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Definition

Evaluation along an auxiliary map

Composition of a function of independent physical and periodic auxiliary variables with a prescribed auxiliary map.

Let F(x,Y)F(x,Y) be smooth and Zk\mathbb Z^k-periodic in the independent auxiliary variable YY, and let Ψ:U⊂Rn→Rk\Psi:U\subset\mathbb R^n\to\mathbb R^k be smooth. Evaluation along the auxiliary map is the

F♯(x)=F(x,Ψ(x)).F^\sharp(x)=F(x,\Psi(x)).

Replacing Ψ\Psi by an integer translate leaves F♯F^\sharp unchanged. Compatible local lifts therefore also define this operation for a map into a torus.

Derivatives after evaluation

The gives

∂xjF♯=(∂xjF+∑a=1k∂xjΨa ∂YaF)Y=Ψ(x).\partial_{x_j}F^\sharp= \left(\partial_{x_j}F+\sum_{a=1}^k \partial_{x_j}\Psi_a\,\partial_{Y_a}F\right)_{Y=\Psi(x)}.

The first derivative on the right holds YY fixed. For Ψ(r,t)=vrh(r)+vtt\Psi(r,t)=v_r h(r)+v_t t, the evaluated radial and time derivatives are represented before restriction by ∂r+h′(r)vr⋅∇Y\partial_r+h'(r)v_r\cdot\nabla_Y and ∂t+vt⋅∇Y\partial_t+v_t\cdot\nabla_Y. Auxiliary periodicity need not yield physical spatial periodicity. Auxiliary averaging is performed before restriction unless explicitly stated otherwise.