= Euler potentials
{c}
{title2=$\mathbf B=\nabla\alpha\times\nabla\beta$}
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.
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