DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)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.

Unit normal fields, orientations, and flux through a regular surface patch

Definition

For a regular patch $(D, φ)$ , the parametrization induces on its interior the unit normal $N_{φ} = \frac{φ_{u} \times φ_{v}}{∥ φ_{u} \times φ_{v} ∥_{2}} .$ The denominator is positive there by regularity and The surface area density is the norm of the cross product of the parameter tangents, and the vector is orthogonal to the tangent plane (The tangent plane of a regular surface patch). Choosing $N_{φ}$ rather than $- N_{φ}$ is an orientation.

For a continuous vector field $F$ , the flux in the orientation induced by $φ$ is $\int_{D} (F \circ φ) \cdot (φ_{u} \times φ_{v})$ . This is the scalar Riemann integral of a continuous function on $D$ (Surface area and scalar surface integrals on a regular patch, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ); replacing the orientation by its negative negates the integrand.

Depends on

Used by

Dependency tree · two levels

22 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

University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)
M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)