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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Coordinate derivations are well-defined derivations

Statement

For every smooth chart $(U, x)$ containing $p$ , each coordinate operator $\partial_{i} ∣_{p}$ from Coordinate derivations at a point is a well-defined derivation at $p$ .

Facts & Assumptions

Given: A smooth chart $(U, x)$ containing $p$ and an index $i$ .

[F1]

Equal germs have equal representatives on some neighbourhood of $p$ (The germ of a smooth function at a point).

[F2]

The coordinate derivation is defined by differentiating a chart representative at the coordinate point $a = x (p)$ (Coordinate derivations at a point).

[F3]

Derivations are linear maps satisfying the Leibniz rule (Derivations at a point and the tangent space).

Proof

technique · direct

1.1

If $[f] = [g]$ , then $f \circ x^{- 1}$ and $g \circ x^{- 1}$ agree on a neighbourhood of $a$ , so their $i$ th partial derivatives at $a$ are equal; hence $\partial_{i} ∣_{p}$ is well defined by [F1] and [F2].

F1F2given

1.2

Linearity is immediate from linearity of partial differentiation, and the usual product rule for partial derivatives gives $\partial_{i} ∣_{p} ([f g]) = f (p) \partial_{i} ∣_{p} ([g]) + g (p) \partial_{i} ∣_{p} ([f])$ .

F2given

2.1

Thus $\partial_{i} ∣_{p}$ satisfies [F3], so it is a derivation at $p$ .

F3step 1.1step 1.2∎

Depends on

Used by

Coordinate derivations form a basis of the tangent space Theorem

Cited to discharge well-definedness by Coordinate derivations at a point.

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)