Chain rule in implicit anisotropic coordinates

Statement

Use implicit anisotropic coordinates $(q,X,\eta)$ with parameter $d$ , and write $\tau=T-t$ . For a smooth $f(X,\eta)$ and a fixed $b\in\mathbb R$ , derivatives at fixed physical spatial coordinates satisfy

\partial_t(q^bf)=\frac{q^{b-1}}{L} \left(-bf+d\eta f_\eta+Xf_X\right),

\partial_z(q^bf)=\frac{q^{b-d}}{L} \left(2b\eta f+(1-\eta^2)f_\eta-2\eta Xf_X\right).

Derivation

Implicit differentiation gives $q_\tau=1/L$ and $q_z=2\eta q^{1-d}/L$ . Consequently $\eta_t=d\eta/(qL)$ , $X_t=X/(qL)$ , $\eta_z=q^{-d}(1-\eta^2)/L$ , and $X_z=-2\eta Xq^{-d}/L$ . Substitute these identities into the product and chain rules. The positive lower bound on $L$ prevents a denominator singularity away from $q=0$ .