7.3.2 Tensor Component Coordinate Dependence

Tensor component values change with coordinate systems, reflecting their dependence on the chosen basis and transformation rules in tensor algebra.

Tensor Component Coordinate Dependence is the property by which the numerical values stored in a tensor's component array change when the coordinate system used to describe the underlying vector space changes, even though the tensor itself, as an abstract multilinear object, remains the same. It captures the precise rule linking components in one coordinate system to components in another, so that the same geometric or physical object can be represented consistently across different frames.

Definition and Scope

The Transformation Law

For a tensor of type ((p,q)), a change of basis described by a matrix (A) and its inverse (A^{-1}) transforms components according to

T_{j}^{'} = A_{k}^{i} {(A^{- 1})}_{j}^{l} T_{l}^{k}

with one factor of (A) applied for each upper index and one factor of (A^{-1}) applied for each lower index, so that upper and lower indices transform in mutually compensating ways.

Why Dependence Exists

Coordinate dependence arises because a component is, by definition, the coefficient of a basis vector or covector in an expansion of the tensor. Since the basis vectors themselves change under a coordinate change, the coefficients needed to reproduce the same underlying object must change correspondingly.

Structural Properties

Contravariant Versus Covariant Behavior

The direction in which components change distinguishes upper indices from lower ones. Contravariant components, carrying upper indices, transform with the same matrix used to relate the new basis vectors to the old, while covariant components, carrying lower indices, transform with the inverse of that matrix, so that the two types of index scale oppositely under a given coordinate change.

Invariants Untouched by Coordinate Dependence

Certain scalar quantities built from a tensor's components are constructed precisely so that the coordinate dependence of the individual entries cancels out. The trace of a ((1,1)) tensor, and more generally full contractions pairing every upper index with a lower index, produce a number that is the same in every coordinate system, even though the individual components entering the sum are different in each system.

trace (T) = \sum_{i = 1}^{n} T_{i}^{i}

Special Coordinate Systems

Coordinate dependence becomes especially visible when comparing Cartesian, curvilinear, and rotated frames. The same physical quantity, such as a stress or a metric, can have a simple diagonal component structure in one coordinate system suited to the geometry of the problem, and a dense, fully populated component structure in another, less well-adapted system, without any change to the underlying tensor.

Role Within Tensor Algebra

Defining What Counts as a Tensor

Coordinate dependence following the specific transformation law above is, in fact, the defining criterion that separates a tensor from an arbitrary indexed array of numbers: an object is a tensor precisely when its components obey this rule under every admissible change of coordinates.

Practical Consequences for Computation

Recognizing the coordinate dependence of a set of components is necessary before comparing, adding, or otherwise combining tensors evaluated in different frames; two component arrays cannot be combined directly unless they are first expressed in a common coordinate system, since a naive entry-by-entry combination of components from different frames does not correspond to any well-defined operation on the underlying tensors.