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.
Surface area and scalar surface integrals on a regular patch
Definition
Let be a regular surface patch and let be a continuous real-valued function on . For a continuous scalar field on the patch image, , and .
Here denotes the patch with its chosen parametrization. Both integrands are bounded and Riemann integrable on the compact Jordan region by continuity (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set). Boundary values are included in the parameter integral but may be changed on the content-zero boundary without changing its value.
Depends on
Used by
- Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
- Finitely patched regular surfaces, their area, scalar integrals, and flux Definition
- Unit normal fields, orientations, and flux through a regular surface patch Definition
- A closed cylinder as a finitely patched oriented surface Example
- Opposite parametrizations preserve area and negate flux Example
- Surface area and flux on a sphere, with scalar integrals on a hemisphere Example
- The surface area of a torus is 4π²ab Example
- Scalar surface integrals on a surface of revolution Theorem
- Surface area and scalar surface integrals are invariant under regular reparametrization Theorem
- Surface area, scalar integrals, and flux over a C¹ graph Theorem
Dependency tree · two levels
17 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, formula 3.2.12 (standard reference, not scraped)
- University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)