数学分析笔记14:多元函数微分学

2020年06月05日 阅读数:695
这篇文章主要向大家介绍数学分析笔记14:多元函数微分学,主要内容包括基础应用、实用技巧、原理机制等方面,希望对大家有所帮助。

偏导数与全微分

偏导数与全微分的概念

如今,咱们把导数和微分的概念,推广到多元函数的情形。只不过,在二维以上,函数的方向十分复杂,毫不只有左导数和右导数两个方向。然而,咱们能够先对某个变元求导数,称为偏导数。html

定义14.1(偏导数) f ( x 1 , x 2 , , x n ) f(x_1,x_2,\cdots,x_n) ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 的某个邻域上有定义,若是对第 i ( 1 i n ) i(1\le i \le n) 的变元,极限
lim Δ x i 0 f ( x 1 0 , , x i 1 0 , x i , x i + 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) Δ x i \lim_{\Delta x_i\to 0}{\frac{f(x_1^0,\cdots,x_{i-1}^0,x_i,x_{i+1}^0,\cdots,x_n^0)-f(x_1^0,\cdots,x_n^0)}{\Delta x_i}} 存在,称该极限为 f ( x 1 , , x n ) f(x_1,\cdots,x_n) ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处对 x i x_i 的偏导数,\记为 f ( x 1 0 , , x n 0 ) x i \frac{\partial f(x_1^0,\cdots,x_n^0)}{\partial x_i} f i ( x 1 0 , , x n 0 ) f_i(x_1^0,\cdots,x_n^0)
web

若是 f ( x 1 , , x n ) f(x_1,\cdots,x_n) 在某个开集 E E 上每一个点对全部变元的偏导数都存在,那么,对各个变元的偏导数,都是这个开集上的一个 n n 元函数,一样能够讨论极限、连续性的等概念。
咱们再一元函数上还有微分的概念,在一元函数上,全微分定义成某点的"切线",在二元函数上,全微分就应该是某点的切平面,在三维以上,就是切“超平面”,只不过,这时咱们没有几何直观能够参考。
一维上的直线能够表为 y = a + b x y=a+bx
二维上的平面可表为 y = a + b 1 x + b 2 x y=a+b_1x+b_2x
推广到 n n 维上,超平面可表为 y = a + k = 1 n b k x k y=a+\sum_{k=1}^{n}{b_kx_k}
所谓全微分,就是在函数在某点附近,能够用一个超平面近似,即:
f ( x ) = f ( x 0 ) + k = 1 n b k Δ x k + o ( Δ x ) f(x)=f(x_0)+\sum_{k=1}^n{b_k\Delta x_k}+o(||\Delta x||) 算法

定义14.2(全微分) f ( x 1 , , x n ) f(x_1,\cdots,x_n) x 0 = ( x 1 0 , , x n 0 ) x_0=(x_1^0,\cdots,x_n^0) 的某个邻域上有定义,若是 f ( x 1 , , x n ) f(x_1,\cdots,x_n) 可表为
f ( x 1 , , x n ) = f ( x 1 0 , , x n 0 ) + k = 1 n A k ( x k x k 0 ) + o ( x x 0 ) f(x_1,\cdots,x_n)=f(x_1^0,\cdots,x_n^0)+\sum_{k=1}^n{A_k(x_k-x_k^0)}+ o(||x-x_0||) 其中 A 1 , , A n A_1,\cdots,A_n Δ x = x x 0 \Delta x = x-x_0 无关,则称 f ( x ) f(x) x 0 x_0 处可微,超平面 k = 1 n A k d x k \sum_{k=1}^n{A_kdx_k} 称为 f ( x ) f(x) x 0 x_0 处的全微分,记为 d f = k = 1 n A k d x k df = \sum_{k=1}^n{A_kdx_k}
数组

定理14.1(可微的必要条件) f ( x 1 , , x n ) f(x_1,\cdots,x_n) x 0 = ( x 1 0 , , x n 0 ) x_0=(x_1^0,\cdots,x_n^0) 的某个邻域上有定义, f ( x 1 , , x n ) f(x_1,\cdots,x_n) x 0 = ( x 1 0 , , x n 0 ) x_0=(x_1^0,\cdots,x_n^0) 上可微,则 f f x 0 = ( x 1 0 , , x n 0 ) x_0=(x_1^0,\cdots,x_n^0) 对各变元可求偏导,而且:
d f = k = 1 n f k ( x 1 0 , , x n 0 ) d x k df = \sum_{k=1}^n{f_k(x_1^0,\cdots,x_n^0)dx_k}
app

这由全微分的定义能够直接验证。其次,容易验证可微必连续。但就算n元函数在某点对各变元可求偏导且连续,也不必定可微。ide

例14.1 f ( x , y ) = { x y x 2 + y 2 x 2 + y 2 > 0 0 x = 0 , y = 0 f(x,y)= \begin{cases} \frac{xy}{\sqrt{x^2+y^2}}&x^2+y^2>0\\ 0&x=0,y=0 \end{cases} f ( x , y ) f(x,y) ( 0 , 0 ) (0,0) 处连续且对各变元可求偏导,然而: lim ( x , y ) ( 0 , 0 ) f ( x , y ) x 2 + y 2 = lim ( x , y ) ( 0 , 0 ) x y x 2 + y 2 \lim_{(x,y)\to(0,0)}{\frac{f(x,y)}{\sqrt{x^2+y^2}}} =\lim_{(x,y)\to(0,0)}{\frac{xy}{x^2+y^2}} 极限不存在,所以, f ( x , y ) f(x,y) ( 0 , 0 ) (0,0) 点不可微svg

那么,知足何种条件可以可微呢?下面咱们给出一个充分条件:函数

定理14.2(可微的充分条件) f ( x 1 , , x n ) f(x_1,\cdots,x_n) x 0 = ( x 1 0 , , x n 0 ) x_0=(x_1^0,\cdots,x_n^0) 的某个邻域上有定义且对各变元可求偏导,而且各偏导在 x 0 x_0 处连续,则 f ( x 1 , , x n ) f(x_1,\cdots,x_n) x 0 x_0 处可微spa

证:
f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) = f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) + k = 1 n 1 ( f ( x 1 0 , , x k 0 , x k + 1 , , x n ) + f ( x 1 0 , , x k 0 , x k + 1 , , x n ) ) = k = 1 n [ f ( x 1 0 , , x k 1 0 , x k , x k + 1 , , x n ) f ( x 1 0 , , x k 0 , x k + 1 , , x n ) ] f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0) =f(x_1,\cdots,x_n)-\\f(x_1^0,\cdots,x_n^0)+\sum_{k=1}^{n-1}{( -f(x_1^0,\cdots,x_k^0,x_{k+1},\cdots,x_n) +f(x_1^0,\cdots,x_k^0,x_{k+1},\cdots,x_n))}\\ =\sum_{k=1}^n{[f(x_1^0,\cdots,x_{k-1}^0,x_k,x_{k+1},\cdots,x_n) -f(x_1^0,\cdots,x_k^0,x_{k+1},\cdots,x_n)]} 由拉格朗日中值定理,存在 ξ k \xi_k 介于 x k x_k x k 0 x_k^0 之间 f ( x 1 , , x n ) = f ( x 1 0 , , x n 0 ) + k = 1 n f k ( x 1 0 , , x k 1 0 , ξ k , x k + 1 , , x n ) Δ x k = k = 1 n f k ( x 1 0 , , x n 0 ) Δ x k + k = 1 n [ f k ( x 1 0 , , x k 1 0 , ξ k , x k + 1 , , x n ) f k ( x 1 0 , , x n 0 ) ] Δ x k f(x_1,\cdots,x_n)=f(x_1^0,\cdots,x_n^0) +\sum_{k=1}^n{f_k(x_1^0,\cdots,x_{k-1}^0,\xi_k,x_{k+1},\cdots,x_n)\Delta x_k}\\ =\sum_{k=1}^n{f_k(x_1^0,\cdots,x_n^0)\Delta x_k} +\sum_{k=1}^n[f_k(x_1^0,\cdots,x_{k-1}^0,\xi_k,x_{k+1},\cdots,x_n)-f_k(x_1^0,\cdots,x_n^0)]\Delta x_k 考察余项: f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) k = 1 n f k ( x 1 0 , , x n 0 ) Δ x k k = 1 n Δ 2 x k = k = 1 n [ f k ( x 1 0 , , x k 1 0 , ξ k , x k + 1 , , x n ) f k ( x 1 0 , , x n 0 ) ] Δ x k i = 1 n Δ 2 x i \frac{f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0)-\sum_{k=1}^n{f_k(x_1^0,\cdots,x_n^0)\Delta x_k}}{\sqrt{\sum_{k=1}^n{\Delta^2 x_k}}}\\ =\sum_{k=1}^n[f_k(x_1^0,\cdots,x_{k-1}^0,\xi_k,x_{k+1},\cdots,x_n)-f_k(x_1^0,\cdots,x_n^0)]\frac{\Delta x_k}{\sqrt{\sum_{i=1}^n\Delta^2 x_i}} Δ x k i = 1 n Δ 2 x i 1 |\frac{\Delta x_k}{\sqrt{\sum_{i=1}^n\Delta^2 x_i}}|\le 1 所以: f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) k = 1 n f k ( x 1 0 , , x n 0 ) Δ x k k = 1 n Δ 2 x k k = 1 n f k ( x 1 0 , , x k 1 0 , ξ k , x k + 1 , , x n ) f k ( x 1 0 , , x n 0 ) |\frac{f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0)-\sum_{k=1}^n{f_k(x_1^0,\cdots,x_n^0)\Delta x_k}}{\sqrt{\sum_{k=1}^n{\Delta^2 x_k}}}|\\ \le \sum_{k=1}^n|f_k(x_1^0,\cdots,x_{k-1}^0,\xi_k,x_{k+1},\cdots,x_n)-f_k(x_1^0,\cdots,x_n^0)| 再由偏导数的连续性,就有 lim ( x 1 , , x n ) ( x 1 0 , , x n 0 ) f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) k = 1 n f k ( x 1 0 , , x n 0 ) Δ x k k = 1 n Δ 2 x k = 0 \lim_{(x_1,\cdots,x_n)\to(x_1^0,\cdots,x_n^0)}{|\frac{f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0)-\sum_{k=1}^n{f_k(x_1^0,\cdots,x_n^0)\Delta x_k}}{\sqrt{\sum_{k=1}^n{\Delta^2 x_k}}}|}=0 所以, f ( x 1 , , x n ) f(x_1,\cdots,x_n) ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微orm

咱们把偏导数连续称为连续可微。这样,可微、可导和连续性的关系能够归纳为:
(1)连续可微必定可微
(2)可微必定可求偏导数
(3)可微必定连续
(4)连续不必定可求偏导
(5)可求偏导不必定可微

多元函数微分法则

为了给出多元情形下的求导和微分法则,咱们首先给出向量值函数的全微分概念

定义14.3 g ( x 1 , x 2 , , x n ) = ( g 1 ( x 1 , , x n ) , , g m ( x 1 , , x n ) ) g(x_1,x_2,\cdots,x_n)=(g_1(x_1,\cdots,x_n),\cdots,g_m(x_1,\cdots,x_n)) n n m m 维向量值函数,在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 附近有定义,若是存在与 ( x 1 , , x n ) (x_1,\cdots,x_n) 无关,仅与 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 有关的 m m n n 列矩阵 A A ,记 Δ x = ( x 1 x 1 0 , , x n x n 0 ) T \Delta x = (x_1-x_1^0,\cdots,x_n-x_n^0)^T ,使得 [ g 1 ( x 1 , , x n ) g 1 ( x 1 0 , , x n 0 ) g m ( x 1 , , x n ) g m ( x 1 0 , , x n 0 ) ] = A Δ x + [ o 1 ( Δ x ) o m ( Δ x ) ] \left[\begin{matrix} g_1(x_1,\cdots,x_n)-g_1(x_1^0,\cdots,x_n^0)\\ \cdots\\ g_m(x_1,\cdots,x_n)-g_m(x_1^0,\cdots,x_n^0) \end{matrix}\right] =A\Delta x + \left[ \begin{matrix} o_1(||\Delta x||)\\ \cdots\\ o_m(||\Delta x||) \end{matrix} \right] 则称向量值函数 g g ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,矩阵 A A 称为 g g ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处的Frechet导数。 A d x Adx 称为 g g ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处的全微分。

实际上,由定义容易得出,若是向量值函数在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微的充要条件是每一个份量函数都在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,而且,Frechet导数就等于:
[ g 1 ( x 1 0 , , x n 0 ) x 1 g 1 ( x 1 0 , , x n 0 ) x n g m ( x 1 0 , , x n 0 ) x 1 g m ( x 1 0 , , x n 0 ) x n ] \left[\begin{matrix} \frac{\partial g_1(x_1^0,\cdots,x_n^0)}{\partial x_1}&\cdots&\frac{\partial g_1(x_1^0,\cdots,x_n^0)}{\partial x_n}\\ \cdots&\cdots&\cdots\\ \frac{\partial g_m(x_1^0,\cdots,x_n^0)}{\partial x_1}&\cdots&\frac{\partial g_m(x_1^0,\cdots,x_n^0)}{\partial x_n} \end{matrix}\right] 为了方便,咱们把Frechet导数记为 g ( x 0 ) g^\prime(x_0) ,对 n n 元函数来讲,Frechet导数就是 n n 维的行向量。实际上,Frechet导数就是一元导数的的一个推广,Frechet可导就等价于可微,在这层意义下,可微和可导是等价的。

定理14.3(线性性质) g 1 , g 2 g_1,g_2 n n m m 维向量值函数而且都在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,则
(1) g 1 + g 2 g_1+g_2 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,而且
g 1 ( x 1 0 , , x n 0 ) + g 2 ( x 1 0 , , x n 0 ) = ( g 1 + g 2 ) ( x 1 0 , , x n 0 ) g_1^\prime(x_1^0,\cdots,x_n^0)+g_2^\prime(x_1^0,\cdots,x_n^0) =(g_1+g_2)^\prime(x_1^0,\cdots,x_n^0) (2) c c 是任意实数, c g 1 cg_1 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,而且
c g 1 ( x 1 0 , , x n 0 ) = ( c g 1 ) ( x 1 0 , , x n 0 ) cg_1^\prime(x_1^0,\cdots,x_n^0) = (cg_1)^\prime(x_1^0,\cdots,x_n^0)

这两个性质按照Frechet导数的定义是显然的。

定理14.4 g g n n m m 维向量值函数,而且在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微, f f n n 元函数,而且在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,则 F = g ( x 1 , , x n ) f ( x 1 , , x n ) F=g(x_1,\cdots,x_n)f(x_1,\cdots,x_n) ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,而且
F ( x 1 0 , , x n 0 ) = g ( x 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) + g ( x 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) F^\prime(x_1^0,\cdots,x_n^0)=g(x_1^0,\cdots,x_n^0)f^\prime(x_1^0,\cdots,x_n^0) +g^\prime(x_1^0,\cdots,x_n^0)f(x_1^0,\cdots,x_n^0)

在理解定理14.4时,须要注意的是 f g fg n n m m 维向量值函数,其Frechet导数是 m m n n 列矩阵,等式右边第一项中: g g m m 行的列向量, f f^\prime n n 列的行向量,而第二项是一个数乘的形式。经过定理14.4,多元导数就和一元导数在乘法运算法则上统一块儿来了。下面证实定理14.4

证:
首先 g ( x 1 , , x n ) f ( x 1 , , x n ) g ( x 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) = g ( x 1 , , x n ) f ( x 1 , , x n ) g ( x 1 0 , , x n 0 ) f ( x 1 , , x n ) + g ( x 1 0 , , x n 0 ) f ( x 1 , , x n ) g ( x 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) g(x_1,\cdots,x_n)f(x_1,\cdots,x_n)-g(x_1^0,\cdots,x_n^0) f(x_1^0,\cdots,x_n^0)\\ =g(x_1,\cdots,x_n)f(x_1,\cdots,x_n)-g(x_1^0,\cdots,x_n^0)f(x_1,\cdots,x_n)\\ +g(x_1^0,\cdots,x_n^0)f(x_1,\cdots,x_n)-g(x_1^0,\cdots,x_n^0) f(x_1^0,\cdots,x_n^0) 其次,由可微性,就有 g ( x 1 , , x n ) g ( x 1 0 , , x n 0 ) = g ( x 1 0 , , x n 0 ) Δ x + o 1 ( Δ x ) g(x_1,\cdots,x_n)-g(x_1^0,\cdots,x_n^0)= g^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_1(||\Delta x||) f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) = f ( x 1 0 , , x n 0 ) Δ x + o 2 ( Δ x ) f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0)= f^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_2(||\Delta x||) 再令 h ( x 1 , , x n ) = f ( x 1 , , x n ) o 1 ( Δ x ) + [ f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) ] Δ x + g ( x 1 0 , , x n 0 ) o 2 ( Δ x ) h(x_1,\cdots,x_n)=f(x_1,\cdots,x_n)o_1(||\Delta x||) \\+[f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0)]\Delta x +g(x_1^0,\cdots,x_n^0)o_2(||\Delta x||) 因为 Δ x k Δ x 1 |\frac{\Delta x_k}{||\Delta x||}|\le 1 再由 f ( x 1 , , x n ) f(x_1,\cdots,x_n)
( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处的连续性,就有 lim Δ x 0 [ f ( x 1 , , x n ) f ( x 1 0 , , x n 0 ) ] Δ x Δ x = 0 \lim_{||\Delta x|| \to 0}{\frac{[f(x_1,\cdots,x_n)-f(x_1^0,\cdots,x_n^0)]\Delta x}{||\Delta x||}}=0 所以 lim Δ x 0 h ( x 1 , , x n ) Δ x = 0 \lim_{||\Delta x|| \to 0}{\frac{h(x_1,\cdots,x_n)}{||\Delta x||}}=0 F ( x 1 , , x n ) F ( x 1 0 , , x n 0 ) = [ g ( x 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) + g ( x 1 0 , , x n 0 ) f ( x 1 0 , , x n 0 ) ] Δ x + h ( x 1 , , x n ) F(x_1,\cdots,x_n)-F(x_1^0,\cdots,x_n^0)=[g(x_1^0,\cdots,x_n^0)f^\prime(x_1^0,\cdots,x_n^0) \\+g^\prime(x_1^0,\cdots,x_n^0)f(x_1^0,\cdots,x_n^0)]\Delta x + h(x_1,\cdots,x_n)

接下来咱们给出多元下的复合函数求导法则:

定理14.5 f f n n m m 维向量函数,在 ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可导, ( y 1 0 , , y m 0 ) = f ( x 1 0 , , x n 0 ) (y_1^0,\cdots,y_m^0)=f(x_1^0,\cdots,x_n^0) , g g m m k k 维向量函数,在 ( y 1 0 , , y m 0 ) (y_1^0,\cdots,y_m^0) 处可导,则 f ( g ( x 1 , , x n ) ) f(g(x_1,\cdots,x_n)) ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可导,而且
( g ( f ( x 1 0 , , x n 0 ) ) ) = g ( f ( x 1 0 , , x n 0 ) ) f ( x 1 0 , , x n 0 ) (g(f(x_1^0,\cdots,x_n^0)))^\prime = g^\prime(f(x_1^0,\cdots,x_n^0))f^\prime(x_1^0,\cdots,x_n^0)

证:
g g ( y 1 0 , , y m 0 ) (y_1^0,\cdots,y_m^0) 可微,则 g ( y 1 , , y m ) g ( y 1 0 , , y m 0 ) = g ( y 1 0 , , y m 0 ) Δ y + o 1 ( Δ y ) (1) \tag{1} g(y_1,\cdots,y_m)-g(y_1^0,\cdots,y_m^0)=\\ g^\prime(y_1^0,\cdots,y_m^0)\Delta y + o_1(||\Delta y||) 再由 f f ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,则 f ( x 1 , , x n ) ( y 1 0 , , y m 0 ) = f ( x 1 0 , , x n 0 ) Δ x + o 2 ( Δ x ) (2) \tag{2} f(x_1,\cdots,x_n)-(y_1^0,\cdots,y_m^0)= f^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_2(||\Delta x||) f ( x 1 , , x n ) ( y 1 , , y m ) Δ x = f ( x 1 0 , , x n 0 ) Δ x + o 2 ( Δ x ) Δ x \frac{||f(x_1,\cdots,x_n)-(y_1,\cdots,y_m)||}{||\Delta x||} =||\frac{f^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_2(||\Delta x||)}{||\Delta x||}|| 由范数的性质,有 f ( x 1 0 , , x n 0 ) Δ x + o 2 ( Δ x ) Δ x f ( x 1 0 , , x n 0 ) Δ x Δ x + o 2 ( Δ x ) Δ x ||\frac{f^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_2(||\Delta x||)}{||\Delta x||}|| \le ||\frac{f^\prime(x_1^0,\cdots,x_n^0)\Delta x}{||\Delta x||}||+||\frac{o_2(||\Delta x||)}{||\Delta x||}|| lim Δ x 0 o 2 ( Δ x ) Δ x = 0 \lim_{||\Delta x||\to 0}{\frac{o_2(||\Delta x||)}{||\Delta x||}} = 0 同时设 f i j ( x 1 0 , , x n 0 ) f_{ij}(x_1^0,\cdots,x_n^0) f f 的第 i i 个份量对第 j j 个变元的偏导数,则 f ( x 1 0 , , x n 0 ) Δ x = [ i = 1 n f 1 i ( x 1 0 , , x n 0 ) Δ x i i = 1 n f m i ( x 1 0 , , x n 0 ) Δ x i ] f^\prime(x_1^0,\cdots,x_n^0)\Delta x= \left[ \begin{matrix} \sum_{i=1}^n{f_{1i}(x_1^0,\cdots,x_n^0)\Delta x_i}\\ \cdots\\ \sum_{i=1}^n{f_{mi}(x_1^0,\cdots,x_n^0)\Delta x_i} \end{matrix} \right] 对任意的 1 i m 1\le i \le m ,都要 j 1 n f i j ( x 1 0 , , x n 0 ) Δ x j j = 1 n f i j 2 ( x 1 0 , , x n 0 ) Δ x |\sum_{j-1}^n f_{ij}(x_1^0,\cdots,x_n^0)\Delta x_j| \le \sqrt{\sum_{j=1}^n{f_{ij}^2(x_1^0,\cdots,x_n^0)}} ||\Delta x|| 所以 f ( x 1 0 , , x n 0 ) Δ x i = 1 m j = 1 n f i j 2 ( x 1 0 , , x n 0 ) Δ x ||f^\prime(x_1^0,\cdots,x_n^0)\Delta x|| \le \sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}{f_{ij}^2(x_1^0,\cdots,x_n^0)}}||\Delta x|| f ( x 1 0 , , x n 0 ) Δ x Δ x i = 1 m j = 1 n f i j 2 ( x 1 0 , , x n 0 ) ||\frac{f^\prime(x_1^0,\cdots,x_n^0)\Delta x}{||\Delta x||}|| \le \sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}{f_{ij}^2(x_1^0,\cdots,x_n^0)}} 所以, f ( x 1 0 , , x n 0 ) Δ x + o 2 ( Δ x ) Δ x ||\frac{f^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_2(||\Delta x||)}{||\Delta x||}|| 局部有界,所以 lim Δ x 0 f ( x 1 0 , , x n 0 ) Δ x + o 2 ( Δ x ) Δ x = 0 \lim_{||\Delta x||\to 0}{||\frac{f^\prime(x_1^0,\cdots,x_n^0)\Delta x + o_2(||\Delta x||)}{||\Delta x||}||}=0 再将(2)代入(1)就能够证得结论

考虑向量函数和多元函数复合的情形: g ( y 1 , , y m ) g(y_1,\cdots,y_m) m m 元函数, f ( x 1 , , x n ) f(x_1,\cdots,x_n) n n m m 维向量函数, g g f ( x 1 0 , , x n 0 ) f(x_1^0,\cdots,x_n^0) 处可微, f f ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微,那么 g ( f ( x 1 , , x n ) ) g(f(x_1,\cdots,x_n)) ( x 1 0 , , x n 0 ) (x_1^0,\cdots,x_n^0) 处可微。
咱们记 h = g ( f ) h=g(f) ,设 f = ( f 1 , , f m ) f=(f_1,\cdots,f_m) ,应用复合函数求导法则,就有: h ( x 1 0 , , x n 0 ) x i = j = 1 m f j ( x 1 0 , , x n 0 ) x i g ( y 1 0 , , y m 0 ) y j \frac{\partial h(x_1^0,\cdots,x_n^0)}{\partial x_i} =\sum_{j=1}^m{\frac{\partial f_j(x_1^0,\cdots,x_n^0)}{\partial x_i} \frac{\partial g(y_1^0,\cdots,y_m^0)}{\partial y_j}} 这称为多元函数求导的链式法则

高阶偏导数与高阶全微分

高阶偏导就是偏导的偏导,只不过,在高维情形下,由求偏导次序能否交换的问题。以二元函数为例, f ( x , y ) f(x,y) 的二阶偏导有四个:
2 f x 2 \frac{\partial^2 f}{\partial x^2} 表示对 x x 求两次偏导, 2 f x y \frac{\partial^2 f}{\partial x\partial y} 表示先对 x x 求偏导,再对 y y 求偏导,其余两个也能够相似写出。
问题在于 2 f x y = 2 f y x \frac{\partial^2 f}{\partial x\partial y}=\frac{\partial^2 f}{\partial y\partial x} 是否成立?下面咱们证实:在高阶偏导数连续的条件下,偏导次序是能够交换的。

定理14.6 f ( x , y ) f(x,y) ( x 0 , y 0 ) (x_0,y_0) 的某个邻域上可求二阶偏导数,而且 2 f x y , 2 f y x \frac{\partial^2 f}{\partial x\partial y},\frac{\partial^2 f}{\partial y\partial x} 都在 ( x 0 , y 0 ) (x_0,y_0) 处连续,则 2 f ( x 0 , y 0 ) x y = 2 f ( x 0 , y 0 ) y x \frac{\partial^2 f(x_0,y_0)}{\partial x\partial y}= \frac{\partial^2 f(x_0,y_0)}{\partial y\partial x}

证:
首先 [ f ( x , y ) f ( x , y 0 ) ] [ f ( x 0 , y ) f ( x 0 , y 0 ) ] ( x x 0 ) ( y y 0 ) = [ f ( x , y ) f ( x 0 , y ) ] [ f ( x , y 0 ) f ( x 0 , y 0 ) ] ( x x 0 ) ( y y 0 ) \frac{[f(x,y)-f(x,y_0)]-[f(x_0,y)-f(x_0,y_0)]}{(x-x_0)(y-y_0)} = \frac{[f(x,y)-f(x_0,y)]-[f(x,y_0)-f(x_0,y_0)]}{(x-x_0)(y-y_0)} 由拉格朗日中值定理,存在 ( 0 , 1 ) (0,1) 之间的正实数 θ 1 \theta_1 θ 2 \theta_2 [ f ( x , y ) f ( x , y 0 ) ] [ f ( x 0 , y ) f ( x 0 , y 0 ) ] ( x x 0 ) ( y y 0 ) = f x ( x 0 + θ 1 ( x x 0 ) , y ) f x ( x 0 + θ 1 ( x x 0 ) , y 0 ) y y 0 = f x y ( x 0 + θ 1 ( x x 0 ) , y 0 + θ 2 ( y y 0 ) ) \frac{[f(x,y)-f(x,y_0)]-[f(x_0,y)-f(x_0,y_0)]}{(x-x_0)(y-y_0)} \\= \frac{ f_x(x_0+\theta_1(x-x_0),y)-f_x(x_0+\theta_1(x-x_0),y_0) }{y-y_0}\\=f_{xy}(x_0+\theta_1(x-x_0),y_0+\theta_2(y-y_0)) h ( x , y ) = [ f ( x , y ) f ( x , y 0 ) ] [ f ( x 0 , y ) f ( x 0 , y 0 ) ] ( x x 0 ) ( y y 0 ) h(x,y)=\frac{[f(x,y)-f(x,y_0)]-[f(x_0,y)-f(x_0,y_0)]}{(x-x_0)(y-y_0)} h ( x , y ) h(x,y) 的全面极限和两个累次极限存在,相等,这样就能够证得偏导次序可交换

咱们假设二元函数在 ( x 0 , y 0 ) (x_0,y_0) 的某个邻域上各阶偏导数都存在,那么,各阶偏导数都在 ( x 0 , y 0 ) (x_0,y_0) 处连续(由于连续可微),考察函数: h ( t ) = f ( x 0 + t , y 0 + t ) h(t)=f(x_0+t,y_0+t) ,则 h ( t ) h(t) t = 0 t=0 处的各阶偏导数:
h ( k ) ( 0 ) = i = 0 k C k i k f ( x 0 , y 0 ) x i y ( k i ) h^{(k)}(0)=\sum_{i=0}^{k}{C_k^i \frac{\partial^k f(x_0,y_0)}{\partial x^i \partial y^{(k-i)}}} 这就和二项式定理相似,其余高阶导数的求法,大多用到数学概括法,这里再也不赘述。

方向导数

在一元状况下,导数有左导数和右导数,而在多元情形下,因为方向远远不止两个,但咱们仍是能够定义出方向导数。
方向向量就定义为 d = ( d 1 , , d n ) d=(d_1,\cdots,d_n) ,其中 k = 1 n d k 2 = 1 \sqrt{\sum_{k=1}^n{d_k^2}}=1 d d 就称为方向向量。方向导数就定义为极限:
f ( x ) d = lim t 0 + f ( x + t d ) f ( x ) t \frac{\partial f(x)}{\partial d}=\lim_{t\to 0^+}{\frac{f(x+td)-f(x)}{t}} 方向导数该如何计算呢?若是 f ( x ) f(x) f ( x 0 ) f(x_0) 处可微,那么
f ( x 0 + t d ) f ( x 0 ) = t f ( x 0 ) d T + o ( t ) f(x_0+td)-f(x_0)=tf^\prime(x_0)d^T+o(t) 这样
f ( x 0 ) d = f ( x 0 ) d T \frac{\partial f(x_0)}{\partial d} = f^\prime(x_0)d^T 实际上就是偏导按照方向进行加权。

高维泰勒公式

高维泰勒公式,就是 f ( x 0 + t ( x x 0 ) ) f(x_0+t(x-x_0)) 0 0 处的泰勒公式,再令 t = 1 t=1 ,高维泰勒公式形式比较复杂,在三阶以上很难写出通常的形式。不过,咱们这里给出零阶,一阶,二阶的泰勒公式的形式,在多元函数极值判断中起到重要的做用。咱们称矩阵 H f ( x 0 ) = [ f 11 ( x 0 ) f 12 ( x 0 ) f 1 n ( x 0 ) f 21 ( x 0 ) f 22 ( x 0 ) f 2 n ( x 0 ) f n 1 ( x 0 ) f n 2 ( x 0 ) f n n ( x 0 ) ] H_f(x_0) = \left[ \begin{matrix} f_{11}(x_0)&f_{12}(x_0)&\cdots&f_{1n}(x_0)\\ f_{21}(x_0)&f_{22}(x_0)&\cdots&f_{2n}(x_0)\\ \cdots&\cdots&\cdots&\cdots\\ f_{n1}(x_0)&f_{n2}(x_0)&\cdots&f_{nn}(x_0) \end{matrix}\right]