TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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.

Derivations of smooth functions are exactly smooth vector fields

Statement

The assignment sending a smooth vector field $X$ to the operator $f \mapsto X f$ defines a bijection between smooth vector fields on $M$ and $R$ -linear derivations $D : C^{\infty} (M) \to C^{\infty} (M)$ .

Facts & Assumptions

Given: An $R$ -linear derivation $D : C^{\infty} (M) \to C^{\infty} (M)$ .

[L1]

Every smooth vector field acts as a derivation of $C^{\infty} (M)$ (A vector field acts as a derivation of smooth functions).

[L2]

A derivation at a point is exactly a tangent vector at that point (Derivations at a point and the tangent space).

[L3]

In a chart, the coordinate derivations form a basis of the tangent space (Coordinate derivations form a basis of the tangent space).

[L4]

A vector field is smooth exactly when its coordinate components are smooth (Smoothness of a vector field is equivalent to smooth coordinate components).

[L5]

For a point inside an open set there is a smooth bump function equal to $1$ on a neighbourhood of that point and supported in the open set (A manifold bump for a compact set inside an open set).

Proof

technique · direct

1.1

The forward map is well defined by [L1]: every smooth vector field $X$ yields an $R$ -linear derivation $f \mapsto X f$ .

1.2

Fix $p \in M$ . If global smooth functions $f$ and $g$ agree on a neighbourhood of $p$ , choose an open set $W$ on which $f = g$ and use [L5] to choose $χ : M \to [0, 1]$ that is $1$ on a neighbourhood of $p$ and has support contained in $W$ . Then $χ (f - g) = 0$ , so evaluating the Leibniz rule for $D$ at $p$ gives $0 = D (χ (f - g)) (p) = χ (p) D (f - g) (p) + (f - g) (p) D (χ) (p) = D (f) (p) - D (g) (p) .$ Thus $f \mapsto D (f) (p)$ depends only on the germ of $f$ at $p$ , and it defines a derivation $D_{p} : C_{p}^{\infty} (M) \to R$ . By [L2], there is a unique tangent vector $X_{p} \in T_{p} M$ with $X_{p} ([f]) = D_{p} ([f])$ .

L2L5given

2.1

Let $p \in M$ , choose a chart $(U, x^{1}, \dots, x^{n})$ around $p$ , and use [L5] again to choose $χ : M \to [0, 1]$ that is $1$ on a neighbourhood $V$ of $p$ and has support contained in $U$ . For each $i$ , let ${\tilde{x}}^{i}$ be the global smooth function that equals $χ x^{i}$ on $U$ and $0$ outside $U$ . Then for every $q \in V$ , the germs of ${\tilde{x}}^{i}$ and $x^{i}$ agree at $q$ , so [L3] writes $X_{q} = \sum_{i} D ({\tilde{x}}^{i}) (q) \frac{\partial}{\partial x^{i}} ∣_{q} .$ Each coefficient function $D ({\tilde{x}}^{i}) ∣_{V}$ is smooth, because $D ({\tilde{x}}^{i})$ is a global smooth function. Hence [L4] makes $X$ smooth on $V$ .

L3L4L5step 1.2given

3.1

Since every point has a neighbourhood $V$ on which step 2.1 makes $X$ smooth, the pointwise-defined tangent vectors $X_{p}$ form a global smooth vector field $X$ on $M$ .

step 1.2step 2.1

4.1

By construction, $X f = D (f)$ for every smooth function $f$ , so the map from smooth vector fields to derivations is surjective. If two smooth vector fields induce the same derivation, then their values at each point agree on every smooth function, hence are equal by [L2]; thus the map is injective.

L2step 1.2step 3.1

5.1

Therefore smooth vector fields and $R$ -linear derivations of $C^{\infty} (M)$ are in bijection.

step 1.1step 4.1∎

Depends on

Used by

Coordinate formula for the Lie bracket Proposition
Related vector fields have related Lie brackets Proposition
Smooth vector fields form a Lie algebra under the Lie bracket Theorem

Dependency tree · two levels

20 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, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)