11.19.3 Tensor Variance Convention Coordinate Direction

Tensor Variance Convention governs how tensor components transform with coordinate direction, defining covariant and contravariant behavior in mathematical physics.

Tensor Variance Convention Coordinate Direction is the specific application of the basis direction convention to coordinate systems rather than abstract basis vectors, fixing which of the two coordinate systems in a change of variables is treated as the source and which as the target when writing the Jacobian matrices that govern covariant and contravariant transformation laws.

Foundational Setting

Coordinates as a Special Case of Basis Change

A change of coordinates $x^{i}$ to ${\tilde{x}}^{i}$ induces, at every point, a corresponding change of basis vectors for the tangent space at that point. The coordinate direction convention extends the same directional choice used for abstract basis vectors to this coordinate setting, fixing the new coordinates as functions of the old ones as the primary direction of the transformation.

The Two Jacobians Involved

Once this direction is fixed, two related Jacobian matrices arise, one built from differentiating the new coordinates with respect to the old, and one from differentiating the old with respect to the new:

J_{j}^{i} = \frac{\partial {\tilde{x}}^{i}}{\partial x^{j}}

K_{i}^{j} = \frac{\partial x^{j}}{\partial {\tilde{x}}^{i}}

Assigning the Convention to Contravariant and Covariant Laws

Contravariant Components Use the Named Direction

By the coordinate direction convention, contravariant components, such as those of a tangent vector, transform using $J$ , the Jacobian built in the designated new-with-respect-to-old direction:

{\tilde{v}}^{i} = \sum_{j} J_{j}^{i} v^{j}

Covariant Components Use the Opposite Direction

Covariant components, such as gradient components, transform using $K$ instead, the Jacobian built in the reversed, old-with-respect-to-new direction:

{\tilde{ω}}_{i} = \sum_{j} K_{i}^{j} ω_{j}

Why the Direction Choice Must Be Fixed Consistently

Ambiguity Without a Fixed Reference

If different parts of a calculation implicitly assumed different coordinate directions, one term treating the new coordinates as the reference and another treating the old coordinates as the reference, the resulting Jacobian factors would not compose correctly, and quantities intended to be invariant would fail to cancel properly.

Ensuring Correct Cancellation in Contractions

Fixing a single coordinate direction throughout guarantees that a contraction between a covariant and contravariant component uses $J$ and $K$ as genuine matrix inverses of each other, so the product correctly reduces to the identity:

\sum_{j} K_{i}^{j} J_{j}^{k} = δ_{i}^{k}

Visual Overview

Diagram of the Coordinate Direction Choice

Consistency Across Successive Coordinate Changes

Chained Transformations Preserve the Direction

When a calculation passes through several coordinate systems in sequence, maintaining the same coordinate direction convention at each step ensures that the composed Jacobian for the overall transformation, from the very first system to the very last, matches the product of the individual Jacobians computed along the way, consistent with the multivariable chain rule.

Reversing the Convention Mid-Calculation

Switching the designated coordinate direction partway through a calculation, without correspondingly relabeling which Jacobian is $J$ and which is $K$ , produces a transformation law with the roles of covariant and contravariant behavior effectively swapped, leading to results inconsistent with the rest of the calculation.

Summary of Key Traits

Defining Characteristics

The coordinate direction convention fixes which coordinate system is treated as old and which as new when constructing Jacobian matrices.
Contravariant components transform with the Jacobian in the designated direction; covariant components transform with the Jacobian in the reversed direction.
The two Jacobians are constructed to be exact matrix inverses of one another, ensuring correct cancellation in contractions.
Maintaining the same directional choice throughout a calculation, including across chained coordinate changes, is required for consistent results.