10.14.2 Tensor Inverse Jacobian Direction Relation

The Tensor Inverse Jacobian Direction Relation links tensor calculus with inverse Jacobian matrices to define directional relationships in multivariable transformations.

Tensor Inverse Jacobian Direction Relation is the correspondence between the columns and rows of the inverse Jacobian matrix and the geometric directions of the original and new coordinate bases, describing how each entry of the inverse Jacobian encodes the projection of one basis direction onto another across a change of basis.

Basis Vectors and Coordinate Directions

Original and New Basis Vectors

At any point, the original coordinate system induces a set of basis vectors $e_{i}$ tangent to the coordinate lines $x^{i}$ , and the new coordinate system induces its own set of basis vectors ${\bar{e}}_{j}$ tangent to the coordinate lines ${\bar{x}}^{j}$ . These two families of directions are generally not aligned, and the inverse Jacobian matrix is the object that records their relative orientation.

Direction Relation Formula

The original basis vectors are expressed as a linear combination of the new basis vectors through the inverse Jacobian entries:

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

Each entry of the inverse Jacobian therefore measures how much of the new basis direction ${\bar{e}}_{j}$ is needed to reconstruct the original basis direction $e_{i}$ , giving the inverse Jacobian a direct geometric reading as a change-of-direction table rather than a purely symbolic array of derivatives.

Complementary Direction Relation of the Forward Jacobian

Opposite Basis Expansion

The forward Jacobian matrix plays the mirrored role, expressing the new basis vectors in terms of the original ones:

{\bar{e}}_{j} = \sum_{i}^{n} J_{i}^{j} e_{i}

Why the Roles Cannot Be Swapped

Substituting one basis expansion into the other and requiring the result to reduce to the identity relation shows that the inverse Jacobian is not an arbitrary alternative to the forward Jacobian, but the unique matrix consistent with expanding original directions in terms of new directions once the forward expansion is fixed, tying the two direction relations together as a single consistent pair.

Direction Relation for Dual Basis Vectors

Covector Directions

The dual, or covector, basis vectors, often denoted with raised indices as one-forms, follow the direction relation of the covariant transformation, so that the new dual basis vectors are expressed in terms of the original dual basis vectors using the inverse Jacobian, while the original dual basis vectors are expressed in terms of the new ones using the forward Jacobian:

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

This mirrors the covariant component transformation law and confirms that the direction relation encoded by the inverse Jacobian governs dual directions in the same crossed fashion it governs covariant components.

Geometric Diagram of Direction Mixing

Non-Orthogonal Basis Change

Directional Interpretation of Rows and Columns

Reading a Row

Fixing the lower index of the inverse Jacobian and letting the upper index vary produces one full row, which lists the components of a single original basis direction as measured along every new basis direction simultaneously, giving a complete decomposition of that one original direction.

Reading a Column

Fixing the upper index and letting the lower index vary produces one full column, which lists how a single new coordinate direction contributes to each of the original basis directions in turn, giving the reciprocal decomposition organized by new direction rather than by original direction.

Degenerate and Aligned Cases

Parallel Directions

When a particular original direction and a particular new direction coincide, the corresponding inverse Jacobian entry approaches one while the other entries in that row approach zero, recovering the direction relation of an unchanged basis vector.

Orthogonal Directions

When an original direction is expanded with no contribution at all from a specific new direction, the corresponding entry vanishes identically, indicating that the two directions do not mix under that particular change of basis, at least locally at the point being considered.