, a single inverse entry can be computed from the cofactor expansion:

{(J^{- 1})}_{j}^{i} = \frac{C_{ij}}{det (J)}

where $C_{ij}$ denotes the cofactor of the forward Jacobian matrix associated with the transposed position, confirming that any single entry of the inverse depends, in general, on every entry of the forward matrix, not merely on the corresponding forward entry.

Illustrative Diagram

One Directional Sensitivity

Special Behavior at Fixed Points

Diagonal Entries

When the index of the original coordinate matches the index of the new coordinate, the entry lies on the diagonal of the inverse Jacobian matrix and typically stays close to one when the coordinate change is a small perturbation of the identity map, while off-diagonal entries stay close to zero, reflecting weak coupling between differently labeled directions.

Vanishing Entries

An entry equal to zero at a point indicates that, at that point, the corresponding original coordinate does not respond, to first order, to changes in that particular new coordinate, which occurs, for instance, whenever the coordinate change treats that direction as fixed or as a function of the other new coordinates only.