Test Biot-Savart derivatives against an analytic circular loop

[rogeriojorge]

Needs the project's external reviewer. ESSOS PRs are not self-merged.

Why

Free-boundary equilibrium workflows differentiate coil fields through third spatial order and with respect to coil parameters. The active field tests currently check construction and cylindrical/Cartesian self-consistency, but they do not compare these quantities with an independent physical oracle.

What this adds

A circular filament in the xy plane has the analytic on-axis field

Bz(z) = mu0 I R^2 / (2 (R^2 + z^2)^(3/2)).

The source-free axisymmetric expansion around the axis determines every Cartesian component through third spatial order:

Bz = F - (x^2 + y^2) F'' / 4 + ...
(Bx, By) = -(x, y) F' / 2
           + (x, y) (x^2 + y^2) F''' / 16 + ...

The test writes the first three derivatives of F explicitly, then compares all entries of B, dB/dX, d2B/dX2, and d3B/dX3 at a nonzero axial point. It also checks two selected Coils.with_dofs JVPs against analytic derivatives: one physical ampere through the normalized current coordinate and a common-radius direction through the circle's two Fourier coefficients.

Absolute tolerances scale with the largest analytic component at each derivative order or sensitivity. This keeps zero-component checks tied to the corresponding physical scale: T, T/m, T/m^2, T/m^3, T/A, or T/m. The relative tolerance is 2e-11.

Scope

This is a full Cartesian tensor check at one symmetry-axis point. It is not an off-axis or full-domain oracle. The parameter test covers two selected directions of the field value; it does not cover the full coil-parameter Jacobian or parameter sensitivities of the spatial derivative tensors.

The oracle found no numerical defect in this tested scope. The change adds regression coverage only and does not modify production code.

Validation

• pytest -q tests/test_fields.py: 6 passed
• Python 3.11, JAX/JAXlib 0.9.2, CPU with 64-bit mode enabled
• git diff --check

Related qualification work: VMEX HINT comparison #302. This PR remains draft for manual review.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.