微积分下基本概念-同步电大进度

以你之姓@ 2022-10-23 14:24 198阅读 0赞

# 基础知识 #

### 求导公式 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70][]

### **范德蒙行列式** ###

![2021030611491264.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 1][]

### 变限积分函数求导公式： ###

![20210310205712284.png][]

# 多元函数微分学 #

**领域**：空间一点的周围很小的部分

**空心领域**：空间一点的周围很小的部分，不含这点

**D**：一片指定的区域

**内点**：一点存在领域在D内

**边界点**：一点所有领域都存在属于D和不属于D，点可在D上D内D外

**边界**：D的边界点集合

**外点**:存在领域不在D内

**聚点**：所有领域都要有D内非P的点

**开集**：D中每个点都是内点

**开区域**：无边

**闭区域**：有边

**n元函数**：R^n的一个子集

**等值线**：相当于地图等高线

**heinz归结原则**：当P以任何方式趋于Po时，极限相同

**区域**：无孤立点

**区域**：相当于一元函数的定义域和值域等

**一元函数**：一条曲线到一个数集上的映射

**多元函数**：平面、空间至更高维度区域到一个数集上的映射

**初等函数**：基本初等函数通过有限次四则运算、复合组成的可用解析式写出的函数

### 多元函数的极限： ###

**两种方法判断不存在**

①以一种解析式接近时极限不存在，y=kx，y=x，![y=\\frac\{1\}\{2\}x][y_frac_1_2_x]，![y=kx^2][y_kx_2]，![x=ky^2][x_ky_2]（目的是让x或者y的次数相同，从而消去x或者y）

②以两种不同的解析式接近时极限不同

**四种方法求出极限**

①极限定义

②无穷小与有限函数的乘积为0

③多元函数换元成一元函数

④夹逼准则

**其它方法求极限**

*  多元函数化成一元函数：同乘同除一个变量，用等价无穷小消去另一个变量
 *  多元初等函数直接代入极限
 *  两种重要极限：化为e

### 多元函数的连续和间断 ###

**连续**：D内聚点，极限=值，同求极限的方法

**不连续**：极限不存在或无定义

**连续定理**：所有初等多元函数在定义域内都是连续的

**间断点**：不在定义域内的一切点

**有界闭区域多元性质：最大最小值定理：**在定义域内最大最小值至少出现一次

**有界闭区域多元性质：介值定理：**在不同两点间，一定存在一点值在这两个值之间

**偏导数**：对x偏导，就将其它字母看成常数，然后求导（注意：如果指数和底数同时存在自变量，要把指数函数两边求In再两边求导）

**混合二阶偏导数**：注意分母的顺序，虽然顺序不影响偏导结果

**一元函数微分，可微**：y=f(x)          ![\\Delta y=A\\Delta x+o(\\Delta x)][Delta y_A_Delta x_o_Delta x]           ![dy=A\\Delta x=f\{\}'(x)dx][dy_A_Delta x_f_x_dx]

**多元函数微分，可微**：z=f(x,y)  ![\\Delta z=A\\Delta x+B\\Delta y+o(\\rho )][Delta z_A_Delta x_B_Delta y_o_rho]    ![\\rho =\\sqrt\{\\Delta x^2+\\Delta y^2\}][rho _sqrt_Delta x_2_Delta y_2]    ![dz=\\frac\{\\delta z\}\{\\delta x\}dx+\\frac\{\\delta z\}\{\\delta y\}dy][dz_frac_delta z_delta x_dx_frac_delta z_delta y_dy]

**判断在一点(a,b)可微的步骤:1.求出在这点的偏导数   2.令  ** ![\\rho =\\sqrt\{x^2+ y^2\}][rho _sqrt_x_2_ y_2]    ![\\Delta z=f(x,y)-f(a,b)][Delta z_f_x_y_-f_a_b]  3.求极限![lim\\varrho ->0\\frac\{\\Delta z-xianxingzhubu\}\{\\varrho \}=0][lim_varrho -_0_frac_Delta z-xianxingzhubu_varrho _0]则可微

**偏增量**：一个自变量不变，另一个变

**偏导数**：定义![\\Delta x->0][Delta x-_0]的极限下，x的偏增量除以![\\Delta x][Delta x]；只有分段函数在分段点的偏导用定义，不然就用求导代值的方法

**用定义求一点偏导**：![f\_\{x\}(0,0)=lim\\frac\{f(\\Delta x,0)-f(0,0)\}\{\\Delta x\}][f_x_0_0_lim_frac_f_Delta x_0_-f_0_0_Delta x]，两点直接代值

**z=f(x,y)+f(x+y)求z对x，z对y的偏导**：相当于一元函数求导，

f（x，y）对x求偏导，f加撇，乘y，对y求偏导，f加撇，乘x；

f（x+y）对x求偏导，对y求偏导，f加撇

f（x,g(y)）对x求偏导，f加撇，对y求偏导，f加撇，乘![g\{\}'(y)][g_y]；

**全偏量**：两个自变量同时发生改变

**全微分：**![dz=\\frac\{\\delta z\}\{\\delta x\}dx+\\frac\{\\delta z\}\{\\delta y\}dy][dz_frac_delta z_delta x_dx_frac_delta z_delta y_dy]

**可微则可导，连续**

一元函数：可微=可导

多元函数：可微-》可导

**在一点存在连续的偏导数则可微**

**偏导在一点的极限不存在，则不连续，则不可微**

**多元初等函数在定义域内可微**

### 函数偏导与连续性的关系 ###

**一元函数：**可导一定连续，连续不一定可导

**多元函数：**可导不一定连续，连续不一定可导

### 偏导数的几何意义 ###

f(x,y)是空间中的不规则曲面，上面存在一点P(![x\_\{0\},y\_\{0\}][x_0_y_0])，关于P点对x的偏导数的几何意义为

平行于xoz平面的平面与f(x,y)交一条曲线

P点对x的偏导数就是这条曲线上P点的切线

**证明连续**：极限的△X△Y趋近0，全增量=0，当y以一种解析式趋近(0,0)，极限全增量，代换，判断存在不存在  
cos1/x和sin1/x不存在  
**证明偏导存在**：偏增量/△X极限存在  
**偏导存在且有界，则连续**

###    多元复合函数的求导 ###

**多元复合显示函数**：只有自变量，不含中间变量，直接求导  
**多元复合抽象函数**：z=f(u，v)，有多少个中间变量，有几项只和  
**求导的链式法则**：对的偏导=两项相加  
**半导数**：只有一个中间变量  
**全导数**：只有一个自变量  
**特殊情况**：z=f(u,v,y)  u=u(x,y)  v=v(x,y)  
fx=两项和  
fy=三项和（第三项∂f/∂y）

**特殊情况**：u=f(x,y,z)    z=g(y,t)    t=h(x,y)    u和z的偏导都要求，将z的偏导代入可得答案

**特殊情况**：z=y/f(g(x,y)) 偏导=商的求导  
**极坐标P29**  
**一元全微分形式不变性**  
**全微分运算公式**①+②×③÷  
**复合函数的高阶偏导数**：两次链式法则，第二次非x的系数不变，若系数含x则为乘法求导法则，f变为![\\frac\{\\vartheta f\}\{\\vartheta x\}][frac_vartheta f_vartheta x]再链式

# 隐函数求导法则 #

**方程的个数，未知数的个数，自变量数，因变量数**

方程的个数=因变量数

未知数的个数，自变量数，因变量数

**一元隐函数求导方法**：左右同时求导，再将带导数的合并同类项，放到等号的一边

**多元隐函数求导方法：**左右同时求导，再将带导数的合并同类项，放到等号的一边

**F(x,y)隐函数存在定理：存在P（![x\_\{0\},y\_\{0\}][x_0_y_0]）领域有连续偏导数，F（![x\_\{0\},y\_\{0\}][x_0_y_0]）=0，![F\_\{y\}(x\_\{0\},y\_\{0\})\\neq 0][F_y_x_0_y_0_neq 0]**

**F(x,y)隐函数求导方法:![\\frac\{dy\}\{dx\}=-\\frac\{F\_\{x\}\}\{F\_\{y\}\}][frac_dy_dx_-_frac_F_x_F_y]  (就是分子分母下标交换)**

**F(x,y,z)隐函数存在定理：存在P（![x\_\{0\},y\_\{0\},z\_\{0\}][x_0_y_0_z_0]）领域有连续偏导数，F（![x\_\{0\},y\_\{0\},z\_\{0\}][x_0_y_0_z_0]）=0，![F\_\{y\}(x\_\{0\},y\_\{0\},z\_\{0\})\\neq 0][F_y_x_0_y_0_z_0_neq 0]**

**F(x,y,z)隐函数求导方法:![\\frac\{\\vartheta z\}\{\\vartheta x\}=-\\frac\{F\_\{x\}\}\{F\_\{z\}\}][frac_vartheta z_vartheta x_-_frac_F_x_F_z]    ![\\frac\{\\vartheta z\}\{\\vartheta y\}=-\\frac\{F\_\{y\}\}\{F\_\{z\}\}][frac_vartheta z_vartheta y_-_frac_F_y_F_z](就是分子分母下标交换)**

![F\_\{x\},F\_\{y\},F\_\{z\}][F_x_F_y_F_z]**求导：**除了下标字母，其它都是常数，（可用![f\_\{1\},f\_\{2\}][f_1_f_2]简化运算）

**方程组隐函数求导存在定理：存在P（![x\_\{0\},y\_\{0\},u\_\{0\},v\_\{0\}][x_0_y_0_u_0_v_0]）领域有连续偏导数，F（![x\_\{0\},y\_\{0\},u\_\{0\},v\_\{0\}][x_0_y_0_u_0_v_0]）=0，G（![x\_\{0\},y\_\{0\},u\_\{0\},v\_\{0\}][x_0_y_0_u_0_v_0]）=0，雅可比式不等于0**

**雅可比式：![J=\\frac\{\\vartheta (F,G)\}\{\\vartheta (u,v)\}=\\begin\{vmatrix\} F\_\{u\} & F\_\{v\} \\\\ G\_\{u\} & G\_\{v\} \\end\{vmatrix\}][J_frac_vartheta _F_G_vartheta _u_v_begin_vmatrix_ F_u_ _ F_v_ _ G_u_ _ G_v_ _end_vmatrix]**

**方程组隐函数求导：**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70 2][]

**特殊方程组隐函数求导：只有三个未知数，一个自变量时，按一元函数的方法求，然后通过线性方程组求解**

# **空间曲线** #

### **参数式** ###

**空间曲线：x=x（t） y=y(t)  z=z(t)**

**光滑曲线：可导且导数不全为零**

**切线方程：![\\frac\{x-x\_\{0\}\}\{x\{\}'(t)\}=\\frac\{y-y\_\{0\}\}\{y\{\}'(t)\}=\\frac\{z-z\_\{0\}\}\{z\{\}'(t)\}][frac_x-x_0_x_t_frac_y-y_0_y_t_frac_z-z_0_z_t]**

**切向量：![n=(x\{\}'(t),y\{\}'(t),z\{\}'(t))][n_x_t_y_t_z_t]**

**法平面：![x\{\}'(t)(x-x\_\{0\})+y\{\}'(t)(y-y\_\{0\})+z\{\}'(t)(z-z\_\{0\})=0][x_t_x-x_0_y_t_y-y_0_z_t_z-z_0_0]**

### 相交式 ###

**两个方程对x求导**

**解出x的两个偏导**

**切向量=（1，![F\_\{y\}(M\_\{0\})][F_y_M_0]，![F\_\{z\}(M\_\{0\})][F_z_M_0]）**

**切线：![\\frac\{x-x\_\{0\}\}\{1\}=\\frac\{y-y\_\{0\}\}\{F\_\{y\}(M\_\{0\})\}=\\frac\{z-z\_\{0\}\}\{F\_\{z\}(M\_\{0\})\}][frac_x-x_0_1_frac_y-y_0_F_y_M_0_frac_z-z_0_F_z_M_0]**

**法平面：![(x-x\_\{0\})+F\_\{y\}(M\_\{0\})(y-y\_\{0\})+F\_\{z\}(M\_\{0\})(z-z\_\{0\})=0][x-x_0_F_y_M_0_y-y_0_F_z_M_0_z-z_0_0]**

# **空间曲面** #

**空间曲面：F(x,y,z)=0**

**切平面法向量**：**![n=(F\_\{x\}(M\_\{0\}),F\_\{y\}(M\_\{0\}),F\_\{z\}(M\_\{0\}))][n_F_x_M_0_F_y_M_0_F_z_M_0]**

**切平面方程：![F\_\{x\}(M\_\{0\})(x-x\_\{0\})+F\_\{y\}(M\_\{0\})(y-y\_\{0\})+F\_\{z\}(M\_\{0\})(z-z\_\{0\})=0][F_x_M_0_x-x_0_F_y_M_0_y-y_0_F_z_M_0_z-z_0_0]**

**法线：![\\frac\{x-x\_\{0\}\}\{F\_\{x\}(M\_\{0\})\}=\\frac\{y-y\_\{0\}\}\{F\_\{y\}(M\_\{0\})\}=\\frac\{z-z\_\{0\}\}\{F\_\{z\}(M\_\{0\})\}][frac_x-x_0_F_x_M_0_frac_y-y_0_F_y_M_0_frac_z-z_0_F_z_M_0]**

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2FpcXExMzY_size_16_color_FFFFFF_t_70]: /images/20221023/3ebeabe8e16f4084a4fdc97823db4fb8.png
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