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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Coordinate formula for the differential of a function

Statement

If $f : M \to R$ is smooth and $(U, x)$ is a smooth chart around $p$ , then $d f_{p} = \sum_{i = 1}^{n} \frac{\partial (f \circ x^{- 1})}{\partial x^{i}} (x (p)) d x_{p}^{i},$ where $d x_{p}^{i} : = d (x^{i})_{p}$ .

Facts & Assumptions

Given: A smooth function $f : M \to R$ and a chart $(U, x)$ around $p$ .

[F1]

The differential of a real-valued smooth function is the linear functional $v \mapsto v ([f])$ (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

1.1L1given

By [L1], every tangent vector has the form $v = \sum_{i} v ([x^{i}]) \partial_{x^{i}} ∣_{p}$ .

2.1F1step 1.1

Applying [F1] to such a vector gives $d f_{p} (v) = \sum_{i} v ([x^{i}]) \partial_{x^{i}} ∣_{p} ([f]) = \sum_{i} v ([x^{i}]) \partial (f \circ x^{- 1}) / \partial x^{i}$ at $x (p)$ .

3.1step 2.1∎

This is exactly the action of the covector $\sum_{i} (\partial (f \circ x^{- 1}) / \partial x^{i}) (x (p)) d x_{p}^{i}$ on every $v$ , so the two covectors are equal.

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)