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Choose two independent labels for the magnetic field lines, constant along each line. These are Euler potentials: $\mathbf{B}\cdot \nabla \alpha =\mathbf{B}\cdot \nabla \beta =0$. Where the two gradients are independent, the vector perpendicular to both is $\nabla \alpha \times \nabla \beta$, so initially writeThe solenoidal condition gives $0=\nabla \cdot \mathbf{B}=(\nabla \alpha \times \nabla \beta )\cdot \nabla f$. Thus $f$ is constant along a field line and is locally a function of its labels. Redefining one label, for example $\tilde{\beta }(\alpha ,\beta )={\int }^{\beta }f(\alpha ,b)db$, absorbs this factor. Dropping the tilde givesThis representation automatically has zero divergence, since it is also $\nabla \times (\alpha \nabla \beta )$.

For an magnetic arcade, assign the labels on one polarity of the footpoint surface and transport them along each line to its other footpoint. They specify both line identity and magnetic flux: on a transverse surface, flux coordinates can be chosen so that the directed flux element is $d\alpha d\beta$. In particular,The boundary labels at paired footpoints must agree, so prescribed boundary functions encode a compatible footpoint mapping, not independently arbitrary values at the two ends.

The construction assumes a smooth family of nonintersecting lines that can be labelled from a transverse boundary patch, with suitable patches at nulls or coordinate singularities. A local Euler-potential representation does not establish the existence of two globally single-valued regular labels for every possible solenoidal field; closed, recurrent or more complicated field topology can obstruct such a global choice. The line-tied magnetic arcade geometry supplies the relevant simplifying assumption here.