第四讲多元函数微分学

一、概念

1.第一种定义法(数学专业,同济六、七版)
- $设f(x,y)的定义域为D，P_0(x_0,y_0)是D的聚点，\\ \forall \varepsilon > 0, \exists \delta > 0, 当P(x,y) \in D \cap \mathring{U}(P_0,\delta)时，\\恒有|f(x,y)-A| < \varepsilon \Rightarrow \\ \lim_{x \to x_0, y \to y_0}f(x,y)=A$
  $聚点的概念:理解为内点+边界点，但边界点不一定是D的$
2.第二种定义法(非数学专业高等数学教材)
- $若二元函数f(x,y)在(x_0,y_0)的去心邻域内有定义，\\且(x,y)以任意方式趋向于(x_0,y_0)时,f(x,y)均 \to A 则\\ \lim_{x \to x_0, y \to y_0}f(x,y)=A$
$\lim_{x \to x_0, y \to y_0} \Leftrightarrow \lim_{(x,y) \to (x_0, y_0)}$ 同时趋向
[注]除了洛必达法则、单调有界准则、穷举法以外，可照搬一元函数求极限的方法。如
- 1.等价无穷小替换
- 2.无穷小 * 有界变量=无穷小
  - $\lim_{x->0^+}x\ln x = 0$
- 3.夹逼准则等

$若\lim_{x \to x_0, y \to y_0}f(x,y)=f(x_0,y_0),称f(x,y)在(x_0,y_0)连续$
- $[注]若"≠",叫不连续，多元时不讨论间断类型$

$z=f(x,y)$
$f'_x(x,y) = z'_x(x,y)= \frac {\partial f(x,y)}{\partial x} = \frac {\partial z(x,y)}{\partial x}$ 偏导的写法
$\frac {\partial f}{\partial x}|_{(x_0,y_0)}= f_x'(x_0,y_0) \triangleq \lim_{\Delta x \to 0 } \frac {f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$
$\frac {\partial f}{\partial y}|_{(x_0,y_0)}=f_y'(x_0,y_0) \triangleq \lim_{\Delta y \to 0 } \frac {f(x_0 ,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}$
$\partial 偏微分符号，拉格朗日发明，法文，读音类似round$

$1. \Delta z = f(x_0+ \Delta x, y_0+ \Delta y) - f(x_0,y_0) 全增量$
$$2. A\Delta x+B\Delta y \left{ \begin{array}{ll}
A = f_x'(x_0,y_0) \
B = f_y'(x_0,y_0) \end{array} \right. 线性增量$$
$3. \lim_{\Delta x \to 0, \Delta y \to 0}\frac{\Delta z - (A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}} = 0 \Rightarrow f(x,y)在c可微$
- $\Delta z - (A\Delta x+B\Delta y) = o(\sqrt{(\Delta x)^2+(\Delta y)^2} )$
- $\Delta z = (A\Delta x+B\Delta y) + o(\sqrt{(\Delta x)^2+(\Delta y)^2} )$ 全增量，线性主部，误差（error）
$微分学中 \Delta x =dx, \Delta y =dy \\得 dz|_{(x_0,y_0)}=f_x'(x_0,y_0)dx+f_y'(x_0,y_0)dy, dz为全微分$
$dz=f_x'(x,y)dx+f_y'(x,y)dy = \frac {\partial f}{\partial x}dx+\frac {\partial f}{\partial y}dy$

$1.用定义求f_x'(x_0,y_0),f_y'(x_0,y_0)$
$2.用公式求f_x'(x,y),f_y'(x,y)$
$3.验证\lim_{x \to x_0, y \to y_0}f'_x(x,y) 是否=f'_x(x_0,y_0) \\且\lim_{x \to x_0, y \to y_0}f'_y(x,y) 是否=f'_y(x_0,y_0) ,\\若均等，则称f(x,y)在(x_0,y_0)处偏导数连续$

$设 z=f(u,v,w),u=u(y),v=v(x,y),w=w(x),\\称x,y叫自变量,u,v,w叫中间变量,z叫因变量$
$复合变量结构图$
有几条路就有几项相加，每条路上有几段就有几个相乘
$\frac {\partial z}{\partial x} = \frac {\partial z}{\partial v} \bullet \frac {\partial v}{\partial x} + \frac {\partial z}{\partial w} \bullet \frac {dv}{dx}$
- $w 到 x 只有一条，所以用\frac {dv}{dx} 不用 \frac {\partial v}{\partial x}$
注意书写规范

$设 z=f(u,v,w),u=u(y),v=v(x,y),w=w(x),\\称x,y叫自变量,u,v,w叫中间变量,z叫因变量$
四种二阶偏导数
- $\frac {\partial(\frac {\partial z}{\partial x})}{\partial x} = \frac {\partial^2 z}{\partial x^2}$
- $\frac {\partial(\frac {\partial z}{\partial x})}{\partial y} = \frac {\partial^2 z}{\partial x \partial y }$
- $\frac {\partial(\frac {\partial z}{\partial y})}{\partial x} = \frac {\partial^2 z}{\partial y \partial x}$
- $\frac {\partial(\frac {\partial z}{\partial y})}{\partial y} = \frac {\partial^2 z}{\partial y^2}$
无论z对谁求导，也无论z已经求了几次导，新函数仍然与原来函数有完全相同的复合结构
[自注]这段没有看明白，到知乎上查了下，解释如下
- 1.偏导数是相对某个具体的坐标系才有意义的
- 2.链式法则是对复合函数才适用
[自注]这里我的理解是所谓中间变量其实是“坐标系的基”，结合线性代数就好理解了。
- 而链式法则一开始就是在转换坐标系，机器人学里的链式法则也是在转换坐标系，所以链式法则只能用于复合函数
- 所以那句“无论求了几次导……”就好理解了，不管怎么求导，也不会改变坐标系的基的，因为在过程中转换坐标系完成求导，但是最终结果还是在原坐标系

Last updated 5 years ago

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