PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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Chain rule for smooth half-space maps

Statement

If $f : U \to V$ and $g : V \to W$ are smooth maps between relatively open half-space sets, then $g \circ f$ is smooth and $D (g \circ f)_{p} = D g_{f (p)} \circ D f_{p}$ .

Facts & Assumptions

Given: Relatively open half-space sets $U, V, W$ , smooth maps $f : U \to V$ and $g : V \to W$ , and a point $p \in U$ .

[L1]

Half-space smoothness supplies smooth Euclidean extensions near every point (Smooth functions on relatively open half-space sets).

[L2]

The derivatives of two Euclidean extensions agreeing on a relatively open half-space set agree on that set (Half-space extensions agreeing on a relatively open set have the same derivatives there).

[L3]

Total derivatives satisfy the Euclidean chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

technique · direct

1.1

By [L1], choose Euclidean extensions near $p$ and $f (p)$ and shrink the first neighbourhood so that its image lies in the domain of the second extension. Their ordinary composite then extends $g \circ f$ near $p$ .

givenL1choose

2.1

Applying [L3] to the extensions from step 1.1 gives the displayed formula, and [L2] makes the resulting derivatives independent of both extension choices. Hence $g \circ f$ is smooth and the formula is intrinsic.

L2L3step 1.1∎

Depends on

Used by

Smooth charts, atlases, and structures with boundary Definition
Boundary smoothness is independent of charts and extensions Lemma
Boundary-defining functions exist locally and detect inward vectors Proposition
Smooth invariance of the manifold boundary Theorem
Tangent and cotangent bundles extend over a boundary Theorem

Dependency tree · two levels

8 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)