Knowlpedia

Theorem

Chain rule for iterated torus variables

Integer matrix iterations amplify auxiliary directional derivatives by their action on the direction vector.

Statement

Let Y(x)∈TnY(x)\in\mathbb T^n be smooth and F(x,H)F(x,H) smooth and periodic in HH. For a fixed integer i≥0i\ge0, define

u(x)=F(x,JiY(x)).u(x)=F(x,J^iY(x)).

The gives

∂xau=(∂xaF+(Ji∂xaY)⋅∇HF)H=JiY(x).\partial_{x_a}u=\left(\partial_{x_a}F+ (J^i\partial_{x_a}Y)\cdot\nabla_HF\right)_{H=J^iY(x)}.

Local lifts of YY give the same answer because JJ is integral and FF is periodic.

Eigenvector directions

If Y(r,t)=vrh(r)+vttY(r,t)=v_rh(r)+v_tt and Jvr=λrvrJv_r=\lambda_rv_r, Jvt=λtvtJv_t=\lambda_tv_t, then the represented derivatives are ∂r+λrih′(r)vr⋅∇H\partial_r+\lambda_r^i h'(r)v_r\cdot\nabla_H and ∂t+λtivt⋅∇H\partial_t+\lambda_t^i v_t\cdot\nabla_H. The discrete level ii and all sheet labels are held fixed. A level selected by a floor function is not a differentiable variable; separate smooth band formulas must be combined with their cutoffs.