1.4.5 Tensor Coordinate Description

Tensor Coordinate Description explains how tensors are expressed in coordinate systems, detailing their components and transformation rules within formal mathematics.

Tensor Coordinate Description is the specification of a tensor's components relative to the particular basis induced by a system of coordinates, x^1, ..., x^n, on the underlying space, using the coordinate basis vectors formed from partial derivatives along each coordinate direction and the coordinate basis covectors formed from the differentials of the coordinate functions themselves. Unlike a general description relative to an arbitrary, unspecified basis, a coordinate description is tied specifically to a labeled system of coordinates, and its transformation rule is expressed directly in terms of the partial derivatives relating one coordinate system to another.

The Coordinate Basis

Coordinate Basis Vectors

Given a system of coordinates x^1, ..., x^n, the coordinate basis vectors are defined as the partial derivative operators along each coordinate direction, conventionally written ∂_i or ∂/∂x^i.

\partial_{i} = \frac{\partial}{\partial x^{i}}

Coordinate Basis Covectors

The coordinate basis covectors, dual to the coordinate basis vectors, are defined as the differentials of the coordinate functions, conventionally written dx^i, satisfying dx^i(∂_j) equal to 1 when i = j and 0 otherwise.

d x^{i} (\partial_{j}) = δ_{j}^{i}

Expressing a Tensor in Coordinate Description

General Tensor Field

A tensor of type (p, q) is described relative to a coordinate system by expanding it in terms of the coordinate basis vectors and covectors, with coefficients the coordinate components.

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

The coefficients T^{i_1...i_p}_{j_1...j_q} are functions of the coordinates when the tensor varies from point to point, in which case the tensor is described not by a fixed array but by an array of functions of position, called a tensor field in coordinate description.

Transformation Between Coordinate Systems

The Jacobian Matrix

When switching from coordinates x^1, ..., x^n to a new coordinate system y^1, ..., y^n, related by functions y^k = y^k(x^1, ..., x^n), the coordinate basis vectors transform using the Jacobian matrix of partial derivatives.

{\tilde{\partial}}_{k} = \frac{\partial x^{i}}{\partial y^{k}} \partial_{i}

and the coordinate basis covectors transform using the inverse Jacobian.

d y^{l} = \frac{\partial y^{l}}{\partial x^{j}} d x^{j}

Coordinate Component Transformation

Combining these substitutions, the coordinate description of a tensor's components transforms according to a formula built entirely from the partial derivatives relating the two coordinate systems, playing exactly the role that the abstract change-of-basis matrix A plays for a general basis.

{\tilde{T}}_{l_{1} \dots}^{k^{1} \dots} = \frac{\partial y^{k^{1}}}{\partial x^{i^{1}}} \dots \frac{\partial x^{j_{1}}}{\partial y^{l_{1}}} \dots T_{j_{1} \dots}^{i^{1} \dots}

Distinction from a General Basis Description

Coordinate Bases Are a Special Case

Every coordinate basis is a basis in the general sense used throughout tensor algebra, but not every basis arises from a coordinate system: the coordinate basis vectors ∂_i satisfy a special compatibility condition, namely that they commute with one another, ∂_i∂_j = ∂_j∂_i, a property called holonomicity that a general, arbitrarily chosen basis need not satisfy.

Advantage of Coordinate Description

Because coordinate basis vectors and covectors are tied directly to a labeled coordinate system, coordinate description allows a tensor's components to be written explicitly as functions of position, T^i_j(x^1, ..., x^n), which is essential for describing how a tensor varies across a space, a role a purely algebraic, point-independent basis description does not directly serve.

Diagrammatic Summary

The diagram represents the coordinate basis vectors as the tangent directions along the curves traced by varying one coordinate while holding the others fixed, illustrating how the coordinate description ties each basis vector directly to the underlying system of coordinates.