✍内容

[[Ordinary Differential Equations - V.I.Arnold(英文版).pdf]] ![[Ordinary Differential Equations - V.I.Arnold(英文版).pdf#page=276&rect=38,388,382,530|Ordinary Differential Equations - V.I.Arnold(英文版), p.272]]

The derivative of ${\displaystyle \mathit{f}:U\subset {\mathbb{R}}^m\to {\mathbb{R}}^n}$ at ${\displaystyle x\in U\subset {\mathbb{R}}^m}$ is a linear operator from one Euclidean space into another

$${\mathit{f}}_{\ast \mathit{x}}:T_x{\mathbb{R}}^m⟶T_{\mathit{f}(\mathit{x})}{\mathbb{R}}^n$$

[!thm] A continuously differentiable mappin ${\displaystyle \mathit{f}}$ satisfies a Lipschitz condition on each convex subset $V$ of the domain $U$ with constant $L$ equal to the supremum of the derivative ${\displaystyle \mathit{f}}$ on $V$:

$$L=\underset{\mathit{x}\in V}{sup}|{\mathit{f}}_{\ast \mathit{x}}|$$