Изменения

Пусть <tex> F: E \subset \mathbb{R}^m \to \mathbb{R}^l, ; \ a \in \operatorname{Int} EIntE, \ F(E) \subset I </tex>,  <tex> G: I \subset \mathbb{R}^l \to \mathbb{R}^n, ; \ b = Ff(a) \in \operatorname{Int} I IntI </tex>,  <tex> F </tex> дифференцируемо — дифф. в <tex> (\cdot) a , G </tex>, — дифф. в <tex> (\cdot) b </tex>; <tex> H = G \circ F \ // H(x) = G (F(x)) </tex> Тогда: <tex> H </tex> дифференцируемо — дифф. в <tex> (\cdot) a; H'(a) = G'(F(a)) \cdot F'(a) </tex>|proof=<tex> F(a + h) = F(a) + F'(a)h + \alpha(h)||h||; \ // \alpha(h) \xrightarrow[h \to 0]{} 0 </tex> <tex> G(b + k) = G(b) + G'(b )k + \beta(k)||k||; \ // \beta(k) \xrightarrow[k \to 0]{} 0 </tex>. Тогда  <tex> H(a + h) = G (F(a + h)) = G(\overbrace{F(a)}^{b} + \circ overbrace{F '(a)h + \alpha(h)||h||}^{k}) = </tex> дифференцируемо в <tex> G(b) + G'(b)(F'(a )h + \alpha(h)||h||) + \beta(k)||k|| = </tex>, и при этом  <tex> = \overbrace{G(F(a)) + G '(F(a) \circ cdot F'(a)h)}^{H(a)} + \overbrace{G'(ab)\alpha(h)||h|| + \beta(k)||k||}^{? o(h) \leftarrow \text{proverim}} </tex> 1. <tex> ||\ G'(b)\alpha(h) \|h\| \ || = \|h\| \cdot ||G'(b)\alpha(h)|| \le \|h\|\cdot C_{G(b)} \cdot ||\alpha(h)|| = o(h) </tex> 2. <tex> \beta(k)||k|| </tex> <tex> \|k\| = || \ F'(a)h + \alpha(h)\|h\| \ || \le \overbrace{||F'(a)h||}^{C_{F'(a)} \cdot \|h\|} + \|\alpha(h)\|\cdot\|h\| \le (C_{F'(a)} + \|\alpha(h)\|\cdot \|h\|) </tex> <tex> ||\ \beta(k) ⋅ \cdot \|k\| \ || \le \overbrace{||\beta{k}||}^{\to 0, h \to 0} \cdot \overbrace{(C_{F'(a)} + ||\alpha(h)||)}^{ogr. pri: \ h \to 0} \cdot \|h\| = o(h)</tex> <tex> F = (f_1(x_1 \ldots x_m), f_2(x_1 \ldots x_m), \ldots, f_l(x_1 \ldots x_m)) </tex> <tex> G = (g_1(y_1 \ldots y_l), \ldots, g_n(y_1 \ldots y_l)) </tex><tex> H = \overbrace{g_1}^{h_1}(f_1(x_1 \ldots x_n), \ldots, f_l(x_1 \ldots x_n)), \ldots, \overbrace{g_n}^{h_n}(f \ldots)) </tex> <tex> \frac{\partial h_i}{\partial x_j}(a) = \frac{\partial g_i}{\partial y_1}(b) \cdot \frac{\partial f_1}{\partial x_j}(a) + \frac{\partial g_i}{\partial y_2}(b) \cdot \frac{\partial f_2}{\partial x_j}(a) + \ldots + \frac{\partial g_i}{\partial y_l}(b) \cdot \frac{\partial f_l}{\partial x_j}(a) </tex>