TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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Coordinate differentials form the dual cotangent basis

Statement

If $(U, x)$ is a smooth chart around $p$ , then the covectors $d x_{p}^{1}, \dots, d x_{p}^{n}$ form the basis of $T_{p}^{*} M$ dual to the tangent basis $\partial_{x^{1}} ∣_{p}, \dots, \partial_{x^{n}} ∣_{p}$ .

Facts & Assumptions

Given: A smooth chart $(U, x)$ around $p$ .

[F1]

For a smooth real-valued function $f$ , the differential is the functional $d f_{p} (v) = v ([f])$ (The differential of a smooth real-valued function).

[L1]

The coordinate derivations form a basis of $T_{p} M$ (Coordinate derivations form a basis of the tangent space).

Proof

technique · direct

1.1

By [F1], one has $d x_{p}^{i} (\partial_{x^{j}} ∣_{p}) = \partial_{x^{j}} ∣_{p} ([x^{i}]) = δ_{i j}$ .

F1given

2.1

Thus $d x_{p}^{1}, \dots, d x_{p}^{n}$ are dual to the basis from [L1].

L1step 1.1

3.1

Therefore they form a basis of $T_{p}^{*} M$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)