10.19.1 Tensor Non Tensorial Extra Term Pattern

The Tensor Non Tensorial Extra Term Pattern reveals how non-tensorial components emerge in tensor algebra through specific structural expansions and transformations.

Tensor Non Tensorial Extra Term Pattern is the detailed internal structure of the additional, inhomogeneous term itself that appears in a non-tensorial transformation law, describing precisely how this extra piece is built from second derivatives of the transformation map combined with inverse Jacobian factors, and how its specific form determines the way it fails to cancel under composition and contraction.

Anatomy of the Extra Term

Second Derivative Core

At the core of the extra term sits a second partial derivative of the target coordinates with respect to two source coordinates, a quantity with no counterpart anywhere in the ordinary tensorial rule:

\frac{\partial^{2} {\bar{x}}^{i}}{\partial x^{q} \partial x^{k}}

This object carries three indices, one upper and two lower, yet it is not itself a tensor, since it measures the curvature of the coordinate map rather than a linear rate of change, and it vanishes identically only when the transformation map is linear in the source coordinates.

Attaching Jacobian Factors

The extra term in a full non-tensorial transformation law is this second derivative multiplied by exactly as many inverse Jacobian factors as needed to lower its free upper index to match the position required by the quantity being transformed:

\sum_{q}^{n} {(J^{- 1})}_{j}^{q} \frac{\partial^{2} {\bar{x}}^{i}}{\partial x^{q} \partial x^{k}}

so that the extra term as a whole carries the same free index pattern as the surrounding tensorial part of the formula, allowing it to be added directly to that part without an index mismatch.

Structural Reason the Extra Term Does Not Cancel

No Partner Available Under a Single Contraction

In the ordinary tensorial rule, a contraction between an upper-index factor and a lower-index factor cancels through the Jacobian product identity, but the second derivative at the core of the extra term has no corresponding inverse object anywhere in the formula that would cancel it in the same way, since there is no such thing as an "inverse second derivative" playing a role analogous to the inverse Jacobian.

Symmetric Index Structure

The extra term is symmetric in its two lower indices, since ordinary partial derivatives commute, meaning the order in which the two source-coordinate derivatives are taken does not matter, and this symmetry is inherited by any tensorial or non-tensorial quantity whose transformation law includes a term of this type, appearing as symmetry in the corresponding pair of indices of the quantity itself.

Cancellation in Special Combinations

Difference of Two Instances

If two non-tensorial quantities are known to carry the identical extra term as part of their individual transformation laws, their difference transforms without any extra term at all, since the two copies of the second-derivative piece cancel directly in the subtraction, leaving only the ordinary tensorial-rule contributions and confirming that the difference is a genuine tensor even though neither individual quantity is.

Antisymmetrization Removing the Symmetric Piece

Because the extra term is symmetric in a specific pair of lower indices, antisymmetrizing a non-tensorial quantity over that same pair of indices removes the extra term entirely, since antisymmetrizing a symmetric quantity produces zero, which is part of why an antisymmetrized combination built from otherwise non-tensorial connection coefficients yields a genuine tensor.

Diagram of the Extra Term's Composition

Building Blocks Assembled

Locating the Extra Term in a Full Transformation Law

Position Relative to the Tensorial Part

In a typical non-tensorial transformation law, the extra term is added to, rather than multiplied into, the ordinary tensorial-rule part built from Jacobian and inverse Jacobian factors alone, so a complete transformation law for such a quantity always has the recognizable two-part additive structure: one tensorial summand and one extra summand carrying the second-derivative structure.

Consistency With Composition

When such a transformation law is applied through two successive coordinate changes, the extra term picks up its own compatibility structure via the chain rule applied to second derivatives, ensuring that composing two such transformations produces a combined extra term consistent with performing the whole transformation in a single step directly between the initial and final coordinate systems.