11.16.2 Tensor Coordinate Change Contravariant Response

Tensor Coordinate Change Contravariant Response explains how tensor components transform under coordinate changes, preserving geometric meaning through specific rules.

Tensor Coordinate Change Contravariant Response is the pattern of adjustment exhibited by contravariant tensor components when the underlying coordinate system is replaced by another, characterized by the use of the inverse Jacobian, or inverse basis-change matrix, so that the geometric object the components describe remains unchanged despite the components themselves taking new numerical values.

Foundational Setting

Coordinates and Components

In a smooth coordinate system, a point is labeled by coordinates $x^{i}$ , and vector-like quantities are represented by components indexed with a superscript, such as $v^{i}$ . When coordinates are replaced by a new set ${\tilde{x}}^{i}$ , the components of any contravariant object must be recomputed to describe the same underlying vector in the new frame.

The Jacobian Matrix

The transformation between coordinate systems is captured locally by the Jacobian matrix of partial derivatives:

J_{i}^{j} = \frac{\partial {\tilde{x}}^{i}}{\partial x^{j}}

The Contravariant Transformation Law

Direct Statement

The contravariant response of tensor components under coordinate change is expressed by applying the Jacobian matrix directly to each upper index:

{\tilde{v}}^{i} = \sum_{j} J_{j}^{i} v^{j}

Why the Term "Contravariant" Applies

The label "contravariant" reflects that these components change in the opposite sense to the coordinate basis vectors. If the basis vectors are scaled down by a change of coordinates, the contravariant components scale up in compensation, so that the physical or geometric vector they jointly represent stays fixed.

Distinction from Covariant Response

Opposing Transformation Directions

Where covariant quantities, such as gradient components, transform using the same matrix that maps old basis vectors to new ones, contravariant quantities require the inverse of that matrix. This opposition is what allows a contraction between a covariant and a contravariant object to be invariant.

\sum_{i} ω_{i} v^{i} = \sum_{i} {\tilde{ω}}_{i} {\tilde{v}}^{i}

Visualizing the Response

Behavior Under Successive Coordinate Changes

Composition of Jacobians

When two coordinate changes are applied in sequence, the contravariant response composes through matrix multiplication of the individual Jacobians, matching the chain rule for partial derivatives:

J_{i}^{k} (total) = \sum_{j} J_{j}^{k} J_{i}^{j}

Consistency Requirement

This composition property ensures that the contravariant response is path-independent: transforming directly from an original coordinate system to a final one gives the same result as transforming through any sequence of intermediate coordinate systems.

Extension to Higher-Rank Tensors

Multiple Upper Indices

For a tensor with several contravariant indices, each index responds independently to the coordinate change, with one Jacobian factor supplied per upper index:

{\tilde{T}}^{i j} = \sum_{k, l} J_{k}^{i} J_{l}^{j} T^{k l}

Practical Significance

This rule underlies the transformation behavior of physical quantities represented as contravariant tensors, such as velocity fields and displacement vectors, guaranteeing that equations formulated with these tensors retain the same form regardless of the coordinate system chosen.

Summary of Key Traits

Defining Characteristics

Contravariant response uses the inverse-type Jacobian matrix rather than the basis-change matrix directly.
Components adjust oppositely to how basis vectors adjust, preserving the represented geometric object.
The response composes consistently across chained coordinate transformations.
Every contravariant index of a tensor of any rank follows this same rule independently.