10.15.5 Tensor Coordinate Transformation Component Effect

Understanding how tensor components transform under coordinate changes and its implications in tensor algebra.

Tensor Coordinate Transformation Component Effect is the concrete numerical change undergone by each individual tensor component as a result of the coordinate transformation process, describing how a specific component's value increases, decreases, mixes with other components, or stays fixed depending on the variance type of the index it carries and the particular Jacobian entries involved.

Effect on a Single Contravariant Component

Mixing With Other Original Components

A contravariant component in the target chart is generally not equal to any single original component but to a weighted sum of all of them, so the effect of the transformation is to blend the original components together according to the forward Jacobian entries:

{\bar{V}}^{j} = \sum_{i}^{n} J_{i}^{j} V^{i}

If the off-diagonal Jacobian entries are non-zero, the effect includes genuine mixing: the new component depends on original components associated with directions other than its own, an effect that is absent only when the change of basis leaves that particular direction unrotated relative to the others.

Effect of Scaling a Single Direction

When the coordinate change simply rescales one coordinate direction, leaving the others alone, the effect on the corresponding contravariant component is a pure multiplicative rescaling by the reciprocal of the same factor, since a longer unit of new coordinate along that direction corresponds to a smaller numerical component for a fixed physical vector.

Effect on a Single Covariant Component

Opposite Scaling Behavior

A covariant component experiences the opposite scaling effect compared to a contravariant component under the same coordinate change, since it is governed by the inverse Jacobian rather than the forward Jacobian:

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

If stretching a coordinate direction causes a contravariant component in that direction to shrink, the covariant component associated with that same direction grows by the reciprocal factor, which is exactly the effect needed to keep a full contraction between the two types of component numerically invariant.

Effect on Mixed Tensor Components

Compounded Effects Per Index

For a tensor with several indices, the component effect compounds: each upper index contributes its own contravariant-style scaling and mixing, and each lower index contributes its own covariant-style scaling and mixing, with the overall new component built as a product of contributions summed over every combination of original indices:

{\bar{T}}_{l}^{k} = \sum_{i}^{n} \sum_{j}^{n} J_{i}^{k} {(J^{- 1})}_{l}^{j} T_{j}^{i}

Effect Depends on Every Original Component

Because of this summation, the effect on a single new mixed-tensor component generally depends on every one of the original tensor's components, not merely on the one occupying the corresponding position, unless the Jacobian matrices involved happen to be diagonal.

Diagram of Component Mixing Effect

Before and After Comparison

Effect on Special Component Patterns

Zero Components Do Not Stay Zero

A component that vanishes in the source chart does not necessarily vanish in the target chart, since it may receive non-zero contributions from other original components through the Jacobian entries, so the property of having a zero component is generally not preserved by a coordinate transformation unless the vanishing reflects an intrinsic geometric feature such as the tensor being identically zero everywhere.

Symmetric and Antisymmetric Patterns

If a tensor is symmetric or antisymmetric in a pair of indices in the source chart, the same symmetry or antisymmetry pattern is preserved in every index pair transformed by the identical Jacobian factors in the target chart, since the summation defining the new components respects the underlying index symmetry of the original tensor, an effect that follows directly from swapping the summed dummy indices in the transformation formula.

Effect at Special Points

Behavior at a Fixed Point of the Map

At a point where the transformation map acts as the identity, meaning source and target coordinates coincide locally, the component effect vanishes entirely to first order, since the Jacobian at such a point reduces to the identity matrix, leaving every component numerically unchanged there even though it may change elsewhere in the domain.