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→Теорема о непрерывно дифференцируемых отображениях
<tex> \forall \epsilon > 0 \exists \delta > 0 : \forall x : |x - \overline{x}| < \delta </tex>
<tex> ||F'(x) - F'(\overline{x})|| < \epsilon </tex>
<tex> ||F'(x) - F'(\overline{x})|| \le \sqrt{\sum(\frac{\partial f_i}{\partial x_j}(x) - \frac{\partial f_i}{\partial x_j}(\overline{x}))^2} </tex>
<tex> \forall \epsilon > 0 </tex> выберем <tex> \delta : |\frac{\partial f_i}{\partial x_j}(x) - \frac{\partial f_i}{\partial x_j}(\overline{x})| < \frac{\epsilon}{\sqrt{mn}}</tex>; при <tex> |x - \overline{x}| < \delta; i = 1 \ldots n; j = 1 \ldots m </tex>
<tex> F'(x)e_i = </tex><tex> \begin{pmatrix} \frac{\partial f_i}{\partial x_1}(x) \\ \ldots \\ \frac{\partial f_i}{\partial x_n}(x) \end{pmatrix}; </tex><tex> \begin{matrix} |F'(x)e_i| \le ||F'(x)|| \cdot 1 \\ |\frac{\partial f_i}{\partial x_j}(x)| \le |F'(x)e_i| \le ||F'(x)|| \end{matrix} </tex>
Точно также: <tex> |\frac{\partial f_i}{\partial x_j}(x) - \frac{\partial f_i}{\partial x_j}(\overline{x})| \le ||F'(x) - F'(\overline{x})|| </tex>
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