多元函数

XantC

概念

定义

\[ u=f(P) \]

\[ \;P(x_1, x_2, \cdots, x_n)\in D, \; D\subset \mathbb{R}^n \]

极限

\[ \lim_{P\to P_0}f(P)=L \]

\[ \iff \forall\varepsilon>0, \exists \delta>0,\text{s.t.} \;\text{if} \;P(x, y)\in D\cap \mathring{U}(P_0,\delta), \text{then}\;|f(P)-L|<\varepsilon \]

连续性

\[ \lim_{P\to P_0}f(P)=f(P_0) \]

偏导数

定义

\[ \frac{\partial z}{\partial x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \]

\[ \frac{\partial z}{\partial y}=\lim_{\Delta y\to 0}\frac{f(x, y+\Delta y)-f(x, y)}{\Delta y} \]

计算

非复合

\[ 把剩余变量视为常量，再按一元函数求导法则来求 \]

复合

一元与多元

\[ z=f(x, y), x=\varphi(t), y=\psi(t) \]

\[ \frac{\mathrm{d}z}{\mathrm{d}t}=\frac{\partial z}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}t}+\frac{\partial z}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}t}\]

证明

\[ \Delta z=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y+\varepsilon_1\Delta x+\varepsilon_2\Delta y \]

\[ (\Delta x \to 0, \Delta y \to 0\colon\; \varepsilon_1 \to 0, \varepsilon_2 \to 0) \]

\[ \frac{\Delta z}{\Delta t}=\frac{\partial z}{\partial x}\frac{\Delta x}{\Delta t}+\frac{\partial z}{\partial y}\frac{\Delta y}{\Delta t}+\varepsilon_1\frac{\Delta x}{\Delta t}+\varepsilon_2\frac{\Delta y}{\Delta t} \]

\[ \lim_{\Delta t\to 0}\frac{\Delta z}{\Delta t}=\lim_{\Delta t\to 0}(\frac{\partial z}{\partial x}\frac{\Delta x}{\Delta t}+\frac{\partial z}{\partial y}\frac{\Delta y}{\Delta t})\]

\[ \frac{\mathrm{d}z}{\mathrm{d}t}=\frac{\partial z}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}t}+\frac{\partial z}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}t} \]

多元与多元

\[ z=f(u, v), u=\varphi(x, y), v=\psi(x, y) \]

\[ \frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x} \]

\[ \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial y} \]

隐函数

一元

\[ F(x, y) = 0, y=y(x) \]

\[ F_x^{'}+F_y^{'}\frac{\mathrm{d}y}{\mathrm{d}x}=0 \]

\[ \frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{F_{x}^{'}}{F_{y}^{'}} \]

二元

\[ F(x, y, z)=0, z=z(x, y) \]

\[ \frac{\partial z}{\partial x}=-\frac{F_{x}^{'}}{F_{z}^{'}}, \frac{\partial z}{\partial y} =-\frac{F_{y}^{'}}{F_{z}^{'}}\]

二元方程组

\[ \begin{cases} F(x, y, u, v)=0\\ G(x, y, u, v)=0 \end{cases} \]

\[ J=\frac{\partial(F, G)}{\partial(u, v)}=\begin{vmatrix} F_u^{'} & F_v^{'} \\ G_u^{'} & G_v^{'} \end{vmatrix} \]

\[ \frac{\partial u}{\partial x}=-\frac{1}{J}\frac{\partial(F, G)}{\partial(x, v)}, \frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial(F, G)}{\partial(u, x)} \]

\[ \frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F, G)}{\partial(y, v)}, \frac{\partial v}{\partial y}=-\frac{1}{J}\frac{\partial(F, G)}{\partial(u, y)} \]

方向导数与梯度

方向导数

定义

\[ \begin{align*} \frac{\partial f}{\partial l}&=\lim_{t\to 0^{+}}\frac{f(x+t\cos \alpha, y+t\cos\beta)-f(x, y)}{t}\\ \end{align*} \]

\[ |(\cos\alpha, \cos\beta)|=1 \]

计算

\[ \frac{\partial f}{\partial l}=f_x^{'}\cos\alpha+f_y^{'}\cos\beta\]

梯度

定义

\[ \nabla f=(f_x^{'}, f_y^{'}) \]

意义

与方向导数

\[ \begin{align*} \frac{\partial f}{\partial l}&=\nabla f \cdot \boldsymbol{e_l} \\ &=|\nabla f|\cos\langle\nabla f,\boldsymbol{e_l}\rangle \end{align*} \]

\[ \nabla f 表示f增长最快的方向和增速大小 \]

与法向量

\[ \boldsymbol{n}=\frac{\nabla f(x_0, y_0)}{|\nabla f(x_0, y_0)|} \]

\[ \therefore \nabla f(x_0, y_0)=\frac{\partial f}{\partial n}\boldsymbol{n} \]

\[ \begin{align*} 注意\colon\frac{\partial f}{\partial n}&=|\nabla f(x_0,y_0)|\\ &\neq f^{'}_x(x_0,y_0)+f^{'}_y(x_0,y_0) \end{align*} \]

极值

无条件极值

\[ \begin{cases} f_x^{'}(x,y)=0\\ f_y^{'}(x,y)=0 \end{cases} \]

\[ D=\begin{vmatrix} f_{xx}^{''}&f_{xy}^{''}\\ f_{xy}^{''}&f_{yy}^{''} \end{vmatrix}= f_{xx}^{''}f_{yy}^{''}-( f_{xy}^{''})^2 \]

\[ D>0,\; f_{xx}^{''}>0\colon 极小值 \]

\[ D>0,\; f_{xx}^{''}<0\colon 极大值 \]

\[ D<0\colon 无极值 \]

\[ D=0\colon 极值情况不确定 \]

证明：正定矩阵

有条件极值（拉格朗日乘数法）

概念

\[ z=f(x,y),\; \varphi(x,y)=0 \]

\[ 令 \;L(x,y)=f(x,y)+\lambda\varphi(x,y) \]

\[ \begin{cases} L_x^{'}(x_0,y_0)=0\\ L_y^{'}(x_0,y_0)=0\\ \varphi(x_0,y_0)=0 \end{cases} \]

\[ (x_0, y_0)为可能极值点 \]

证明

\[ \varphi(x_0,y_0)=0确定y=y(x) \]

\[ \begin{cases} \frac{\partial}{\partial x}f(x_0, y(x_0))=0\\ \varphi(x_0,y(x_0))=0 \end{cases} \]

\[ \begin{cases} \displaystyle f_x^{'}(x_0,y_0)+f_y^{'}(x_0,y_0)\frac{\mathrm{d}y}{\mathrm{d}x}\bigg|_{x=x_0}=0\\ \displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}\bigg|_{x=x_0}=-\frac{\varphi_x^{'}(x_0,y_0)}{\varphi_y^{'}(x_0,y_0)} \end{cases} \]

\[ \Rightarrow f_x^{'}(x_0, y_0)-f_y^{'}(x_0,y_0)\frac{\varphi_x^{'}(x_0,y_0)}{\varphi_y^{'}(x_0,y_0)}=0 \]

\[ 令\; \lambda = -\frac{f_y^{'}(x_0,y_0)}{\varphi_y^{'}(x_0,y_0)} \]

\[ \therefore \begin{cases} \displaystyle f_x^{'}(x_0, y_0)+\lambda\varphi_x^{'}(x_0,y_0)=0\\ \displaystyle f_y^{'}(x_0, y_0)+\lambda\varphi_y^{'}(x_0,y_0)=0\\ \displaystyle\varphi(x,y)=0 \end{cases} \]

全微分

\[ \mathrm{d}z=\frac{\partial z}{\partial x}\mathrm{d}x+\frac{\partial z}{\partial y}\mathrm{d}y \]

\[ \iff \Delta z=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y+\mathcal{o}(\rho) \]

一元向量值函数

定义

\[ \boldsymbol{r}=\boldsymbol{f}(t)=f_1(t)\boldsymbol{i}+f_2(t)\boldsymbol{j}+f_3(t)\boldsymbol{k} \]

极限

定义

\[ \lim_{t\to t_0}\boldsymbol{f}(t)=\boldsymbol{r_0} \]

\[ \iff \forall\varepsilon>0, \exists \delta>0,\text{s.t.} \;\text{if} \;0<|t-t_0|<\delta, \text{then}\;|\boldsymbol{f}(t)-\boldsymbol{r_0}|<\varepsilon \]

计算

\[ \lim_{t\to t_0}\boldsymbol{f}(t)=(\lim_{t\to t_0}f_1(t), \lim_{t\to t_0}f_2(t), \lim_{t\to t_0}f_3(t)) \]

导数

定义

\[ \lim_{\Delta t\to 0}\frac{\Delta \boldsymbol{r}}{\Delta t}=\lim_{\Delta t\to 0}\frac{\boldsymbol{f}(t+\Delta t)-\boldsymbol{f}(t)}{\Delta t} \]

计算

\[ \boldsymbol{f}^{'}(t)=f_1^{'}(t)+f_2^{'}(t)+f_3^{'}(t) \]

应用

几何

曲线

参数方程形式

\[ \begin{cases} x=\varphi(t)\\ y=\psi(t)\\ z=\omega(t) \end{cases} \]

\[ 切线：\frac{x-x_0}{\varphi^{'}(t_0)}=\frac{y-y_0}{\psi^{'}(t_0)}=\frac{z-z_0}{\omega^{'}(t_0)} \]

\[ 法平面：\varphi^{'}(t_0)(x-x_0)+\psi^{'}(t_0)(y-y_0)+\omega^{'}(t_0)(z-z_0)=0 \]

曲面交线形式

\[ \begin{cases} F(x,y,z)=0\\ G(x,y,z)=0 \end{cases} \]

\[ 切线：\frac{x-x_0}{\begin{vmatrix} F_y&F_z\\ G_y&G_z \end{vmatrix}}_M= \frac{y-y_0}{\begin{vmatrix} F_z&F_x\\ G_z&G_x \end{vmatrix}}_M= \frac{z-z_0}{\begin{vmatrix} F_x&F_y\\ G_x&G_y \end{vmatrix}}_M \]

\[ 法平面：{\begin{vmatrix} F_y&F_z\\ G_y&G_z \end{vmatrix}}_M(x-x_0)+{\begin{vmatrix} F_z&F_x\\ G_z&G_x \end{vmatrix}}_M(y-y_0)+{\begin{vmatrix} F_x&F_y\\ G_x&G_y \end{vmatrix}}_M(z-z_0)=0 \]

曲面

\[ F(x,y,z)=0 \]

\[ 切平面：F_x^{'}(x_0, y_0, z_0)(x-x_0)+F_y^{'}(x_0, y_0, z_0)(y-y_0)+F_z^{'}(x_0, y_0, z_0)(z-z_0)=0 \]

\[ 法线：\frac{x-x_0}{F_x^{'}(x_0, y_0, z_0)}=\frac{y-y_0}{F_y^{'}(x_0, y_0, z_0)}=\frac{z-z_0}{F_z^{'}(x_0, y_0, z_0)} \]

物理

向心加速度

\[ \boldsymbol{f}(t)=(r\cos\omega t, r\sin\omega t) \]

\[ \boldsymbol{v}(t)=\boldsymbol{f}^{'}(t)=(-r\omega\sin\omega t, r\omega\cos\omega t) \]

\[ \boldsymbol{a}(t)=\boldsymbol{f}^{''}(t)=(-r\omega^2\cos\omega t, -r\omega^2\sin\omega t) \]

\[ |\boldsymbol{a}(t)|=r\omega^2 \]

\[ 所以加速度始终与速度方向垂直，大小为r\omega^2 \]