多元函数
XantC
概念
定义
u = f(P)
P(x1, x2, ⋯, xn) ∈ D, D ⊂ ℝn
极限
limP → P0f(P) = L
⇔ ∀ε > 0, ∃δ > 0, s.t. if P(x, y) ∈ D ∩ Ů(P0, δ), then |f(P) − L| < ε
连续性
limP → P0f(P) = f(P0)
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 = φ(t), y = ψ(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
$$ \Delta z=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y+\varepsilon_1\Delta x+\varepsilon_2\Delta y $$
(Δx → 0, Δy → 0: ε1 → 0, ε2 → 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 = φ(x, y), v = ψ(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 α, cos β)| = 1
计算
$$ \frac{\partial f}{\partial l}=f_x^{'}\cos\alpha+f_y^{'}\cos\beta$$
梯度
定义
∇f = (fx′, fy′)
意义
与方向导数
$$ \begin{align*} \frac{\partial f}{\partial l}&=\nabla f \cdot \boldsymbol{e_l} \\ &=|\nabla f|\cos\langle\nabla f,\boldsymbol{e_l}\rangle \end{align*} $$
∇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, fxx″ > 0: 极小值
D > 0, fxx″ < 0: 极大值
D < 0: 无极值
D = 0: 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), φ(x, y) = 0
令 L(x, y) = f(x, y) + λφ(x, y)
$$ \begin{cases} L_x^{'}(x_0,y_0)=0\\ L_y^{'}(x_0,y_0)=0\\ \varphi(x_0,y_0)=0 \end{cases} $$
(x0, y0)为可能极值点
证明
φ(x0, y0) = 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
r = f(t) = f1(t)i + f2(t)j + f3(t)k
极限
定义
limt → t0f(t) = r0
⇔ ∀ε > 0, ∃δ > 0, s.t. if 0 < |t − t0| < δ, then |f(t) − r0| < ε
计算
limt → t0f(t) = (limt → t0f1(t), limt → t0f2(t), limt → t0f3(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
f′(t) = f1′(t) + f2′(t) + f3′(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：φ′(t0)(x − x0) + ψ′(t0)(y − y0) + ω′(t0)(z − z0) = 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：Fx′(x0, y0, z0)(x − x0) + Fy′(x0, y0, z0)(y − y0) + Fz′(x0, y0, z0)(z − z0) = 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
f(t) = (rcos ωt, rsin ωt)
v(t) = f′(t) = (−rωsin ωt, rωcos ωt)
a(t) = f″(t) = (−rω2cos ωt, −rω2sin ωt)
|a(t)| = rω2
所以加速度始终与速度方向垂直，大小为rω2