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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources