PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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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.

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The tangent plane is invariant under regular reparametrization

Statement

Regular reparametrizations preserve the tangent plane at corresponding interior parameter points.

Precisely, if $ψ = φ \circ h$ and $(s, t)$ is interior, then $span {ψ_{s}, ψ_{t}} = span {φ_{u}, φ_{v}} ∣_{h (s, t)} .$

Facts & Assumptions

Given: A regular reparametrization $ψ = φ \circ h$ and an interior parameter point $(s, t)$ .

[L1]

The tangent plane is the span of the two parameter derivatives (The tangent plane of a regular surface patch), and a regular reparametrization is induced by a parameter diffeomorphism (Surface reparametrizations and their orientation sign).

[L2]

The chain rule gives $D ψ = D φ \circ D h$ , and total derivatives applied to standard basis vectors are the parameter partial derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , A total derivative computes every directional derivative, and its matrix is the Jacobian).

Proof

technique · direct

1.1

By [L2], each of $ψ_{s}$ and $ψ_{t}$ is a linear combination of $φ_{u}$ and $φ_{v}$ at $h (s, t)$ , so the new tangent span is contained in the old one.

givenL1L2

2.1

Apply the same argument to the inverse parameter diffeomorphism $h^{- 1}$ ; it expresses $φ_{u}$ and $φ_{v}$ as linear combinations of $ψ_{s}$ and $ψ_{t}$ , giving the reverse containment.

step 1.1L1L2

3.1

The two spans are equal. Invertibility of $D h$ ensures neither independent tangent pair loses rank.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)