10.22.4 Tensor Inverse Jacobian Factor Notation

Tensor Inverse Jacobian Factor Notation expresses inverse Jacobian matrices using tensor algebra, enabling coordinate transformations in multivariable systems.

Tensor Inverse Jacobian Factor Notation is the symbolic convention used to represent the partial-derivative coefficients that express old coordinates as functions of new coordinates during a change of basis. It is the counterpart of the direct Jacobian factor notation, and it is the specific set of coefficients required to transform covariant tensor components correctly.

Definition and Basic Form

The Inverse Factor as a Partial Derivative

The inverse Jacobian factor is the derivative of a source coordinate with respect to a target coordinate.

J_{i^{'}}^{i} = \frac{\partial x^{i}}{\partial x^{i^{'}}}

Index Placement

The primed index, belonging to the new coordinate system, is placed as a subscript, while the unprimed index, belonging to the old coordinate system, is placed as a superscript. This placement is opposite to that of the direct Jacobian factor, and it reflects the fact that the inverse factor governs covariant, rather than contravariant, transformation behavior.

Derivation From the Direct Factor

Matrix Inversion Relationship

If the direct Jacobian factors are collected into a matrix, the inverse Jacobian factors form the matrix inverse of that same matrix, provided the coordinate transformation is invertible in the region under consideration.

J_{i}^{k^{'}} J_{k^{'}}^{j} = δ_{j}^{i}

Existence Condition

The inverse Jacobian factors exist only where the determinant of the direct Jacobian matrix is nonzero, since a vanishing determinant means the coordinate map is not locally invertible and no consistent inverse factor set can be defined at that point.

Role in Covariant Transformation

Transforming a Covariant Vector

A covariant vector, also called a one-form component, transforms by contraction with a single inverse Jacobian factor.

W_{i^{'}} = J_{i^{'}}^{i} W_{i}

Transforming Higher-Rank Covariant Tensors

For a purely covariant tensor of rank two, one inverse Jacobian factor is attached to each index.

T_{i^{'} j^{'}} = J_{i^{'}}^{i} J_{j^{'}}^{j} T_{i j}

Relationship to Basis Vectors

Transforming the Basis Itself

The natural basis vectors of a coordinate system transform using the direct Jacobian factor, not the inverse one, because the basis vectors are built from partial derivatives of the position with respect to the coordinates.

Transforming the Dual Basis

The dual basis, formed from the differentials of the coordinate functions, transforms using the inverse Jacobian factor, which mirrors the fact that covariant tensor components transform the same way as the dual basis one-forms.

Practical Computation

Solving for the Inverse Factors Directly

When the coordinate transformation is given in closed form as new coordinates in terms of old coordinates, the inverse Jacobian factors are typically obtained by first inverting the coordinate relations to express old coordinates in terms of new coordinates, and then differentiating directly, rather than attempting to invert the direct Jacobian matrix symbolically.

Local Versus Global Validity

Because the inverse coordinate relations may only be well defined on a restricted region, the inverse Jacobian factor notation is understood to hold locally, valid wherever the coordinate map is a diffeomorphism, and it may take different symbolic forms on different coordinate patches.

Consistency Checks

Chain Rule Verification

A valid inverse Jacobian factor set must satisfy the chain rule identity linking it back to the direct factors through the Kronecker delta, and this identity serves as the standard check that a computed inverse factor set has been derived correctly.

Symmetry Under Repeated Transformation

Applying the direct Jacobian factor followed by the inverse Jacobian factor for the same pair of coordinate systems must return every tensor component to its original value, and this round-trip consistency is the defining property that distinguishes a true inverse factor from an unrelated coefficient set.