9.16.3 Tensor Basis Dependent Component Transformation

Tensor Basis Dependent Component Transformation is the formal law specifying exactly how the numerical components of a tensor must change under an arbitrary change of basis, expressed as a fixed rule determined entirely by the tensor's type. It is the precise mathematical statement underlying the more general behavior by which components depend on the choice of basis.

Statement of the Transformation Law

The General Transformation Formula

For a tensor of type (p, q), the transformation law states that each new component is obtained from the old components by applying the inverse transformation matrix once for every contravariant index and the transformation matrix once for every covariant index, summed appropriately over all old index values.

Determined Solely by Tensor Type

The transformation law depends only on the number of upper and lower indices carried by the tensor, not on the specific numerical values of its components. Two different tensors of the same type obey exactly the same transformation law, differing only in the specific numbers being transformed.

Derivation of the Law

Consequence of Basis Vector Transformation

The transformation law for components is not an independent postulate; it follows necessarily from the transformation rule for basis vectors, combined with the requirement that the tensor's summation form must yield the same tensor regardless of which basis is used.

Role of the Dual Basis Transformation

Because the dual basis must transform in the way that preserves the Kronecker pairing with the new basis vectors, the transformation law for covariant components is fixed to match the transformation of the dual basis, while the law for contravariant components is fixed to match the transformation of the basis vectors inversely.

Verifying the Law

Substitution into the Summation Form

The correctness of the transformation law can be confirmed by substituting it, together with the transformation of the basis vectors and dual basis covectors, into the full summation form and checking that all transformation factors cancel, leaving the original tensor unchanged.

Special Case of Scalars

Applying the transformation law to a tensor with no indices at all, a scalar, yields no transformation whatsoever, correctly reflecting that scalars do not depend on the choice of basis, which serves as a basic consistency check on the general law.

Applying the Law in Practice

Tensors of Low Order

For vectors and covectors, the transformation law reduces to a single matrix or inverse matrix multiplication applied to the entire component array at once, making these the simplest cases to which the general law applies.

Tensors of Higher Order

For tensors with several indices, the law requires applying the transformation matrix or its inverse separately to each index, effectively transforming the component array along each of its index directions in turn, consistent with treating the array as a multidimensional generalization of a matrix.

Significance of the Law

A Universal Rule Across All Applications

Because the transformation law is fixed by tensor type alone, it applies uniformly across every context in which tensors appear, from abstract algebra to physical and geometric applications, without needing to be re-derived for each new setting.

Criterion for Recognizing a Tensor

Conversely, an array of numbers that fails to obey this transformation law under a change of basis does not qualify as the components of a genuine tensor of the corresponding type, making the transformation law a defining criterion as much as a consequence of tensorial structure.