10.15 Tensor Coordinate Transformation Process

Tensor Coordinate Transformation Process is the ordered sequence of steps by which the components of a tensor, given in one coordinate system, are converted into the components representing the same geometric or physical object in a different coordinate system, using the Jacobian matrix and its inverse to relate corresponding directions and rates of change between the two systems.

Preliminary Setup

Identifying the Two Systems

The process begins by fixing an original coordinate system with coordinates $$x^i$$ and a new coordinate system with coordinates $${\overline{x}}^j$$, together with an explicit or implicit smooth invertible map connecting them over the region of interest.

Confirming Invertibility

Before any component is transformed, the map between the two systems must be checked for local invertibility, which amounts to verifying that the Jacobian determinant at the point in question is non-zero:

If this condition fails at a point, the coordinate transformation process cannot proceed there, since neither the forward Jacobian's inverse nor a well-defined new coordinate chart exists at that point.

Building the Transformation Coefficients

Assembling the Forward Jacobian

The next step computes every partial derivative of each new coordinate with respect to each original coordinate, filling in the full forward Jacobian matrix entry by entry:

Assembling the Inverse Jacobian

Where covariant components are involved, the inverse Jacobian matrix must also be assembled, either by direct differentiation of an explicit inverse map or by numerically or symbolically inverting the forward Jacobian matrix as a linear algebra operation.

Applying the Transformation Law

Selecting the Correct Law by Variance Type

Each tensor component is transformed according to its variance type: contravariant indices are contracted with the forward Jacobian, covariant indices are contracted with the inverse Jacobian, and mixed tensors receive one factor for every index, of the appropriate matrix, all multiplied together. For a general mixed tensor of one upper and one lower index:

Carrying Out the Summation

For each free index combination in the new system, the process sums the products of transformation coefficients and original components over every repeated index, following the Einstein summation convention, until every new component has been computed from the full set of original components.

Process Flow Diagram

Step Sequence

Verification Steps

Checking the Result

Once the new components are computed, the process is typically verified by re-applying the reverse transformation to the newly obtained components and confirming that the original components are recovered exactly, which relies on the Jacobian product identity between the forward and inverse matrices.

Confirming Invariance of Scalars

A further check consists of forming any full contraction available in the tensor expression, such as a covariant-contravariant pairing, in both coordinate systems and confirming that the resulting scalar value is identical, which serves as an independent consistency test on the whole transformation process.

Special Cases in the Process

Linear Coordinate Changes

When the map between the two coordinate systems is linear, the Jacobian matrix is constant throughout the domain, so the transformation coefficients need to be computed only once and can then be reused for every point, simplifying the process considerably compared to a general curvilinear change of coordinates.

Composed Transformations

When more than two coordinate systems are involved, the process can be carried out either by transforming directly between the two systems of interest using their own Jacobian matrices, or by transforming through an intermediate system in two stages, and the chain rule guarantees both routes produce the same final components.