PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-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.

The gradient is characterized by inner products

Statement

The gradient is the unique smooth vector field $Y$ satisfying $g (Y, X) = X f$ for every smooth vector field $X$ .

Facts & Assumptions

Given: A Riemannian manifold and a smooth real function $f$ .

[F1]

Riemannian gradient: For a smooth real function $f$ , its Riemannian gradient is ${grad}_{g} f = (d f)^{♯}$ . prop-exterior-derivative-of-a-function-is-its-differential identifies $d f (X) = X f$ . The smooth bundle isomorphism in thm-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.

Proof

technique · direct

1.1

By the inverse pairing defining $♯$ , $g ({grad}_{g} f, X) = g ((d f)^{♯}, X) = d f (X) = X f$ . Thus the gradient has the required property.

F1given

2.1

If $Y$ also has the property, set $Z = Y - {grad}_{g} f$ , itself a smooth global vector field. Subtracting the two identities and taking $X = Z$ gives $g (Z, Z) = 0$ . Positive definiteness makes $Z = 0$ at every point, so $Y = {grad}_{g} f$ . This also covers zero-dimensional and empty manifolds.

F1step 1.1∎

Source locator

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

Depends on

Riemannian gradient

Used by

The euclidean metric and its musical maps Example
Riemannian divergence theorem Theorem

Dependency tree · two levels

3 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, second edition (standard reference, not scraped)