Section01_基本概念

极限

• 一元 $\displaystyle \forall\ \epsilon >0, \exists\ \delta>0,\ \text{when } 0<\vert x-x_{0}\vert <\delta,\ \vert f(x)-A\vert < \epsilon$，记为$\displaystyle \lim_{x\to x_{0}}f(x) = A$
  • $\displaystyle \exists\ \lim_{x\to x_{x}}f(x)\Leftrightarrow f(x_{0}-0) = f(x_{0}+0)$
• 二元 $f(x,y)$ 在 $M_{0}(x_{0},y_{0})$ 的去心邻域内有定义 $\displaystyle \forall\ \epsilon>0,\exists\ \delta>0,\text{when } 0<\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}<\delta,\vert f(x,y)-A\vert < \epsilon$，记为 $\displaystyle \lim_{\substack{x\to x_{0}\\y\to y_{0}}} f(x,y) =A$

  例题

• $\displaystyle \lim_{\substack{x\to 0\\ y \to 2}}\frac{\sqrt{1+xy}-\sqrt{1-xy}}{x}$ $$\begin{split} & \lim_{\substack{x\to 0\\y\to 2}} \frac{\sqrt{1+xy}-\sqrt{1-xy}}{x} = \lim_{\substack{x\to 0\\y\to 2}} y \frac{\sqrt{1+xy}-\sqrt{1-xy}}{xy} \\ = & 2 \lim_{\substack{x\to 0\\ y\to 2}} \frac{\sqrt{1+xy}-\sqrt{1-xy}}{xy} \xlongequal{t=xy} 2\lim_{t\to 0} \frac{\sqrt{1+t}-\sqrt{1-t}}{t} \\ = & 2\lim_{t\to 0} \frac{1}{\sqrt{1+t}+\sqrt{1-t}} \cdot \frac{2t}{t} = 2 \end{split}$$
• $f(x,y)=\begin{cases}\displaystyle \frac{xy}{x^{2}+y^{2}}& (x,y)\ne (0,0)\\\displaystyle 0& (x,y) = (0,0)\end{cases}$求$\displaystyle \lim_{\substack{x\to 0\\ y\to 0}}f(x,y)$ $$\begin{array}{ll} \because & \displaystyle \lim_{\substack{x\to 0\\ y=x}} =\frac{1}{2} \ne \lim_{\substack{x\to 0 \\ y=-x}} = -\frac{1}{2} \\ \therefore & \displaystyle \lim_{\substack{x\to 0\\y\to0}}f(x,y)\ \nexists\\ \end{array}$$

连续

• 一元 若 $\displaystyle \lim_{x\to x_{0}} f(x) = f(x_{0})\Leftrightarrow f(x_{0}-0)=f(x_{0}+0)=f(x_{0})$ 则称$f(x)$在$x=x_{0}$处连续
• 二元 若 $\displaystyle \lim_{\substack{x\to x_{0}\\y\to y_{0}}}f(x,y) = f(x_{0},y_{0})$ 则称 $f(x,y)$ 在 $(x_{0},y_{0})$ 处连续

偏导数

$$z=f(x,y)\quad (x,y)\in \mathbb{D},M_{0}(x_{0},y_{0})\in\mathbb{D}$$

增量

1. $\Delta z_{x}$ $f(x,y)$在$M_{0}$处关于$x$的偏增量 $$\Delta z_{x} = f(x_{0}+\Delta x,y_{0}) - f(x_{0},y_{0}) = f(x, y_{0}) - f(x_{0},y_{0})$$
2. $\Delta z_{y}$ $f(x,y)$在$M_{0}$处关于$y$的偏增量 $$\Delta z_{y} = f(x_{0},y_{0} + \Delta y) - f(x_{0},y_{0}) = f(x_{0}, y) - f(x_{0},y_{0})$$
3. $\Delta z$ $f(x,y)$在$M_{0}$处的全增量 $$\Delta z = f(x_{0}+\Delta x,y_{0} + \Delta y) - f(x_{0},y_{0}) = f(x, y) - f(x_{0},y_{0})$$

偏导数

1. 若 $\displaystyle \lim_{\Delta x\to 0} \frac{\Delta z_{x}}{\Delta x} = \lim_{x\to x_{0}}\frac{f(x,y_{0})-f(x_{0},y_{0})}{x-x_{0}}\ \exists$ 则称 $f(x,y)$ 在 $M_{0}$ 处关于 $x$ 可偏导，记为$\displaystyle f_{x}(x,y)\text{或}\left.\frac{\partial z}{\partial x}\right\vert_{M_{0}}$
2. 若 $\displaystyle \lim_{\Delta y\to 0} \frac{\Delta z_{y}}{\Delta y} = \lim_{y\to y_{0}}\frac{f(x_{0},y)-f(x_{0},y_{0})}{y-y_{0}}\ \exists$ 则称 $f(x,y)$ 在 $M_{0}$ 处关于 $y$ 可偏导，记为$\displaystyle f_{y}(x,y)\text{或}\left.\frac{\partial z}{\partial y}\right\vert_{M_{0}}$

可（全）微

• 一元 $y = f(x), x\in\mathbb{D}, x_{0}\in\mathbb{D}$，若 $\Delta y = A\Delta x + o(\Delta x)$ 则称 $f(x)$ 在 $x_{0}$ 处可微，$A\Delta x = Adx \triangleq dy\vert_{x=x_{0}}$
  • 可导 <=> 可微
  • $A=f'(x_{0})$
  • $y=f(x)$ 可导，则 $dy = df(x) = f'(x)\cdot dx$
• 二元 $z=f(x,y), (x,y)\in\mathbb{D}, M_{0}(x_{0},y_{0})\in\mathbb{D}$，若 $\Delta z = A\Delta x+B\Delta y+o(\rho)\quad \rho = \sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}$ 则称 $f(x,y)$ 在 $(x_{0},y_{0})$ 处可全微，$A\Delta x + B\Delta y = Adx + Bdy \triangleq dz\vert_{M_{0}}$ 为 $z=f(x,y)$ 在 $M_{0}$ 处的全微分