to be unity. Hence the metric tensor g$_{ij}$ satisfies det g$_{ij}$ ≡ g = 1. The function χ is found by the constraint that θ is cyclic on the unit interval where η is a simple cyclic coordinate on the flux surface in the poloidal plane (e.g., path length on the curve, or the physical angle position about the magnetic axis with respect to any branch cut). Then q is found where the angle brackets give the flux surface average defined as Now, ψ$_{,V}$ = qχ$_{,V}$ is also defined and the toroidal angle coordinate is given by which completes the prescription. The field aligning is a one-to-one and onto coordinate transform from { V, θ, ζ } to { x, y, s } given by where a is a reference minor radius and α$_{k}$ is an arbitrary function of x which is chosen to make the off diagonal perpendicular metric element g xy k vanish at a particular location. This is called the shifted metric procedure [38]. The point is that at any position in s where perpendicular drifts or ∇ 2 ⊥ is evaluated, the coordinate elements are rectangular, and the information contained in magnetic shear enters only in derivatives in the parallel coordinate. The magnetic field components now satisfy at all locations in space for any choice of α$_{k}$ . Hence although the perpendicular coordinate elements are rectangular only for one particular location in s the field aligning is exactly satisfied everywhere. χ − 1 ,V = ∮ dη B · ∇ η and θ = χ$_{,V}$ ∫ dη B · ∇ η (29) q = I 2 πχ$_{,V}$ 〈 1 R 2 〉 (30) 〈 1 R 2 〉 = ∮ dθ R 2 = χ$_{,V}$ ∮ dη R $^{2}$B · ∇ η (31) ζ = ϕ 2 π + I 2 πχ$_{,V}$ ∫ dθ (〈 1 R 2 〉 − 1 R 2 ) (32) x = V /a 3 y$_{k}$ = qθ − ζ − α$_{k}$ ( x ) s = θ (33) B x = 0 B y k = 0 B s = χ$_{,V}$ (34) 10