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.
The cross product in
Definition
For and in , define .
This is the right-handed cross product. The displayed coordinates are those of the standard basis (The standard list with and for is an ordered basis of ; hence , and is the zero space with basis and dimension ), and inner products and norms are those of The Euclidean inner product on .
Depends on
- The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
- The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$
Used by
- A degenerate two-parameter map can collapse its image to a curve Counterexample
- Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
- Regular parametrized surface patches on compact Jordan parameter regions 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
- FALSE: flux is independent of the parametrization without an orientation condition False statement
- The cross product is bilinear, alternating, and orthogonal to both factors Lemma
- The squared cross-product norm is the Gram determinant of two vectors Theorem
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)
- University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)