10.19 Tensor Non Tensorial Transformation Pattern

Tensor Non Tensorial Transformation Pattern is the recognizable way that certain indexed quantities, despite carrying upper and lower indices arranged in a manner visually resembling a tensor, actually transform under a change of basis by picking up one or more additional terms beyond the standard tensorial rule, most often an inhomogeneous term built from second derivatives of the transformation map, marking such a quantity as not being a genuine tensor.

Recognizing the Pattern

The Extra Inhomogeneous Term

A quantity following the non-tensorial pattern transforms according to a formula that includes the ordinary tensorial rule terms plus at least one additional term that does not fit the standard index-factor correspondence:

using the transformation of the Christoffel symbols as the standard example: the first sum reproduces the ordinary tensorial rule with one forward and two inverse Jacobian factors, while the second sum is the extra term involving second derivatives, absent from any genuine tensorial transformation.

Second Derivatives as the Telltale Sign

The presence of second, rather than only first, derivatives of the transformation map in a transformation formula is a strong signal of the non-tensorial pattern, since the ordinary tensorial rule is built entirely from first-derivative Jacobian factors, and any dependence on curvature of the coordinate map itself, rather than merely its local linear approximation, points to an inhomogeneous, non-tensorial transformation law.

Why the Pattern Arises

Correcting for a Varying Basis

Quantities following this pattern typically exist specifically to correct for the fact that the coordinate basis vectors change from point to point in a curvilinear coordinate system, and the extra term in their transformation law is precisely the piece needed to compensate for the second-order variation of the basis itself, a compensation that a genuine tensor, transforming purely multiplicatively, does not need to make.

Vanishing in Special Cases

The extra inhomogeneous term vanishes identically whenever the coordinate transformation is linear, since a linear map has zero second derivatives everywhere, which is why a quantity following the non-tensorial pattern can appear to transform tensorially when only linear changes of basis are considered, revealing its true non-tensorial character only once a genuinely curvilinear transformation is tested.

Consequences of the Pattern

Loss of Coordinate-Independent Meaning

Because a non-tensorial quantity's value depends on more than just the point and the tensor itself, a single component of such a quantity cannot be assigned any coordinate-independent geometric meaning on its own, unlike a tensor component, whose full collection of components across all bases represents one single coordinate-independent object.

Combinations That Restore Tensorial Behavior

Certain combinations built from non-tensorial quantities can cancel the extra inhomogeneous terms and produce a genuine tensor, such as the difference of two connections sharing the same inhomogeneous correction term, or the curvature tensor built from derivatives and products of Christoffel symbols, illustrating that the non-tensorial pattern in an individual quantity does not prevent tensorial quantities from being constructed out of it.

Diagram of the Pattern

Tensorial Part Plus Extra Term

Testing Whether a Quantity Follows the Pattern

Direct Transformation Check

The most reliable test is deriving the transformation law of the quantity directly from its defining formula, using the chain rule on every derivative it involves, and checking whether any term survives beyond those matching the standard tensorial index-factor correspondence; if an extra term remains, the quantity follows the non-tensorial pattern.

Behavior Under a Single Special Transformation

A quicker, though less conclusive, check applies a single curvilinear coordinate change, such as a transformation to polar-type coordinates, and observes whether the transformed quantity contains terms that cannot be accounted for by ordinary Jacobian and inverse Jacobian factors alone, since a genuine tensor never produces such extra terms under any admissible coordinate change, curvilinear or otherwise.

Common Examples Following This Pattern

Connection Coefficients

The Christoffel symbols used to define covariant differentiation are the most commonly cited example of the non-tensorial transformation pattern, since their entire purpose, correcting ordinary partial derivatives so that the result of differentiating a tensor is itself a tensor, requires them to absorb exactly the inhomogeneous, second-derivative-dependent piece that would otherwise spoil the tensorial character of a naively differentiated tensor field.