Questions in category: 多元函数 (Multivariate functions)

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## 1. 设 $z=z(x,y)$ 可微, 且满足 $x^2\frac{\partial z}{\partial x}+y^2\frac{\partial z}{\partial y}=z^2$.

Posted by haifeng on 2023-04-25 13:25:09 last update 2023-04-25 13:38:33 | Answers (1) | 收藏

$\begin{cases} u&=x,\\ v&=\frac{1}{y}-\frac{1}{x}, \end{cases}\quad\text{及}\quad w=\frac{1}{z}-\frac{1}{x},$

## 2. 重极限存在无法推出累次极限也存在的例子.

Posted by haifeng on 2023-03-30 22:35:57 last update 2023-03-30 22:37:23 | Answers (0) | 收藏

$\lim_{(x,y)\rightarrow(0,0)}y\sin\frac{1}{x}=0,$

$\lim_{x\rightarrow 0}\lim_{y\rightarrow 0}y\sin\frac{1}{x}=\lim_{x\rightarrow 0}0=0,$

$\lim_{x\rightarrow 0}y\sin\frac{1}{x}$

## 3. 混合偏导数相等的条件

Posted by haifeng on 2023-03-30 21:54:16 last update 2023-03-30 22:30:40 | Answers (0) | 收藏

$\begin{split} &\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{xy}\\ =&\frac{1}{xy}\Bigl[\bigl(f(x,y)-f(x,0)\bigr)-\bigl(f(0,y)-f(0,0)\bigr)\Bigr]\\ =&\frac{1}{x}\Bigl(f_y(x,\theta y)-f_y(0,\theta_1 y)\Bigr)\\ =&\frac{1}{x}\Bigl(f_y(x,\theta y)-f_y(0,\theta y)+f_y(0,\theta y)-f_y(0,\theta_1 y)\Bigr)\\ =&f_{xy}(\sigma x,\theta y)+\frac{f_y(0,\theta y)-f_y(0,\theta_1 y)}{x} \end{split}$

$\begin{split} &\lim_{x\rightarrow 0}\lim_{y\rightarrow 0}\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{xy}\\ =&\lim_{x\rightarrow 0}\varlimsup_{y\rightarrow 0}f_{xy}(\sigma x,\theta y)+\lim_{x\rightarrow 0}\lim_{y\rightarrow 0}\frac{f_y(0,\theta y)-f_y(0,\theta_1 y)}{x}\\ =&f_{xy}(0,0)+\lim_{x\rightarrow 0}\frac{f_y(0,0)-f_y(0,0)}{x}\\ =&f_{xy}(0,0) \end{split}$

$\begin{split} &\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{yx}\\ =&\frac{1}{yx}\Bigl[\bigl(f(x,y)-f(0,y)\bigr)-\bigl(f(x,0)-f(0,0)\bigr)\Bigr]\\ =&\frac{1}{y}\Bigl(f_x(\sigma x,y)-f_x(\sigma_1 x,0)\Bigr)\\ =&\frac{1}{y}\Bigl(f_x(\sigma x,y)-f_x(\sigma x,0)+f_x(\sigma x,0)-f_x(\sigma_1 x,0)\Bigr)\\ =&f_{yx}(\sigma x,\theta y)+\frac{f_x(\sigma x,0)-f_x(\sigma_1 x,0)}{y} \end{split}$

$\begin{split} &\lim_{y\rightarrow 0}\lim_{x\rightarrow 0}\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{xy}\\ =&\lim_{y\rightarrow 0}\varlimsup_{x\rightarrow 0}f_{yx}(\sigma x,\theta y)+\lim_{y\rightarrow 0}\lim_{x\rightarrow 0}\frac{f_x(\sigma x, 0)-f_x(\sigma_1 x,0)}{y}\\ =&f_{yx}(0,0)+\lim_{y\rightarrow 0}\frac{f_x(0,0)-f_x(0,0)}{y}\\ =&f_{yx}(0,0). \end{split}$

$\lim_{(x,y)\rightarrow (0,0)}\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{xy}=f_{xy}(0,0).$

$\begin{split} &\lim_{y\rightarrow 0}\lim_{x\rightarrow 0}\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{xy}\\ =&\lim_{x\rightarrow 0}\lim_{y\rightarrow 0}\frac{f(x,y)-f(x,0)-f(0,y)+f(0,0)}{xy}.\\ \end{split}$

References:

[1]  梅加强  著  《数学分析》

## 4. 设 $2\sin(x+2y-3z)=x+2y-3z$, 证明: $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=1$.

Posted by haifeng on 2023-03-29 13:16:42 last update 2023-03-29 13:16:42 | Answers (2) | 收藏

## 5. 设 $w=f(x+y+z,xyz)$, $f$ 具有二阶连续偏导数, 求 $\frac{\partial^2 w}{\partial x\partial z}$.

Posted by haifeng on 2023-03-29 13:07:13 last update 2023-03-29 13:07:13 | Answers (1) | 收藏

## 6. 设 $z=xy+yf(\frac{x}{y})$, 其中 $f(u)$ 为可导函数, 证明: $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=xy+z$.

Posted by haifeng on 2023-03-29 12:57:44 last update 2023-03-29 12:57:44 | Answers (1) | 收藏

## 7. 计算下列数的近似值.

Posted by haifeng on 2023-03-25 08:29:46 last update 2023-03-25 08:29:46 | Answers (1) | 收藏

(1)  $(1.007)^{2.98}$

## 8. 求下列函数的全微分

Posted by haifeng on 2023-03-25 08:16:36 last update 2023-03-25 08:55:53 | Answers (1) | 收藏

(4)  $z=\frac{2x-y}{x+2y}$

(5)  $u=x(x+y^2+z^3)$

## 9. 设 $r=\sqrt{x^2+y^2+z^2}$, 证明: $\frac{\partial^2(\ln r)}{\partial x^2}+\frac{\partial^2(\ln r)}{\partial y^2}+\frac{\partial^2(\ln r)}{\partial z^2}=\frac{1}{r^2}$.

Posted by haifeng on 2023-03-25 08:03:12 last update 2023-03-25 08:03:12 | Answers (1) | 收藏

## 10. 证明下列极限不存在.

Posted by haifeng on 2023-03-17 22:39:05 last update 2023-03-17 22:39:05 | Answers (1) | 收藏

1.

$\lim\limits_{(x,y)\rightarrow(0,0)}\frac{x+y}{x-y}$

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