10.14 Tensor Inverse Jacobian Matrix Role

Tensor Inverse Jacobian Matrix Role is the function played by the inverse of the Jacobian matrix in mediating how tensor components transform when passing from a new coordinate system back to an original one, or equivalently, how contravariant and covariant components must be scaled oppositely to preserve the invariant, basis-independent meaning of a tensor under a change of basis.

Position Within Change of Basis

Forward and Backward Maps

When coordinates change from an original system with coordinates $$x^i$$ to a new system with coordinates $${\overline{x}}^i$$, the Jacobian matrix collects the partial derivatives of the new coordinates with respect to the old ones:

The inverse Jacobian matrix reverses this correspondence, collecting the partial derivatives of the old coordinates with respect to the new ones:

Complementary Pairing

The Jacobian and its inverse are not independent constructions; they are tied together by the requirement that composing the forward and backward maps returns the identity transformation, expressed through the Kronecker delta:

This pairing is what allows a single change of basis to be described consistently in both directions, and it is the reason the inverse Jacobian is treated as a distinct object with its own transformation role rather than as a mere algebraic byproduct.

Transformation of Tensor Components

Contravariant Components

Contravariant tensor components, indexed with superscripts, transform using the Jacobian matrix of the new coordinates with respect to the old ones. For a contravariant vector component:

The inverse role appears immediately when the same component must be recovered in the original system from the new one:

Covariant Components

Covariant tensor components, indexed with subscripts, reverse this assignment: they transform using the inverse Jacobian matrix directly, while their recovery in the original system uses the forward Jacobian. For a covariant vector component:

This crossed assignment of the Jacobian and its inverse to contravariant and covariant indices respectively is the mechanism that keeps a fully contracted expression, such as a scalar formed by pairing a covariant and a contravariant component, invariant under the change of basis.

Mixed Tensors and Repeated Application

General Rule

For a mixed tensor with any number of upper and lower indices, the inverse Jacobian matrix appears once for every upper index being transformed, and the forward Jacobian matrix appears once for every lower index. A rank-two mixed tensor transforms as:

Here the forward Jacobian matches the upper index of the tensor, and the inverse Jacobian matches the lower index, reflecting that each index type calls on a specific one of the two matrices, never on the other.

Diagram of Index Assignment

Determinant and Density Weight

Jacobian Determinant

The determinant of the Jacobian matrix and the determinant of its inverse are reciprocal:

Role in Tensor Densities

Tensor densities, which pick up an extra factor of the Jacobian determinant raised to some weight during a change of basis, rely on the inverse Jacobian determinant whenever the density is expressed in the original coordinates after being defined in the new ones. This makes the inverse Jacobian matrix responsible not only for the ordinary index-by-index reversal of coordinate transformation but also for correctly re-scaling densities and pseudo-tensors when moving between coordinate charts.

Consistency Requirements

Non-Singularity

The inverse Jacobian matrix exists only where the Jacobian matrix is non-singular, meaning its determinant does not vanish. At points where this fails, the coordinate change is not invertible, and the notion of transforming covariant components through the inverse Jacobian breaks down locally, since there is no well-defined backward map at that point.

Chain Rule Compatibility

Because the inverse Jacobian is built from partial derivatives of the old coordinates with respect to the new ones, it automatically satisfies the multivariable chain rule linking any third coordinate system to the original and new ones, which ensures that transforming a tensor through an intermediate coordinate system, using the appropriate forward and inverse Jacobian matrices at each stage, yields the same result as transforming directly between the two end systems.