Alphabeta MathHow statement and proof provenance work
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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
FALSE: continuity alone makes the regular-patch surface-area formula applicable
Statement
Every continuous injective parametrization on a compact Jordan region satisfies the regular-patch cross-product surface-area formula.
Facts & Assumptions
Given: The closed unit disc and .
The Euclidean norm is continuous for the Euclidean metric, and its definition restricts on the horizontal axis to (The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for , The Euclidean inner product on ).
A partial derivative is a one-variable derivative along a coordinate line (Directional derivatives and partial derivatives of a map ), while a regular patch must be on a neighbourhood and have a nonzero parameter cross product in the interior (Regular parametrized surface patches on compact Jordan parameter regions).
Refutation
By [L1], is continuous, and its first two coordinates make it injective on .
Along , the third component is . Its right difference quotient at is and its left difference quotient is , so the -partial derivative of does not exist at the interior point by [L2].
Thus the cross-product integrand required by [L2] is unavailable at an interior point despite continuity and injectivity, refuting the statement. A polar cone parametrization moves the apex failure to a parameter-boundary point and is a different parametrization.
Depends on
- Regular parametrized surface patches on compact Jordan parameter regions
- Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
- The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
- The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
42 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
- S. Cañez, Northwestern Math 320-3 lecture notes, cone example (standard reference, not scraped)
- University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)