Let $F \colon V \to L(V,W)$, $X_t = (X_t^1, \ldots, X_t^d)$, $V = \mathbb{R}^d, W=\mathbb{R}^m$, \begin{align*} F(X_t) &= (F_{ij}(X_t))_{1\leq i \leq m, 1 \leq j \leq d}. \end{align*} How should I interpret $DF(X_t)$? Is it a collection of matrices \begin{equation*} \left( \frac{\partial}{\partial x_l} F_{ij}(X_t) \right)_{1\leq i \leq m, 1 \leq j \leq d} \end{equation*} for each $l=1,\ldots,d$?
2026-04-22 09:39:37.1776850777
How to interpret $DF(X_s)$ for matrix-valued $F$?
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