4.22.4 Tensor Multilinear Coordinate Dependence

Tensor Multilinear Coordinate Dependence is the sensitivity of every explicit representation of a multilinear map, its component array, its matrix or hypermatrix, to the specific coordinate system or basis chosen for the underlying vector spaces, extending beyond simple linear changes of basis to the more general setting of curvilinear coordinates on manifolds, where the transformation between coordinate systems is governed by a Jacobian matrix rather than a single fixed linear map.

Coordinate Dependence at the Level of a Vector Space

Linear Change of Basis

For a fixed vector space V, changing basis via an invertible matrix P, so that new coordinates v' relate to old coordinates v by v = Pv', transforms the component array of a multilinear map by contracting each input index against P (or its transpose, depending on convention) and each output index against P⁻¹; this is the coordinate dependence already encountered for bilinear forms (congruence) and for general multilinear maps of any arity, one contraction per index.

Why the Map Itself Is Unaffected

The underlying multilinear map does not change when the basis changes; only the numerical array representing it changes, in a precisely prescribed way, so that the value f(v,w) computed from the new coordinates and the new array exactly reproduces the value computed from the old coordinates and the old array.

Coordinate Dependence on Manifolds

Curvilinear Coordinates and the Jacobian

On a smooth manifold, a change from one coordinate chart x^i to another x'^i is a general (nonlinear) diffeomorphism, and the local linear approximation to this change, at any given point, is given by the Jacobian matrix ∂x'^i/∂x^j; multilinear objects defined pointwise on the manifold, such as the metric tensor or the curvature tensor, transform using this Jacobian in place of the fixed matrix P used for a plain change of basis on a single vector space.

Position-Dependent Transformation

Because the Jacobian generally varies from point to point (unless the coordinate change happens to be affine), the coordinate-dependence of a multilinear tensor field on a manifold is itself a function of position: the same tensor field is represented by different component arrays at different points even within a single choice of coordinate chart, and by different arrays again under a change of chart, layering two distinct sources of coordinate dependence, position within one chart and choice of chart, on top of each other.

Covariant Derivative as a Correction Term

Because ordinary partial differentiation of tensor components does not, by itself, transform correctly under a change of curvilinear coordinates (Jacobian derivatives introduce extra terms), a covariant derivative, incorporating connection coefficients that absorb exactly these extra terms, is required to differentiate a tensor field while preserving its coordinate-independent meaning; this is a direct consequence of the more intricate coordinate dependence present once curvilinear, rather than merely linear, coordinate changes are considered.

Distinguishing Genuine Invariants From Coordinate Artifacts

Invariant Quantities

Certain quantities computed from a multilinear map's component array remain the same regardless of the coordinate system used to compute them: the rank of a bilinear form's matrix, the trace of a type (1,1) tensor, the value of a fully contracted scalar built from a multilinear map and matching dual vectors, all such quantities are coordinate-independent precisely because the transformation rules for the indices involved cancel out in the final combination.

Coordinate-Dependent Artifacts

Other quantities, an individual entry T_{23} of a component array, the specific numerical value of one component in one particular basis, carry no independent meaning and change arbitrarily under a change of coordinates; recognizing which quantities are invariant and which are coordinate artifacts is essential before drawing any conclusion from a computation performed in a specific coordinate system.

Practical Handling of Coordinate Dependence

Choosing Convenient Coordinates

Since the underlying multilinear map does not depend on coordinates, coordinates can always be chosen for computational convenience, such as an orthonormal basis diagonalizing a symmetric bilinear form, or locally flat (geodesic normal) coordinates simplifying curvature computations at a point, without altering any coordinate-independent conclusion drawn afterward.

Verifying Coordinate-Independence of a Construction

When a new operation on multilinear maps is defined via an explicit formula in coordinates, confirming it is a legitimate coordinate-independent construction requires checking that the formula transforms correctly, that is, that applying the coordinate change to the inputs and then computing, or computing first and then applying the coordinate change to the output, give the same result; a construction failing this check is merely an artifact of the coordinate system used to state it, not a genuine operation on the underlying multilinear objects.

Contrast With Purely Algebraic Basis Dependence

Linear Versus General Coordinate Changes

The coordinate dependence encountered for a multilinear map on a fixed vector space is entirely governed by linear algebra, a single fixed change-of-basis matrix applied uniformly; the coordinate dependence encountered for a tensor field on a manifold additionally involves the calculus of the transition functions between charts, since the relevant "change of basis" (the Jacobian) itself varies from point to point. The manifold setting subsumes the vector-space setting as the special case where the coordinate change happens to be linear and the same at every point.

Why Both Cases Share the Same Underlying Principle

Despite this added complexity, the guiding principle is identical in both settings: the multilinear object itself is coordinate-independent, and only its explicit numerical representation depends on the coordinates chosen; recognizing this shared principle is what allows techniques developed for coordinate dependence in linear algebra, congruence transformations, index contraction rules, symmetric or alternating simplifications, to carry over, with appropriate modification for position-dependence, to the differential-geometric setting of tensor fields on manifolds.