Gradient should have zero curl; harmonic has zero local curl but nonzero periods; the positive control has nonzero curl.
One smooth field · several discrete addresses
Curl is circulation.
But around which cell?
Run the same manufactured tangent field through a primal triangle loop and a barycentric dual loop. Then separate that measurement question from the choice of tangent connection.
START HEREChoose one known field. The page runs it immediately.
Triangle and dual-cell circulation return scalars at different mesh locations. “Truth error” compares each with the analytic curl.
One resolution is a picture. The bottom chart is the actual consistency experiment.
ONE VALUE PER FACETriangle boundary circulation
(Σ αᵢⱼ) / Af- truth error
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- curl RMS
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- samples
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ONE VALUE PER VERTEXBarycentric dual-cell circulation
(∮ X♭) / A★v- truth error
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- curl RMS
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- samples
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LOCAL ≠ GLOBALA curl test cannot see every torus loop.
- u period
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- v period
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- analytic curl RMS
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CONNECTION MICROSCOPEEmbedding rotation versus one-ring polar baseline
The analytic torus Levi–Civita angle is the manufactured reference. These are connection errors, not curl values.
- extrinsic RMS
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- intrinsic RMS
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- edge family
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| grid | triangle curl error | dual curl error | extrinsic angle error | intrinsic angle error |
|---|
OPERATOR LEDGERKeep representation transfers explicit.
αᵢⱼ = ½(Xᵢ + Xⱼ) · (pⱼ − pᵢ)Endpoint trapezoids turn tangent samples into oriented line integrals.
curlf = Σ∂f α / AfThree boundary values produce one face-centered scalar.
curlv = ∮∂v★ X♭ / A★vMidpoints and barycenters bound one positive vertex dual cell.
DᵢⱼX = Xⱼ − QᵢⱼXᵢTransport compares vectors across tangent planes; circulation compares scalar integrals.