1.4.2 Coordinate Dependent Tensor Description

Coordinate Dependent Tensor Description explains how tensors transform with coordinates, crucial for physics and geometry.

Coordinate Dependent Tensor Description is the representation of a tensor as an explicit indexed array of numerical components, obtained once a specific basis of the underlying vector space has been chosen, together with the rule that specifies exactly how that array must change when a different basis is chosen instead. This description ties the abstract, basis-independent tensor to a concrete arithmetic object, an array of numbers, that can be directly computed with, compared, and displayed, at the price of requiring the transformation law to be tracked and respected whenever the reference basis is altered.

Producing a Coordinate Description

Fixing a Basis

Given a basis e_1, ..., e_n of V and its associated dual basis e^1, ..., e^n of V*, the coordinate description of a tensor T of type (p, q) is the array of scalars obtained by evaluating T, regarded as a multilinear map, on every combination of these basis elements.

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = T (e^{i^{1}}, \dots, e^{i^{p}}, e_{j_{1}}, \dots, e_{j_{q}})

Displaying the Array

For a tensor of rank up to two, the resulting array can be displayed as a list or as a rectangular grid resembling an ordinary matrix; for higher-rank tensors, the array is instead recorded symbolically through its indexed components, since it cannot be laid out conveniently on a flat page.

The Transformation Law as Part of the Description

Not a Standalone Array

A coordinate description is not fully specified by a single array of numbers alone; it must be accompanied by the specific basis relative to which the array was computed, and by the understanding that switching to a different basis requires applying the tensor's transformation law rather than leaving the array unchanged.

{\tilde{T}}_{l_{1} \dots l_{q}}^{k^{1} \dots k^{p}} = A_{i_{1}}^{k^{1}} \dots {(A^{- 1})}_{l_{q}}^{j^{q}} T_{j_{1} \dots}^{i^{1} \dots}

A Family of Related Arrays

Because the same tensor produces different arrays in different bases, the coordinate description is best understood not as a single fixed array but as an entire family of arrays, one for each admissible basis, all linked to one another by the transformation law, with any one member of the family sufficient to reconstruct every other member once the relevant change-of-basis matrix is known.

When the Coordinate Description Is Useful

Explicit Numerical Computation

Whenever a tensor equation must be evaluated to produce an actual number, such as computing the value of a bilinear form on two specific vectors, the coordinate description supplies the concrete components needed to carry out the arithmetic.

g (u, v) = g_{i j} u^{i} v^{j}

Computer Representation

Numerical software and symbolic computation systems represent tensors internally as coordinate arrays, since abstract multilinear maps are not directly storable as data; every computational implementation of tensor algebra necessarily works with some coordinate description, tied to whatever basis the implementation adopts.

Comparison Against a Preferred Frame

In applications where a particular basis carries special physical or geometric significance, such as a rest frame or a principal axis frame, expressing a tensor's coordinate description relative to that specific basis reveals information, such as diagonal structure or vanishing components, that may not be apparent in an arbitrary basis.

Risks of the Coordinate Description

Mistaking Basis-Dependent Values for Invariant Facts

A common source of error when working with coordinate descriptions is treating a numerical fact about the components in one particular basis, such as a component being zero or two components being equal, as though it were true in every basis, when in fact the equality or vanishing may hold only accidentally in the chosen frame.

The Need to Verify Consistency

Whenever an equation is written entirely in terms of coordinate components, its validity as a statement about the underlying tensors, rather than as an artifact of the chosen basis, must be checked by confirming that both sides transform identically under a change of basis, restoring exactly the guarantee that the coordinate-independent description provides automatically.

Diagrammatic Summary

The diagram shows that a coordinate description of a tensor is inseparable from the basis used to produce it, with the array of components meaningful only in reference to that basis and requiring the transformation law to be applied when the basis is changed.