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

Statement

Let $F : M \to N$ be smooth, let $(U, x)$ and $(V, y)$ be smooth charts with $p \in U$ and $F (p) \in V$ , and let $\tilde{F} : = y \circ F \circ x^{- 1}$ . Then $d F_{p} (\partial_{x^{i}} ∣_{p}) = \sum_{j = 1}^{m} \frac{\partial {\tilde{F}}^{j}}{\partial x^{i}} (x (p)) \partial_{y^{j}} ∣_{F (p)} .$

Facts & Assumptions

Given: A smooth map $F : M \to N$ , charts $(U, x)$ and $(V, y)$ , and the coordinate representative $\tilde{F}$ .

[F1]

The differential acts on a derivation by precomposing target germs with $F$ (The differential of a smooth map).

[L1]

Coordinate derivations form bases of the tangent spaces (Coordinate derivations form a basis of the tangent space).

Proof

technique · direct

1.1

By [L1], it is enough to compute the values of $d F_{p} (\partial_{x^{i}} ∣_{p})$ on the coordinate germs $[y^{j}]$ at $F (p)$ .

L1given

2.1

By [F1], one has $d F_{p} (\partial_{x^{i}} ∣_{p}) ([y^{j}]) = \partial_{x^{i}} ∣_{p} ([y^{j} \circ F]) = \partial {\tilde{F}}^{j} / \partial x^{i}$ at $x (p)$ .

F1step 1.1

3.1

The displayed linear combination of the $\partial_{y^{j}} ∣_{F (p)}$ has exactly the same values on all coordinate germs $[y^{j}]$ , so it equals $d F_{p} (\partial_{x^{i}} ∣_{p})$ .

L1step 2.1∎

Depends on

Used by

The differential of a map between spheres in stereographic coordinates Example
The differential is intrinsic, not merely a Jacobian matrix False statement
The global differential of a smooth map is smooth Theorem

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)