TheoremStatement: 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.
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The global differential of a smooth map is smooth

Statement

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: 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]

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

1.1

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 $\tilde{y} \circ d F \circ {\tilde{x}}^{- 1} (a, v) = (\tilde{F} (a), D \tilde{F} (a) v)$ .

F1L1L2given

2.1

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.

step 1.1

3.1

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

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)