DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

How 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.

The first fundamental form, Gram matrix, and area density of a surface patch

Definition

For a regular patch $(D, φ)$ and an interior parameter point, whose tangent plane is defined by The tangent plane of a regular surface patch, put $E = ⟨ φ_{u}, φ_{u} ⟩, F = ⟨ φ_{u}, φ_{v} ⟩, G = ⟨ φ_{v}, φ_{v} ⟩ .$ The matrix $G_{φ} = (\begin{array}{cc} E & F \\ F & G \end{array})$ is the Gram matrix of the parameter tangents, and the quadratic form $E d u^{2} + 2 F d u d v + G d v^{2}$ is the first fundamental form (The Gram matrix $G (v_{0}, \dots, v_{r - 1}) = (⟨ v_{i}, v_{j} ⟩)_{i, j < r}$ and Gram determinant, with empty value $1$ ).

The surface area density is $J_{φ} = \sqrt{\det G_{φ}}$ , where $G_{φ}$ is the Gram matrix of $φ_{u}$ and $φ_{v}$ . The determinant is nonnegative for all parameter points and positive in the interior because the two tangents are independent there (A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent), so the nonnegative square root exists and is unique.

Depends on

Used by

Dependency tree · two levels

16 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, formulas 3.2.7-3.2.13 (standard reference, not scraped)
R. Sjamaar, Manifolds and Differential Forms, Section 8.1 (standard reference, not scraped)