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 differential is intrinsic, not merely a Jacobian matrix
Statement
False claim: the differential of a smooth map is literally its Jacobian matrix.
Facts & Assumptions
Given: A smooth map .
The differential is intrinsically defined as a map on tangent derivations (The differential of a smooth map).
A Jacobian matrix is only the coordinate representation of the differential in chosen bases (Coordinate formula for the differential).
Refutation
By [F1], is a linear map between tangent spaces defined without coordinates.
By [L1], the Jacobian matrix appears only after choosing source and target charts, so it represents rather than being identical to the intrinsic object.
Therefore the claim is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
4 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
- John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
- Will J. Merry, Differential Geometry (standard reference, not scraped)
- Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)