2.15.2 Tensor Finite Coordinate System

The Tensor Finite Coordinate System offers a structured way to handle tensors in finite spaces, key for algebraic operations.

Tensor Finite Coordinate System is a labeling scheme that assigns to each vector, or to each point of a finite-dimensional space, an ordered n-tuple of real or complex numbers, built from a chosen finite basis or from a chosen local parametrization, so that tensor equations can be written and computed as relations among finitely many coordinate functions and their finitely many partial derivatives. It generalizes the fixed linear basis of a vector space to the broader setting of curved or parametrized finite-dimensional spaces, while retaining the essential finiteness that makes explicit index computation possible.

From Linear Basis to Coordinate System

Linear Coordinates on a Vector Space

The simplest tensor finite coordinate system arises directly from a basis e_1, ⋯, e_n of a vector space V: every vector v receives the coordinates (v^1, ⋯, v^n) defined by

v = \sum_{i = 1}^{n} v^{i} e_{i}

These are called linear coordinates because the coordinate functions v ↦ v^i are themselves linear functionals, namely the dual basis covectors.

Curvilinear and Local Coordinates

On a more general finite-dimensional manifold or parametrized region, a coordinate system is instead a map x = (x^1, ⋯, x^n) from an open region into F^n, not necessarily linear. At each point, the coordinate functions still produce a finite n-tuple, and at each point the partial derivatives ∂/∂x^i form a finite basis of the tangent space, so that the same finite index machinery used for linear coordinates applies pointwise, even though the coordinate functions themselves may vary nonlinearly from point to point.

Coordinate Bases Induced by a Coordinate System

Tangent Basis Vectors

A coordinate system x^1, ⋯, x^n induces, at each point, a natural basis for the tangent space consisting of the partial derivative operators ∂_1, ⋯, ∂_n, where ∂_i = ∂/∂x^i. This coordinate basis plays the same role that a fixed linear basis e_i plays for an ordinary vector space, and tensors can be expressed in components relative to it.

Coordinate Differentials as Dual Basis

The differentials dx^1, ⋯, dx^n of the coordinate functions form the dual basis to ∂_1, ⋯, ∂_n, satisfying dx^i(∂_j) = δ^i_j. A general tensor field can then be written in coordinate components as

T = T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} \partial_{i_{1}} \otimes \dots \otimes d x^{j_{q}}

with each index again ranging over the finite set 1 through n.

Change of Coordinates and the Jacobian

The Jacobian Matrix as a Finite Transition Matrix

When passing from one coordinate system x^1, ⋯, x^n to another x'^1, ⋯, x'^n, the coordinate bases are related by the Jacobian matrix, whose entries are the finitely many partial derivatives ∂x'^i / ∂x^j. Because the space is finite-dimensional, this matrix is an ordinary finite n × n array, and, where it is invertible, it plays exactly the role that a linear transition matrix plays for a fixed basis.

Tensor Transformation Law in Coordinates

Tensor components transform under a coordinate change by the same rule as under a linear change of basis, but with the Jacobian matrix and its inverse in place of a constant matrix:

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

using the Einstein summation convention over the finite index range. This finite sum of finitely many derivative terms is only well posed because both the old and new coordinate systems have exactly n components.

Why Finiteness Is Essential to the Coordinate Picture

Finite Degrees of Freedom

A tensor finite coordinate system presupposes that the space being coordinatized has finite dimension n, so that exactly n independent coordinate functions are needed to locate a point or specify a vector, no more and no fewer. This finiteness is what guarantees the Jacobian is a square, invertible matrix rather than an infinite or ill-defined array of derivatives.

Computability of Tensor Expressions

Because every coordinate index runs over a finite range, expressions such as contractions, covariant derivatives, and coordinate transformations reduce to finite sums and finite matrix operations that can be evaluated explicitly at each point, which is precisely what allows tensor calculus to be carried out concretely in coordinates rather than remaining a purely abstract, coordinate-free formalism.

Diagrammatic Summary

The diagram represents two finite coordinate systems on the same underlying space, related by a Jacobian matrix built from finitely many partial derivatives, which carries tensor components from one coordinate description to the other.