Euler potentials ($\mathbf{B}=\nabla \alpha \times \nabla \beta$)

Two scalar labels constant along a magnetic field line are Euler potentials when their crossed gradients give the magnetic field. The magnetic vector potential $\mathbf{A}=\alpha \nabla \beta$ proves that this representation is a solenoidal vector field. Conversely, two independent local line labels yield $\mathbf{B}=f(\alpha ,\beta )\nabla \alpha \times \nabla \beta$; replacing $\beta$ by ${\int }^{\beta }f(\alpha ,b)db$ absorbs the factor. This is a local construction; a global regular single-valued pair requires compatible magnetic field line topology. Fixing boundary labels preserves their paired-footpoint connections in an ideal variation.