DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Riemannian gradient

For a smooth real function $f$ , its Riemannian gradient is ${grad}_{g} f = (d f)^{♯}$ .

The exterior derivative of a function is its differential identifies $d f (X) = X f$ . The smooth bundle isomorphism in The musical maps are smooth inverse bundle isomorphisms therefore makes the gradient a smooth vector field. In coordinates $({grad}_{g} f)^{i} = \sum_{j} g^{i j} \partial_{j} f$ . Constants, and all functions in dimension zero, have zero gradient.

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 13, pp.328–332 and 341–342.

Depends on