10.16.4 Tensor Passive Transformation Component Change

Tensor Passive Transformation Component Change explains how tensor components adjust under coordinate changes while maintaining their physical meaning.

Tensor Passive Transformation Component Change is the precise rule, within the passive interpretation, that specifies how much and in what direction a tensor's numerical components shift when the coordinate basis is replaced, expressed as a contragredient relationship in which components move oppositely to the basis vectors so that the fixed underlying tensor is preserved.

The Contragredient Principle

Opposite Direction of Change

If the basis vectors are stretched by a certain factor along some direction, a contravariant component associated with that direction must shrink by the same factor, and if the basis vectors are compressed, the corresponding contravariant component must grow, a relationship summarized as components changing contragrediently, meaning oppositely, to the basis:

{\bar{e}}_{j} = \sum_{i}^{n} J_{i}^{j} e_{i} while {\bar{V}}^{j} = \sum_{i}^{n} {(J^{- 1})}_{j}^{i} V^{i}

Why Opposition Is Necessary

The opposition is not a matter of convention but a direct requirement of object preservation: since the fixed vector equals the sum of components times basis vectors in either coordinate system, any factor multiplying the basis vectors must be exactly canceled by the reciprocal factor multiplying the corresponding components, or the two sums would no longer represent the same vector.

Component Change for Covariant Quantities

Same Direction as the Basis

Covariant components change in the same direction as the primary basis vectors, since they are naturally associated with the dual basis, which itself changes oppositely, in the contragredient sense, to the primary basis:

{\bar{W}}_{j} = \sum_{i}^{n} {(J^{- 1})}_{j}^{i} W_{i}

If the primary basis vectors grow in magnitude, the dual basis vectors shrink to preserve their pairing relation with the primary basis, and covariant components, which are paired with the dual basis in the invariant sum, correspondingly change in step with the primary basis rather than against it.

Magnitude of the Component Change

Proportional to the Local Jacobian

The size of the component change at any point depends entirely on the local values of the Jacobian and inverse Jacobian entries at that point, so a coordinate change that stretches space heavily in some region produces a large component change there, while a region left nearly untouched by the coordinate change produces a component change close to zero:

\frac{{\bar{V}}^{j}}{V^{i}} \approx {(J^{- 1})}_{j}^{i}

when only a single dominant direction contributes, giving a rough sense of the component change as a local scale factor.

Compounding Across Multiple Indices

For a tensor with several indices, the component change compounds multiplicatively across all of them, so a rank-two tensor's components can change by roughly the square of a single-direction scale factor when both indices align with the same stretched or compressed direction, illustrating why higher-rank tensor components are generally more sensitive to a given coordinate change than vector components are.

Diagram of Opposing Changes

Basis Grows, Component Shrinks

Component Change and Physical Units

Analogy With Unit Conversion

A familiar instance of contragredient component change occurs in ordinary unit conversion: if the basis unit of length is enlarged from centimeters to meters, a fixed physical length requires a smaller numerical component to represent it, since the ratio between the physical length and the new, larger basis unit is smaller, illustrating the contravariant component change rule with an everyday example.

Density and Weighted Quantities

Quantities that behave like tensor densities experience a component change with an additional multiplicative factor built from the Jacobian determinant, on top of the ordinary per-index contragredient change, reflecting that such quantities carry information about volume or measure alongside their directional components, and so respond to a coordinate change through both mechanisms simultaneously.

Verifying a Component Change Computation

Cross-Check via Invariance

A completed component change computation can be checked by forming any available full contraction of the tensor with another tensor or covector and confirming the same scalar value results in both coordinate systems; if a discrepancy appears, it signals either an arithmetic error in computing the Jacobian entries or an incorrect assignment of which transformation matrix, forward or inverse, was applied to a given index.