9 Tensor Bases and Coordinates

Tensor Bases and Coordinates is the body of concepts concerning how a vector space's basis and a manifold's coordinate system provide the concrete scaffolding — basis vectors, dual basis covectors, and coordinate functions — against which abstract tensors are expressed as component arrays, together with the rules governing how that scaffolding may be chosen, changed, and related to genuinely coordinate-independent tensorial content. It is the foundational layer beneath tensor index notation: before indices can label components at all, a basis or coordinate system must be established, and understanding how that establishment works, and how it can vary, is prerequisite to everything built on top of it.

Basis Vectors and the Dual Basis

A Basis Fixes How Vectors Are Expressed as Numbers

A basis {e₁, ..., eₙ} of an n-dimensional vector space V is a linearly independent spanning set, and every vector v ∈ V is expressed uniquely as v = vⁱeᵢ (summed), with the coefficients vⁱ constituting the vector's components in that basis. Choosing a different basis produces a different set of coefficients for the same vector, related to the first by the standard change-of-basis transformation.

The Dual Basis Pairs With the Basis to Extract Components

Given a basis {eᵢ} of V, the dual basis {eⁱ} of the dual space V* is the unique basis satisfying eⁱ(eⱼ) = δⁱⱼ; it is what allows a vector's components to be recovered by direct evaluation, vⁱ = eⁱ(v), rather than by any indirect computation, and it is the object that ultimately licenses writing covector components with a lower index while vector components carry an upper one.

Coordinate Systems on a Manifold

Coordinate Functions Assign Numbers to Points

A coordinate system on a manifold is a collection of functions x¹, ..., xⁿ assigning n real numbers to each point in some region, providing a local identification of that region with a portion of ℝⁿ. At each point, the partial derivative operators ∂/∂xⁱ form a natural basis of the tangent space at that point, called the coordinate basis, tying the manifold's coordinate functions directly to a basis of vectors usable for tensor components.

Coordinate Bases Versus General Bases

A basis of vectors at a point need not arise from any coordinate system at all; an arbitrary linearly independent set of tangent vectors, not necessarily equal to any ∂/∂xⁱ, is still a perfectly valid basis for expressing tensor components, though it is then called a non-coordinate (or anholonomic) basis. The distinction matters because coordinate bases automatically satisfy certain commutation properties between basis vectors that a general, non-coordinate basis need not satisfy, a fact that affects formulas involving derivatives of tensor fields.

Changing Basis and Changing Coordinates

The Jacobian as the Bridge Between Two Coordinate Systems

Given two coordinate systems xⁱ and x̄ⁱ covering an overlapping region, the Jacobian matrix ∂x̄ⁱ/∂xʲ and its inverse relate the two associated coordinate bases and dictate exactly how tensor components must transform to describe the same underlying tensor in both systems:

This transformation rule is the single fact from which the entire distinction between upper (contravariant) and lower (covariant) index behavior is derived.

Curvilinear Versus Cartesian Coordinates

Coordinate systems whose basis vectors have constant length and orientation everywhere, such as standard Cartesian coordinates on flat space, are especially convenient because the coordinate basis coincides everywhere with a single fixed orthonormal frame; curvilinear coordinate systems, such as polar, cylindrical, or spherical coordinates, instead have basis vectors that vary in direction, length, or both from point to point, requiring the position-dependence of the basis itself to be tracked when differentiating tensor fields expressed in such coordinates.

Diagram of Basis Vectors Varying With Coordinates

The Metric's Relationship to the Chosen Basis

Basis-Dependent Numerical Form of an Invariant Structure

The metric tensor, though itself a coordinate-independent object, takes different numerical component forms in different bases: it equals the identity matrix in an orthonormal Cartesian frame but takes a nontrivial, position-dependent form in curvilinear coordinates, encoding exactly how lengths and angles computed from raw coordinate differences must be corrected in a non-Cartesian system.

Orthonormal Bases as a Simplifying Special Case

An orthonormal basis, in which the metric takes the identity form, is what causes upper and lower components of the same tensor to coincide numerically, which is why elementary treatments of vectors and tensors, restricted to Cartesian coordinates, often do not need to distinguish upper from lower indices at all — a simplification that ceases to hold as soon as a curvilinear or otherwise non-orthonormal basis or coordinate system is introduced.