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

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Regular level surfaces have local regular parametrizations with the same tangent plane

Statement

Let $F : U \subseteq R^{3} \to R$ be $C^{k}$ , $k \geq 1$ , and let $c$ be a regular value. Every point of a regular level surface in $R^{3}$ lies in the relative interior of a regular surface patch, and the patch tangent plane is the level-set tangent space.

If $F^{- 1} (c)$ is empty, the assertion is vacuous.

Facts & Assumptions

Given: The map $F$ , regular value $c$ , and a point $p \in F^{- 1} (c)$ .

[L1]

Near $p$ , the level is ${p + u + g (u) : u \in P}$ for a $C^{k}$ map $g$ on a neighbourhood $P$ of $0$ in $K = \ker D F (p)$ with $g (0) = 0$ and $D g (0) = 0$ , and $\dim K = 2$ ; a regular patch has nonzero parameter cross product in the interior and no interior parameter point shares its image with another point of the parameter region; its tangent plane is the span of the parameter derivatives (A regular level set is locally a $C^{k}$ graph of dimension $m - n$ , The tangent space to a regular level set, Regular parametrized surface patches on compact Jordan parameter regions, The tangent plane of a regular surface patch).

[L2]

Equal-dimensional finite-dimensional vector spaces are linearly isomorphic, and partial derivatives are total derivatives applied to the standard coordinate vectors (Two finite-dimensional vector spaces over $F$ are linearly isomorphic if and only if they have the same dimension, A total derivative computes every directional derivative, and its matrix is the Jacobian).

Proof

technique · direct

1.1

By [L1] write the level near $p$ as $p + u + g (u)$ for $u$ near $0$ in $K$ , with $D g (0) = 0$ . By [L2], choose a linear isomorphism $L : R^{2} \to K$ .

givenL1L2choose

2.1

Define $φ (s, t) = p + L (s, t) + g (L (s, t))$ and restrict it to a sufficiently small closed rectangle $D$ about $0$ . The graph representation makes $φ$ injective, and $D φ (0) = L$ has independent columns. By continuity, after shrinking the rectangle the parameter cross product stays nonzero in its interior, so [L1] makes $(D, φ)$ a regular patch.

step 1.1L1L2construct

3.1

The image of $D φ (0)$ is $K$ , so [L1] makes the patch tangent plane $K$ and also identifies $K$ with the level-set tangent space. Also $p = φ (0)$ lies in the relative interior of the patch image.

step 2.1L1

4.1

The construction works at every point of a nonempty regular level, and there is nothing to choose or prove for an empty level.

step 3.1∎

Depends on

Used by

A compact regular level surface is covered by finitely many regular surface patches Corollary

Dependency tree · two levels

28 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

M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)
J. M. Lee, Introduction to Smooth Manifolds, Regular Level Set Theorem (standard reference, not scraped)