10.7 Tensor Component Transformation Law

Tensor Component Transformation Law is the general rule specifying how the components of a tensor of arbitrary rank must change when the basis of the underlying vector space is changed, expressed as a product of one change-of-basis matrix factor for every index of the tensor, with contravariant indices contracted against the forward matrix and covariant indices contracted against its inverse. It is the master formula from which the transformation behavior of scalars, vectors, covectors, and higher-rank tensors is derived as special cases, and it functions as the defining criterion for what qualifies as a tensor in the first place.

Statement of the Law

General Form for a Mixed Tensor

For a tensor with a given number of upper indices and a given number of lower indices, the component transformation law states that the new-basis components are obtained by contracting the old-basis components with one factor of the forward matrix for every upper index and one factor of the inverse matrix for every lower index.

Special Cases Contained Within the Law

Setting the number of upper and lower indices to specific small values recovers the familiar transformation rules for scalars, which have no indices and hence no matrix factors, vectors, which have a single upper index and a single inverse matrix factor, and covectors, which have a single lower index and a single forward matrix factor.

Justification of the Law

Derivation From Invariance of the Underlying Object

The component transformation law is not an arbitrary postulate; it follows from demanding that the tensor, reconstructed by pairing components with basis vectors and dual basis covectors, remain the same object regardless of which basis is used. Requiring this invariance while allowing the basis vectors to transform under the forward matrix forces the components to transform under the pattern described by the law.

Uniqueness of the Transformation Pattern

Given the transformation of the basis vectors and dual basis covectors, the invariance requirement determines the component transformation law uniquely: no other assignment of forward and inverse matrix factors to the indices would preserve the reconstructed tensor across every possible change of basis.

The Law as a Defining Criterion

Tensors Versus Arbitrary Indexed Arrays

Not every array of numbers indexed by several labels obeys the component transformation law. The law serves as the precise test distinguishing genuine tensors, whose components transform according to this rule under every change of basis, from indexed quantities that merely happen to look similar in one particular basis but fail to transform consistently.

Verification Procedure

To verify that a given indexed quantity is a tensor, it suffices to check that its components under an arbitrary change of basis satisfy the component transformation law with the correct assignment of forward and inverse matrix factors to each index, based on whether that index is written as upper or lower.

Structural Features

Multiplicative Composition Under Successive Changes

Applying the component transformation law for a change from one basis to a second, and then again for a change from the second basis to a third, produces the same result as applying the law once for a direct change from the first basis to the third, with the intermediate matrices combining by ordinary matrix multiplication.

Index-by-Index Independence

Each index of a tensor transforms according to the component transformation law independently of the other indices, in the sense that the matrix factor attached to one index does not depend on whether other indices are upper or lower, even though all factors are applied simultaneously within the same formula.

Schematic Representation

The diagram represents the general component transformation law as a single arrow connecting an old-basis component array to a new-basis component array, with the transformation governed by as many matrix factors as the tensor has indices.