10.18 Tensorial Transformation Rule

Tensorial Transformation Rule is the general defining law specifying how the components of a tensor of arbitrary rank must transform under any admissible change of basis, expressed as a single unified formula in which every upper index of the tensor contributes one factor of the forward Jacobian matrix and every lower index contributes one factor of the inverse Jacobian matrix, and it is this rule, rather than any list of individual component values, that constitutes the actual mathematical definition of what it means for an object to be a tensor.

Statement of the General Rule

The Full Formula

For a tensor with $$p$$ upper indices and $$q$$ lower indices, the tensorial transformation rule reads:

One Factor Per Index

The rule assigns exactly one Jacobian-type factor to each index of the tensor, forward Jacobian for each upper index and inverse Jacobian for each lower index, with the total number of factors equal to the total rank of the tensor, and every summed index runs independently over all coordinate directions.

The Rule as the Definition of a Tensor

Tensors Defined by How They Transform

An array of numbers indexed by several labels is called a tensor precisely when it obeys this transformation rule under every admissible change of basis; an array that fails to obey the rule, changing according to some other law when the basis changes, is not a tensor even if it happens to carry the same number of indices arranged in the same pattern.

Rank-Zero and Rank-One Special Cases

Applying the general rule with zero indices produces the transformation law for a scalar, which is simply invariance:

and applying it with a single upper or single lower index recovers the ordinary contravariant or covariant vector transformation laws as the simplest non-trivial instances of the same general rule.

Properties Guaranteed by the Rule

Preservation of Linear Combinations

Because the rule is linear in the tensor's components, a linear combination of two tensors of the same rank and index pattern transforms component-wise according to the identical rule applied to each tensor separately, so the sum of two tensors is again a tensor, and the tensorial transformation rule is what guarantees this closure property.

Preservation Under Contraction

If one upper and one lower index of a tensor obeying the rule are contracted together, summed against each other, the resulting object of lower rank automatically obeys the tensorial transformation rule appropriate to its remaining indices, since the forward and inverse Jacobian factors attached to the contracted pair cancel through the Jacobian product identity, leaving the correct rule for the surviving indices.

Diagram of the Rule's Structure

Index-by-Index Factor Assignment

Verifying Tensorial Character

The Quotient Rule Test

A practical test for whether an unknown array is a genuine tensor uses the quotient rule: if contracting the array against every possible tensor of a given type always produces a tensor of the expected resulting rank, the array itself must obey the tensorial transformation rule, providing an indirect but reliable check when a direct transformation computation is inconvenient.

Common Non-Tensor Arrays

Arrays such as the Christoffel symbols used in covariant differentiation carry indices in the same visual pattern as a tensor but do not obey the tensorial transformation rule, instead picking up an additional inhomogeneous term involving second derivatives of the transformation map, which is precisely why they are described as transformation coefficients rather than as the components of a tensor.

Applicability Across Coordinate Systems

Validity for Any Admissible Change of Basis

The tensorial transformation rule applies uniformly to any pair of coordinate charts related by a sufficiently smooth, invertible transformation map, whether the change of basis is linear or curvilinear, and whether it connects two charts on a flat space or two charts on a general curved manifold, making the rule the single unifying criterion underlying every specific instance of tensor component transformation discussed elsewhere in the study of tensor change of basis.