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

Definition

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.

Source locator

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

Depends on

Used by

A conformal metric on the plane Example
Gradient hessian and divergence connection formulas Proposition
The gradient is characterized by inner products Proposition

Dependency tree · two levels

7 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

Ved Datar, Lectures on Riemannian Geometry, September 2025 (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)