CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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A curl-free field has zero circulation around the induced boundary chain of a $C^{2}$ patch

Statement

Let $(D, φ)$ be a $C^{2}$ patch over a finite elementary Green region and let $F$ be a $C^{1}$ vector field on an open set $U \subseteq R^{3}$ containing $φ [D]$ , with $curl F = 0$ at every point of $U$ . Then

$\oint_{φ (\partial D)} F \cdot d r = 0.$

The curl must vanish on an open set containing the whole patch image, not merely along the induced boundary chain. Equivalently, by A $C^{1}$ field on an open subset of $R^{3}$ is closed exactly when its curl vanishes, the hypothesis is that $F$ be closed on $U$ .

Facts & Assumptions

Given: The $C^{2}$ patch $(D, φ)$ over a finite elementary Green region, the open $U \supseteq φ [D]$ , and the $C^{1}$ field $F$ on $U$ with $curl F = 0$ throughout $U$ .

[F1]

The curl of a $C^{1}$ field on an open subset of $R^{3}$ is $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ (Divergence and curl of a $C^{1}$ vector field).

[F2]

The circulation of $F$ around the induced boundary chain is the finite sum of the vector line integrals along the arcs $φ \circ σ_{l}$ (The induced boundary chain and circulation of a $C^{2}$ patch over a finite elementary Green region), and integration over a bounded Jordan measurable set is integration of the zero extension over a bounding rectangle (The Riemann integral of a bounded function over a bounded Jordan measurable set).

[L1]

For a $C^{2}$ patch over a finite elementary Green region and a $C^{1}$ field $F$ on an open set containing the patch image, the circulation around the induced boundary chain equals the flux of the curl in the induced orientation, $\oint_{φ (\partial D)} F \cdot d r = \int_{D} ⟨ (curl F) \circ φ, φ_{u} \times φ_{v} ⟩$ (The classical Stokes theorem for a $C^{2}$ patch over a finite elementary Green region).

[L2]

A $C^{1}$ field on an open subset of $R^{3}$ is closed if and only if its curl vanishes identically (A $C^{1}$ field on an open subset of $R^{3}$ is closed exactly when its curl vanishes).

[L3]

For integrable $f, g$ on a nondegenerate rectangle and scalars $α, β$ , the function $α f + β g$ is integrable with integral $α \int f + β \int g$ (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

Proof

technique · direct

1.1

Since $φ [D] \subseteq U$ and $curl F$ vanishes at every point of $U$ by hypothesis and [F1], the integrand $⟨ (curl F) \circ φ, φ_{u} \times φ_{v} ⟩$ is identically zero on $D$ . Its zero extension to a bounding rectangle is the zero function, which by [L3] with $α = β = 0$ is integrable with integral $0$ , so $\int_{D} ⟨ (curl F) \circ φ, φ_{u} \times φ_{v} ⟩ = 0$ by [F2].

givenF1F2L3

2.1

By [L1] the circulation around the induced boundary chain equals that integral, hence is $0$ . By [L2] the hypothesis $curl F = 0$ on $U$ is the same as $F$ being closed on $U$ , so the corollary may be read either way.

step 1.1F2L1L2∎

Remarks

A closed field can still have nonzero circulation around a loop. What this corollary rules out is a nonzero circulation around the induced boundary chain of a $C^{2}$ patch whose whole image lies where the curl vanishes. A closed field on a domain that carries no such patch spanning the loop may circulate: the companion examples page gives a field with circulation $2 π$ around a circle encircling the excluded axis, and no patch over a finite elementary Green region has image inside that domain and that circle as its induced boundary.

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Sources

J. Feldman, A. Rechnitzer and E. Yeager, CLP-4 Vector Calculus (University of British Columbia), section 4.4 (standard reference, not scraped)

A curl-free field has zero circulation around the induced boundary chain of a C2C2 patch