TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Conservative, path-independent, and zero-closed-loop conditions are equivalent

Statement

Let $U \subseteq R^{n}$ be nonempty, open, and piecewise- $C^{1}$ path-connected, and let $F : U \to R^{n}$ be continuous. The following are equivalent:

$F$ is conservative;
$F$ is path-independent;
every closed piecewise- $C^{1}$ path $γ$ in $U$ satisfies $\int_{γ} F \cdot d r = 0$ .

When condition 2 holds, choosing $a \in U$ gives the normalized potential $ϕ (x) = \int_{a}^{x} F \cdot d r$ with $ϕ (a) = 0$ .

Facts & Assumptions

Given: The domain and field in the Statement.

[L1]

Every conservative field is path-independent and has zero integral around every closed path (Conservative fields are path-independent and have zero integral around every closed path).

[L2]

On a piecewise- $C^{1}$ path-connected open set, path independence is equivalent to zero integral around every closed piecewise- $C^{1}$ path (Path independence is equivalent to zero integral around every closed piecewise-C1 path).

[L3]

A continuous path-independent field on such a nonempty domain has the normalized line-integral potential stated above (A continuous path-independent field has a potential constructed by line integrals).

Proof

technique · direct

1.1

Condition 1 implies condition 2, and also condition 3, by [L1].

givenL1

1.2

Conditions 2 and 3 imply each other by [L2].

givenL2

1.3

Condition 2 implies condition 1 by [L3], which also supplies the displayed normalized potential.

givenL3

2.1

Thus each of the three conditions implies the other two, proving their equivalence and the final assertion.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Dependency tree · two levels

9 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

J. Lebl, Basic Analysis II, Theorem 9.3.3 (standard reference, not scraped)