10.19.5 Tensor Non Tensorial Distinction from Tensor Rule

The Tensor Non Tensorial Distinction from Tensor Rule explains how objects transform under coordinate changes, defining tensors versus non-tensorial entities.

Tensor Non Tensorial Distinction from Tensor Rule is the direct, side-by-side comparison between the general tensorial transformation rule and the non-tensorial transformation pattern, laid out criterion by criterion so that the two can be told apart reliably whenever an unfamiliar indexed quantity's transformation behavior is being examined.

Comparing the Defining Formulas

The Tensorial Rule in Isolation

The tensorial rule consists purely of a product of Jacobian-type factors, one per index, multiplying the original component, with nothing added on:

{\bar{T}}_{l}^{k} = \sum_{i}^{n} \sum_{j}^{n} J_{i}^{k} {(J^{- 1})}_{l}^{j} T_{j}^{i}

The Non-Tensorial Pattern by Contrast

The non-tensorial pattern contains this same multiplicative structure as one summand, plus an additional summand built from second derivatives that has no counterpart in the tensorial rule:

{\bar{Γ}}_{jk}^{i} = (tensorial-rule part) + (extra second-derivative term)

so the structural distinction reduces to whether the formula is purely multiplicative in the Jacobian factors, or additive with an extra piece included.

Comparing Behavior Under Contraction

Tensorial Rule Guarantees Cancellation

Under the tensorial rule, any full contraction between an upper and a lower index collapses to the Kronecker delta by the Jacobian product identity, guaranteeing the object invariance signal described elsewhere: fully contracted scalars agree across every coordinate system.

Non-Tensorial Pattern Breaks Cancellation

Under the non-tensorial pattern, the extra term generally has no partner available to cancel it in a contraction, so scalars formed by contracting a non-tensorial quantity typically fail to agree across coordinate systems, marking the absence of the object invariance signal as a diagnostic distinguishing the two cases.

Comparing Coordinate Dependence at a Point

Tensorial Rule: Coordinate-Independent Vanishing

A genuine tensor's vanishing at a point is a coordinate-independent fact, since the purely multiplicative tensorial rule maps zero to zero under any change of basis.

Non-Tensorial Pattern: Coordinate-Dependent Vanishing

A non-tensorial quantity's vanishing at a point can be created or destroyed by an appropriate local choice of coordinates, since the additive extra term can shift a zero value to non-zero or vice versa, a distinction already detailed as non-tensorial coordinate dependence.

Side-by-Side Diagram

Direct Comparison Table

Comparing Behavior Under Linear Transformations

Both Patterns Can Look Alike

Restricted to purely linear changes of basis, the extra term in the non-tensorial pattern vanishes identically, since linear maps have zero second derivatives, so a non-tensorial quantity can pass every check available under linear transformations alone and be mistaken for a genuine tensor; only testing against a genuinely curvilinear change of basis reveals the distinction reliably.

Necessity of a Curvilinear Test

Because of this coincidence under linear transformations, definitively distinguishing the tensorial rule from the non-tensorial pattern for an unfamiliar quantity requires testing its transformation behavior under at least one non-linear, curvilinear change of coordinates, since a linear-only test is structurally incapable of exposing the extra term even when one is present.

Practical Summary of the Distinction

Quick Decision Guide

Given an indexed quantity of unknown status, checking whether its known transformation law is purely built from Jacobian and inverse Jacobian factors, one per index and nothing more, is the fastest available test: an exact match with this pattern indicates the tensorial rule applies, while any additional surviving term, however small it may appear, indicates the non-tensorial pattern and rules out treating the quantity as a genuine tensor.