1.6.2 Coordinate Transformation Invariance

Coordinate Transformation Invariance ensures mathematical properties remain consistent across different coordinate systems through tensor algebra and transformation rules.

Coordinate Transformation Invariance is the property of tensor equations and tensor fields defined on a manifold that their content remains valid and their form remains unchanged under an arbitrary smooth change of coordinates, extending the linear basis-change invariance of tensors on a fixed vector space to the setting in which the transformation itself may vary continuously from point to point, as described by the Jacobian matrix of the coordinate map.

From Linear Basis Change to Coordinate Transformation

Coordinates Versus Bases

On a manifold, a coordinate system assigns n real numbers to each point in a region, and at every point it induces a basis of the tangent space, formed by the partial derivative operators associated with the coordinates. A change of coordinates therefore induces a change of basis at every point, but unlike a fixed linear change of basis on a single vector space, this induced change can differ from point to point.

The Jacobian Matrix as the Local Change-of-Basis Matrix

Given coordinates x^i and new coordinates x′^i, both expressed as functions of one another, the Jacobian matrix of partial derivatives plays the role that the constant matrix A played for linear basis changes, except that its entries are themselves functions of position.

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

The Tensor Transformation Law Under General Coordinates

Components of a Tensor Field

A tensor field of type (p, q) assigns a tensor of that type to the tangent space at every point of the manifold, and in a coordinate system its components are n^{p+q} functions of the coordinates rather than constant numbers.

The Pointwise Transformation Rule

Under a change of coordinates, the components of a tensor field transform at each point using the Jacobian matrix and its inverse evaluated at that point, applied to the upper and lower indices exactly as in the linear case, but now varying smoothly across the manifold.

T_{' l_{1} \dots}^{' k^{1} \dots} (x^{'}) = \frac{\partial x^{'} k^{1}}{\partial x^{i}} \dots \frac{\partial x^{j}}{\partial x^{'} l_{1}} \dots T_{j_{1} \dots}^{i^{1} \dots} (x)

Consistency at Each Point

Because the Jacobian is evaluated locally, the transformation law is applied independently at each point of the manifold, which is what allows coordinate systems to be related by transformations that are nonlinear globally while remaining linear, through the Jacobian, at the level of each individual tangent space.

Invariance of Tensor Equations

Form Invariance

A relation between tensor fields, such as one tensor field being equal to another, or a tensor field satisfying a differential equation built from tensor operations, is called coordinate transformation invariant, or generally covariant, if the relation holds in one coordinate system whenever it holds in any other, with the same symbolic form in each.

S_{j}^{i} = T_{j}^{i} \Rightarrow S_{'} j i = T_{'} j i

Why Non-Tensorial Quantities Break This Property

An array of functions that does not obey the tensor transformation law, such as an ordinary partial derivative of a vector field's components, generally fails to transform consistently under a nonlinear change of coordinates, so equations built from such quantities can hold in one coordinate system and fail in another, in contrast to genuinely tensorial equations.

Curvilinear Coordinates as the Motivating Case

Cartesian to Curvilinear

The transition from Cartesian coordinates to curvilinear coordinates, such as polar, cylindrical, or spherical coordinates, is the paradigmatic example of a coordinate transformation whose Jacobian varies from point to point, since the local relationship between the two coordinate grids changes with position.

Field Equations Written in Any Coordinate System

Because tensor fields expressed in curvilinear coordinates obey coordinate transformation invariance, physical field equations, differential geometric identities, and other relations formulated with tensors can be written once and then evaluated in whichever coordinate system, Cartesian, polar, or otherwise, is most convenient for a given problem, without altering their content.

Diagrammatic Summary

The diagram shows coordinate transformation invariance as the guarantee that a tensor equation established in one coordinate system, connected to any other by a position-dependent Jacobian, continues to hold in that other system with the same form.