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Iterated differentials
The smooth counterpart of the iterated increment \(\Delta^K\) E0009: the iterated differential \(D^\kappa\) applies the Fréchet derivative recursively, indexed by higher multi-indices \(\KM_+^m\) E0007.
(1) Definition. Let \(f_1, \dots, f_m\) be \(C^n\) maps between Banach spaces with \(f_r: X_{r-1} \to X_r\), \(x \in X_0\), \(v_1, \dots, v_k \in X_0\), and \(x_r = (f_r \circ \cdots \circ f_1)(x)\).
-
The iterated differential for \(\alpha \in \KM_+(k)\) and \(m = 1\) is \(D^\alpha(f_1;\, x;\, v_\bullet) = D^{|\alpha|}(f_1;\, x;\, v_\bullet^{\times \alpha})\) (E0020). For \(m \geq 2\) and \(\kappa \in \KM_+^m(k)\), define recursively:
\[ D^\kappa(f_1, \dots, f_m;\, x;\, v_\bullet) := D(f_m;\, x_{m-1};\, (D^\lambda(f_1, \dots, f_{m-1};\, x;\, v_\bullet) ^{\times \kappa(\lambda)})_{\lambda \in \supp(\kappa)}). \] -
The Boolean iterated differential for \(H \in \Part_m(k)\) with \(m \geq 2\) is
\[ D^H(f_1, \dots, f_m;\, x;\, v_\bullet) := D(f_m;\, x_{m-1};\, (D^L(f_1, \dots, f_{m-1};\, x;\, v_\bullet))_{L \in H}). \]The Boolean case is the partition restriction of the multi-index case, analogous to \(\Delta^K\) vs \(\Delta^\kappa\) in E0009.
(2) Validation (AI review, 2026-07-19, claude-fable-5, pass). Recursion typing and well-foundedness; set-indexed families justified by symmetry of \(D^{|H|}\) E0020; partition restriction verified against the embedding (\(D^{\nu_{\mathrm{id}}(H)} = D^H\) at \(m = 2\)). Added the \(C^n\) hypothesis (differentials do not exist for bare maps) and aligned the \(m = 1\) case with \(D^{|\alpha|}\) E0020. No other issues found.
Used by: