微积分下基础题型

待我称王封你为后i 2023-01-11 08:49 173阅读 0赞

**目录**

偏导

多元函数求偏导

求多元函数的二阶偏导

求多元复合函数的偏导

求多元隐函数的偏导

多元函数的全微分

多元复合函数的全微分

已知全微分,求未知数

多元函数求极值

多元隐函数求极值

多元函数求最值

四个小知识点

空间向量

求向量的长度

求向量的点乘

求向量之间的夹角

求一个向量在另一个向量方向上的投影

向量垂直

向量平行

空间几何(上)

求过三点的平面方程

判断面与面、面与向量的关系

已知面过一点和其法向量,求面

求点到面的距离

求两个面的交线方程

线与线、线与面的关系

已知线过一点和其方向向量,求线

求点到线的距离

求x=x(t)  y=y(t)  z=z(t)形式的曲线在某点处的切线和法平面

求x、y、z写在一起的曲线在某点处的切线与法平面

求曲面在某点处的切平面与法线

二重积分

计算形式的二重积分

交换积分次序

计算格式的二重积分

积分区域与圆有关的二重积分

积分区域对称的二重积分

三重积分

第一类曲线积分

函数在圆上的积分

函数在y=y(x)上的积分

利用性质计算

第二类曲线积分

两个函数在圆上的积分

两个函数在线上的积分

利用性质计算

第一类曲面积分

已知面z=z(x,y)，计算某一个函数在这个面上的积分

已知面x=x(y,z)，计算某一个函数在这个面上的积分

已知面y=y(x,z)，计算某一个函数在这个面上的积分

函数在体的表面的积分

第二类曲面积分函数

在某一个曲面上关于dydz的积分

函数在某一个曲面上关于dxdy的积分

函数在某一个曲面上关于dxdz的积分

判断级数的敛散性

判断正项级数的敛散性

判断交错级数的敛散性

判断绝对收敛/条件收敛

幂级数

已知幂级数在某点收敛/发散，判断其在另一点的敛散性

求幂级数的收敛域/收敛区间

求幂级数的收敛半径

求幂级数在收敛域内的和函数

展成幂级数

--------------------

# 偏导 #

### 多元函数求偏导 ###

是把下方的字母当未知数，其它字母当常数，对总函数求导

在（a,b,c）点的偏导=分别求xyz的偏导，将x=a，y=b，z=c代入求出的三个数

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 1][]

### 求多元函数的二阶偏导 ###

先对分母的前一项求偏导，再对偏导结果在求偏导

偏导对象不变，偏导顺序不改变最终结果

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 2][]

### 求多元复合函数的偏导 ###

把复杂部分设为u，v等    --->   套公式

一元函数用d，多元函数用![\\partial][partial]

栗子1

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 3][]

栗子2

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 4][]

栗子2变式

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 5][]

### 求多元隐函数的偏导 ###

无法提出z=。。。，则为多元隐函数

已知。。。=0

1.  设F=。。。
2.  分别求F对x，y，z的偏导
3.  然后代公式，公式如下

![20210126112433526.png][]

例子1）求一阶偏导

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 6][]

例子2）求二阶偏导

z包含x，就不能把z当成常数

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 7][]

然后代入红黄框就OK了

全微分及偏导的应用

### 多元函数的全微分 ###

全微分dZ，有几项就由未知数的个数决定

![dZ=\\frac\{\\partial z\}\{\\partial x\}dx+\\frac\{\\partial z\}\{\\partial y\}dy][dZ_frac_partial z_partial x_dx_frac_partial z_partial y_dy]

例子1

![20210126113359992.png][]

例子2

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 8][]

### 多元复合函数的全微分 ###

公式：![dZ=\\frac\{\\partial z\}\{\\partial x\}dx+\\frac\{\\partial z\}\{\\partial y\}dy][dZ_frac_partial z_partial x_dx_frac_partial z_partial y_dy]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 9][]

### 已知全微分,求未知数 ###

公式![dZ=\\frac\{\\partial z\}\{\\partial x\}dx+\\frac\{\\partial z\}\{\\partial y\}dy][dZ_frac_partial z_partial x_dx_frac_partial z_partial y_dy]

公式![\\frac\{\\partial ^2z\}\{\\partial x\\partial y\}=\\frac\{\\partial ^2z\}\{\\partial y\\partial x\}][frac_partial _2z_partial x_partial y_frac_partial _2z_partial y_partial x]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 10][]

### 多元函数求极值 ###

方法：

1.  求出同时满足z对x的偏导=0和z对y的偏导=0的一对（x,y）
2.  求![A=\\frac\{\\partial ^2z\}\{\\partial x^2\}][A_frac_partial _2z_partial x_2]     ![B=\\frac\{\\partial ^2z\}\{\\partial x\\partial y\}][B_frac_partial _2z_partial x_partial y]    ![C=\\frac\{\\partial ^2z\}\{\\partial y^2\}][C_frac_partial _2z_partial y_2]  的 值
3.  ![B^2-AC<0][B_2-AC_0]    A<0极大值点  A>0极小值点
4.  ![B^2-AC=0][B_2-AC_0 1]不确定
5.  ![B^2-AC>0][B_2-AC_0 2]不是极值点

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 11][]

![20210126114750801.png][]

### 多元隐函数求极值 ###

方法：

1.  求出同时满足     z对x的偏导=0     和     z对y的偏导=0   和   原方程    的一对（x,y,z）
2.  求![A=\\frac\{\\partial ^2z\}\{\\partial x^2\}][A_frac_partial _2z_partial x_2]     ![B=\\frac\{\\partial ^2z\}\{\\partial x\\partial y\}][B_frac_partial _2z_partial x_partial y]    ![C=\\frac\{\\partial ^2z\}\{\\partial y^2\}][C_frac_partial _2z_partial y_2]  的 值
3.  ![B^2-AC<0][B_2-AC_0]    A<0极大值点  A>0极小值点
4.  ![B^2-AC=0][B_2-AC_0 1]不确定
5.  ![B^2-AC>0][B_2-AC_0 2]不是极值点

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 12][]

![20210126114933971.png][]

### 多元函数求最值 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 13][]

![2021012611514635.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 14][]

![20210126115235736.png][]

### 四个小知识点 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 15][]

# 空间向量 #

### 求向量的长度 ###

![20210126115527191.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 16][]

### 求向量的点乘 ###

![20210126115541765.png][]

![20210126115553484.png][]

例子1

![20210126115607852.png][]

例子2

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 17][]

### 求向量之间的夹角 ###

公式：![20210126115809818.png][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 18][]

### 求一个向量在另一个向量方向上的投影 ###

公式：

![20210126115941940.png][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 19][]

### 向量垂直 ###

若向量a和向量b垂直，则其点乘为0

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 20][]

向量叉乘

若向量![\\vec\{c\}=\\vec\{a\}\\times \\vec\{b\}][vec_c_vec_a_times _vec_b]，则c与a垂直，c与b垂直，c与ab所在平面垂直

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 21][]

### 向量平行 ###

当向量a和向量b平行，![\\vec\{a\}\\times \\vec\{b\}=0][vec_a_times _vec_b_0]，每一项成比例

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 22][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 23][]

# 空间几何(上) #

### 求过三点的平面方程 ###

![Ax+By+Cz+D=0][Ax_By_Cz_D_0]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 24][]

### 判断面与面、面与向量的关系 ###

公式

![20210126120921918.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 25][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 26][]

### 已知面过一点和其法向量,求面 ###

![20210126121036826.png][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 27][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 28][]

### 求点到面的距离 ###

![20210126121242588.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 29][]

### 求两个面的交线方程 ###

![20210126121359699.png][]

### 线与线、线与面的关系 ###

![2021012612145326.png][]

![20210126121526257.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 30][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 31][]

### 已知线过一点和其方向向量,求线 ###

![20210126121738829.png][]

例子1

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 32][]

例子2

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 33][]

### 求点到线的距离 ###

方法一

![20210126121929922.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 34][]

方法二

点线距可用叉乘来求

1.  取直线上一点和直线外所求点构成向量
2.  求出直线方向向量
3.  两向量做叉乘，叉乘结果求长度后除以方向向量长度

### 求x=x(t)  y=y(t)  z=z(t)形式的曲线在某点处的切线和法平面 ###

![20210126122859413.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 35][]

### 求x、y、z写在一起的曲线在某点处的切线与法平面 ###

![20210126123048518.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 36][]

### 求曲面在某点处的切平面与法线 ###

![20210126123205912.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 37][]

# 二重积分 #

### 计算![\\int dx\\int dy][int dx_int dy]形式的二重积分 ###

1.  把未知数集中到后边
2.  计算后半部分积分
3.  将计算结果代入前半部分中间

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 38][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 39][]

### 交换积分次序 ###

例子1

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 40][]

例子2

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 41][]

例子3

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 42][]

### 计算![\\int \\int d\\sigma][int _int d_sigma]格式的二重积分 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 43][]

### 积分区域与圆有关的二重积分 ###

![x=rcos\\theta][x_rcos_theta]   ![y=rsin\\theta][y_rsin_theta]  ![dxdy=rd\\theta dr][dxdy_rd_theta dr]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 44][]

### 积分区域对称的二重积分 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 45][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 46][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 47][]

# 三重积分 #

**方法**

*  **第一步：**结合表,表示出![\\Omega][Omega]，并用z=?的形式表示出下表面与上表面
 *  下表面：从z坐标轴向上看的面
 *  上表面：从z坐标轴向上看的面
 *  **第二步：**求出![\\int\_\{down\}^\{up\}f(x,y,z)dz][int_down_up_f_x_y_z_dz]，结果记为g(x,y)  
 *  up     为  z上表面  
 *  down为  z下表面
 *  **第三步：**求出![\\Omega][Omega]在xoy面的投影D
 *  **第四步：**计算![\\int \\int g(x,y)d\\sigma][int _int g_x_y_d_sigma]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 48][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 49][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 50][]

#  第一类曲线积分 #

### 函数在圆上的积分 ###

![20210126134545411.png][]

L为曲线，f为函数

公式

![20210126134750708.png][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 51][]

### 函数在y=y(x)上的积分 ###

![20210126134854176.png][]

公式

![20210126134911686.png][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 52][]

综合变式子

![20210126135114211.png][]

### 利用性质计算![\\int\_\{L\}f(x,y)ds][int_L_f_x_y_ds] ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 53][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 54][]

# 第二类曲线积分 #

### 两个函数在圆上的积分 ###

![2021012613555399.png][]

公式

![20210126135724634.png][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 55][]

### 两个函数在线上的积分 ###

![20210126135800334.png][]

公式![20210126135906740.png][]

例子1

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 56][]

例子2

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 57][]

### 利用性质计算![\\int \_\{L\}P(x,y)dx+Q(x,y)dy][int _L_P_x_y_dx_Q_x_y_dy] ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 58][]

例子

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 59][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 60][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 61][]

变式

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 62][]

# 第一类曲面积分 #

### 已知面z=z(x,y)，计算某一个函数在这个面上的积分 ###

![20210127102121339.png][]

**公式**

![20210127102858980.png][]

**做题步骤**

①画出![\\sum][sum]，并表示出Dxy  
②将![\\sum][sum]表示成z=?的形式，并求出![f(x,y,z(x,y))][f_x_y_z_x_y]  
③求出![\\frac\{\\partial z\}\{\\partial x\}][frac_partial z_partial x]   ![\\frac\{\\partial z\}\{\\partial y\}][frac_partial z_partial y]  
④代入公式，求出答案

**例题1**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 63][]

**例题2**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 64][]

### 已知面x=x(y,z)，计算某一个函数在这个面上的积分 ###

**公式**

![20210127103842884.png][]

**做题步骤**

①画出![\\sum][sum]，并表示出Dyz  
②将![\\sum][sum]表示成x=?的形式，并求出![f(x(y,z),y,z)][f_x_y_z_y_z]  
③求出![\\frac\{\\partial x\}\{\\partial y\}][frac_partial x_partial y]   ![\\frac\{\\partial x\}\{\\partial z\}][frac_partial x_partial z]  
④代入公式，求出答案

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 65][]

### 已知面y=y(x,z)，计算某一个函数在这个面上的积分 ###

**公式**

![20210127104041353.png][]

**做题步骤**

①画出![\\sum][sum]，并表示出Dxz  
②将![\\sum][sum]表示成y=?的形式，并求出![f(x,y(x,z),z)][f_x_y_x_z_z]  
③求出![\\frac\{\\partial y\}\{\\partial x\}][frac_partial y_partial x]   ![\\frac\{\\partial y\}\{\\partial z\}][frac_partial y_partial z]  
④代入公式，求出答案

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 66][]

### 函数在体的表面的积分 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 67][]

# 第二类曲面积分函数 #

### 在某一个曲面上关于dydz的积分 ###

![20210127104520971.png][]

**公式**

![20210127104512885.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 68][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 69][]

**例题**

![20210127104847171.png][]

### 函数在某一个曲面上关于dxdy的积分 ###

**公式**

![20210127105102355.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 70][]

### 函数在某一个曲面上关于dxdz的积分 ###

**公式**

![20210127105159851.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 71][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 72][]

# 判断级数的敛散性 #

### 判断正项级数的敛散性 ###

![20210127105528463.png][]

正项级数：好多项相加，每一项都是正的，项项间有关系

**公式：**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 73][]

进一步判断

![20210127110011167.png][]

**例题**

![20210127105534513.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 74][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 75][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 76][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 77][]

### 判断交错级数的敛散性 ###

![202101271102449.png][]

**公式**

![20210127110259536.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 78][]

### 判断绝对收敛/条件收敛 ###

**公式**

**正项级数**

*  若级数发散，则发散;
 *  若级数收敛，则绝对收敛。

**交错级数**

*  若级数发散，则发散;
 *  若级数收敛，则去掉(-1)的n次方项，变为正项级数:

**交错级数变为正项级数**

*  正项级数收敛，则绝对收敛;
 *  正项级数发散，则条件收敛。

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 79][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 80][]

#  幂级数 #

### 已知幂级数在某点收敛/发散，判断其在另一点的敛散性 ###

**公式**

![20210127111049683.png][]

**例题**

![20210127111158978.png][]

**例题**

![20210127111226499.png][]

### 求幂级数的收敛域/收敛区间 ###

**公式**

![20210127111304824.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 81][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 82][]

### 求幂级数的收敛半径 ###

**公式**

![20210127111847754.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 83][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 84][]

### 求幂级数在收敛域内的和函数 ###

**公式**

![20210127112306462.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 85][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 86][]

**例题：乘x或乘x^2**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 87][]

**例题：提1/x或1/（x^2）**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 88][]

**例题：多个n  ->  多次求导或积分**

**公式**

![20210127112726227.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 89][]

### 展成幂级数 ###

**方法**

![20210127112814872.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 90][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 91][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 92][]

![20210127112935843.png][]

**例题**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 93][]

![2021012711301930.png][]

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70]: /images/20221119/6a16748204214c0b86ed5fef1d3301ca.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 1]: /images/20221119/4a7383fb89ad4c09844fdc7980f351f0.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 2]: /images/20221119/fc461e804d22489eb4038cb3263b512a.png
[partial]: https://private.codecogs.com/gif.latex?%5Cpartial
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 3]: /images/20221119/15e217098b644094966d932b0e41c1b7.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 4]: /images/20221119/863d4dfc5d8b4f3582e31722a6061413.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 5]: /images/20221119/e43730cba7364f15a76d479e1fe0a768.png
[20210126112433526.png]: /images/20221119/e8114e7ffe094671857054a275652116.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 6]: /images/20221119/d7364fdbbd764ddea2019f3058ae9a86.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 7]: /images/20221119/7d00a10fde3a464a8c9ec224f1964865.png
[dZ_frac_partial z_partial x_dx_frac_partial z_partial y_dy]: https://private.codecogs.com/gif.latex?dZ%3D%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x%7Ddx&plus;%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20y%7Ddy
[20210126113359992.png]: /images/20221119/d42758c421e44800ac2d2ce38e2732f6.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 8]: /images/20221119/4b9de6e2ba02426d80acea14180fcd20.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 9]: /images/20221119/db10e55ea9c64afb9318365be72e360c.png
[frac_partial _2z_partial x_partial y_frac_partial _2z_partial y_partial x]: https://private.codecogs.com/gif.latex?%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20y%5Cpartial%20x%7D
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 10]: /images/20221119/8035cc2fb86747dc944f97f371486445.png
[A_frac_partial _2z_partial x_2]: https://private.codecogs.com/gif.latex?A%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5E2%7D
[B_frac_partial _2z_partial x_partial y]: https://private.codecogs.com/gif.latex?B%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5Cpartial%20y%7D
[C_frac_partial _2z_partial y_2]: https://private.codecogs.com/gif.latex?C%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20y%5E2%7D
[B_2-AC_0]: https://private.codecogs.com/gif.latex?B%5E2-AC%3C0
[B_2-AC_0 1]: https://private.codecogs.com/gif.latex?B%5E2-AC%3D0
[B_2-AC_0 2]: https://private.codecogs.com/gif.latex?B%5E2-AC%3E0
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 11]: /images/20221119/f0b4d478c9af4d94bf2cfe9f63eb8970.png
[20210126114750801.png]: /images/20221119/1773b5914aae493486cd5b87698b050d.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 12]: /images/20221119/0781b66d42ae4de8b36e9a523344fb0d.png
[20210126114933971.png]: /images/20221119/2184d5c352f1400e8aa4eb43bd75286d.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 13]: /images/20221119/996535b9f756475fb3b7700273a1a911.png
[2021012611514635.png]: /images/20221119/6d94fd30927d4af5b09b55c76ba336f5.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 14]: /images/20221119/8577fa4cefc84098b370c06bec4b25c2.png
[20210126115235736.png]: /images/20221119/54dfde77f993460ebb32a1a70eab5c5e.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 15]: /images/20221119/984610a3fb8745259bb4e62c49c06aae.png
[20210126115527191.png]: /images/20221119/d3d5c461d8be4868be22da81529367b2.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 16]: /images/20221119/3b211a0270c543a69d53e64f8f7419c5.png
[20210126115541765.png]: /images/20221119/4db9d3436da3404bbbab9169a02e10bf.png
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