11.6.4 Tensor Contravariant Object Coordinate Dependence

Tensor Contravariant Object Coordinate Dependence is the precise characterization of which aspects of a contravariant object change when the coordinate system is changed and which aspects remain fixed, separating the coordinate-dependent numerical component array from the coordinate-independent underlying vector-space element that those components describe.

What Depends on the Coordinate System

The Numerical Component Values

The specific numbers making up a contravariant object's component array are entirely coordinate dependent, changing according to the direct-Jacobian-factor transformation rule whenever the coordinate system is changed, so no single component value can be quoted as meaningful without also specifying the coordinate system it belongs to.

The Associated Coordinate Basis

The specific ordinary basis vectors used to expand a contravariant object also depend on the coordinate system, since they are constructed directly from partial derivatives of position with respect to whichever coordinates are in use, changing along with the coordinates themselves.

What Remains Independent of the Coordinate System

The Underlying Vector-Space Element

The contravariant object itself, understood as a specific element of the vector space or tangent space, does not depend on the coordinate system at all; it is the same geometric direction and magnitude regardless of which coordinates are used to describe it numerically.

Any Full Contraction Against a Covariant Object

Because pairing a contravariant object with a fixed covariant object produces a scalar, and scalars are coordinate independent, the numerical result of this pairing remains fixed, even though the component arrays of both the contravariant object and the covariant object individually depend on the coordinate system.

The Precise Nature of the Dependence

A Linear, Not Arbitrary, Dependence

The coordinate dependence of a contravariant object's components is not an arbitrary distortion but a precisely linear relationship governed by the direct Jacobian factor, meaning the dependence is fully predictable once the coordinate transformation relating the two systems is known.

Local Dependence at a Single Point

For a contravariant tensor field defined throughout a region, the coordinate dependence of its components at any one point involves only the Jacobian factor evaluated at that same point, not at neighboring points, so the coordinate dependence discussed here is a pointwise phenomenon rather than one requiring information about the field's behavior nearby.

Consequences of Recognizing This Dependence

Avoiding Misinterpretation of Raw Component Values

Recognizing that component values alone are coordinate dependent prevents the common error of comparing or combining contravariant components taken from calculations performed in two different, unstated coordinate systems, since such a comparison is meaningless without first transforming both sets of components into a common coordinate system.

Justifying Coordinate-Free Statements of Physical Law

Because the underlying vector-space element and any full contraction against a covariant object remain coordinate independent, physical laws expressed in terms of a full contraction involving a contravariant object, such as work computed from force and displacement, can be stated without reference to any particular coordinate system, relying on this coordinate dependence analysis to guarantee the statement's validity in every admissible coordinate system.