Изменения
→Полиномиальная формула
Замечание: Для <tex> F : E \rightarrow \mathbb{R}^l </tex> — дифференцируемо в точке <tex> a </tex>; <tex>F'(a) = ({\partial f_i\over\partial x_j})_{i = 1 \ldots l; j = 1 \ldots m} </tex>
|proof=
<tex>f(a + h) = f(a) = + f'(a) \cdot h + o(h)</tex>
<tex> h := (0, \ldots, 0, t, 0, \ldots, 0) </tex>
<tex> (\lambda f_i)'(a)h = (\lambda'(a)(h))f_i(a) + \lambda(a)(f'_i(a)h) </tex> — <tex>i</tex>-ая коорд. док. ф-лы; <tex> ]f_i \leftrightarrow f </tex>
<tex> \lambda(a + h)f(a + h) - \lambda(a)f(a) = (\lambda(a + h) - \lambda(a))f(a + h) + \lambda(a)(f(a + bh) - f(a)) =
(\lambda'(a)h + o(h))f(a + h) + \lambda(a)(f'(a)h + o(h)) = </tex>
<tex> (a_1 + ... + a_m)^{r + 1} = (a_1 + ... + a_m) \cdot \sum \frac{r!}{k_1! ... k_m!} \cdot a_1^{k_{1}} ... a_m^{k_{m}} = </tex>
<tex> = \sum \frac{r!}{k_1! ... k_m!} \cdot a_1^{k_{1}+1} ... a_m^{k_{m}} + \sum \frac{r!}{k_1! ... k_m!} \cdot a_1^{k_{1}} a_2^{k_2 + 1} ... a_m^{k_{m}} + </tex><tex> \sum \frac{r!}{k_1! ... k_m!} \cdot a_1^{k_{1}} ... a_{m-1}^{k_{m - 1}} a_m^{k_{m } + 1}} = </tex>
<tex> = \sum_{\beta : |\beta| = r + 1; \beta_1 \ge 1} \frac{r! \beta_1}{\beta_1!\beta_2!...\beta_m!} \cdot a_1^{\beta_1}...a_m^{\beta_m} + \sum_{\beta : |\beta| = r + 1; \beta_2 \ge 1} \frac{r! \beta_2}{\beta_1!\beta_2!...\beta_m!} \cdot a_1^{\beta_1}...a_m^{\beta_m} + </tex> <ещё <tex> m - k </tex> суммы> = <tex> \sum_{|b| = r + 1} \frac{r! (b_1 + ... + b_m)}{b_1! ... b_m!} \cdot a_1^{\beta_1}...a_m^{\beta_m} </tex>;
