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

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Assuming countable choice, the global differential of a smooth map is smooth

Statement

Assume ${A C}_{ω}$ . If $F : M \to N$ is smooth, then the global differential $d F : T M \to T N$ is a smooth map.

Facts & Assumptions

Given: The axiom ${A C}_{ω}$ and a smooth map $F : M \to N$ .

[F1]

The global differential sends $v \in T_{p} M$ to $d F_{p} (v)$ (The global differential or tangent map).

[L1]

In bundle charts, the fiber coordinates of $d F_{p}$ are given by the Jacobian matrix of the coordinate representative of $F$ (Coordinate formula for the differential).

[L2]

Assuming ${A C}_{ω}$ , tangent bundles carry the smooth structures induced by bundle charts (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).

Proof

technique · direct

1.1F1L1L2given

Choose charts $(U, x)$ on $M$ and $(V, y)$ on $N$ with $F (U) \subseteq V$ , and let $\tilde{F} : = y \circ F \circ x^{- 1}$ . In the induced bundle charts, [L1] gives $\tilde{y} \circ d F \circ {\tilde{x}}^{- 1} (a, v) = (\tilde{F} (a), D \tilde{F} (a) v)$ .

2.1step 1.1

The map $a \mapsto \tilde{F} (a)$ is smooth, the matrix entries of $D \tilde{F} (a)$ are smooth, and matrix-vector multiplication is polynomial in those entries and the components of $v$ . Therefore the displayed local formula is smooth.

3.1L2step 2.1∎

Since this holds in bundle charts, [L2] implies that $d F : T M \to T N$ is smooth.

Depends on

Used by

Formal immersion between smooth manifolds Definition
Gauss frame map of an immersion into Euclidean space Definition
Pushforwards and pullbacks of vector fields by a diffeomorphism Definition
The derivative map is continuous Lemma
Euclidean formal immersions are homotopy equivalent to Stiefel-bundle sections Proposition
Local diffeomorphisms carry distributions and integral manifolds Proposition
Translations are diffeomorphisms and their differentials trivialize the tangent bundle Proposition

Dependency tree · two levels

11 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)