微积分A(2)期末复习笔记

微积分A(2)期末复习笔记,仅供参考。 PDF版本链接

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多元函数定义

内积: x,yRnx,y \in \mathbb R^n, x,y=i=1nxiyi\langle x,y \rangle = \sum_{i=1}^n x_iy_i.

范数: N:RnR\forall N: \mathbb R^n \to \mathbb R s.t.: λR,x,yRn\forall \lambda \in \mathbb R,x,y \in \mathbb R^n,

  1. N(x)0N(x) \ge 0N(x)=0x=0N(x)=0 \Leftrightarrow x=0.
  2. N(λx)=λN(x)N(\lambda x) = \vert \lambda \vert N(x).
  3. N(x+y)N(x)+N(y)N(x+y) \le N(x)+N(y).

nn维空间范数等价性:
N,MN,MRn\mathbb R^n中任意两个范数, α,βR\exists \alpha, \beta \in \mathbb R s.t.: xRn\forall x \in \mathbb R^n,

αM(x)N(x)βM(x)\alpha M(x) \le N(x) \le \beta M(x)

因此通常只考虑2-范数: x=x,x\| x \| = \sqrt{\langle x,x \rangle}.

三角不等式:

x+yx+y\| x+y \| \le \| x \| + \| y \|

Cauchy-Schwarz不等式:

x,yxy\vert \langle x,y \rangle \vert \le \| x \| \cdot \| y \|


距离: x,yRn,Ω,ΛRnx,y \in \mathbb R^n, \Omega, \Lambda \subseteq \mathbb R^n,

  1. dis(x,y)=xy\text{dis}(x,y) = \| x-y \|
  2. dis(Ω,y)=inf{xyxΩ}\text{dis}(\Omega,y) = \inf \left\lbrace \| x-y \| \Big \vert x \in \Omega \right \rbrace
  3. dis(Ω,Λ)=inf{xyxΩ,yΛ}\text{dis}(\Omega,\Lambda) = \inf \left\lbrace \| x-y \| \Big \vert x \in \Omega, y \in \Lambda \right \rbrace

直径: diam(Ω)=sup{xyx,yΩ}\text{diam}(\Omega) = \text{sup} \left\lbrace \| x-y \| \Big \vert x, y \in \Omega \right \rbrace.

邻域: x0Rn,δRx_0 \in \mathbb R^n, \delta \in \mathbb R,

  1. B(x0,δ)={xdis(x,x0)<δ}B(x_0,\delta) = \lbrace x \vert \text{dis}(x,x_0) \lt \delta \rbrace.
  2. B0(x0,δ)={x0<dis(x,x0)<δ}B_0(x_0,\delta) = \lbrace x \vert 0 \lt \text{dis}(x,x_0) \lt \delta \rbrace.

补集: Ωc=RnΩ\Omega^c = \mathbb R^n \setminus \Omega.

内点: δ>0,B(x0,δ)Ω\exists \delta \gt 0, B(x_0,\delta) \subseteq \Omega, 则x0Ω0x_0 \in \Omega_0.

外点: δ>0,B(x0,δ)Ω=\exists \delta \gt 0, B(x_0,\delta) \bigcap \Omega = \varnothing, 则x0Ω0c=Ωx_0 \in \Omega_0^c = \overline \Omega.

孤立点: x0Ω,δ>0,B0(x0,δ)=x_0 \in \Omega, \exists \delta \gt 0, B_0(x_0,\delta) = \varnothing, 则x0x_0为孤立点.

边界点: δ>0,B(x0,δ)Ω and B(x0,δ)ΩB(x0,δ)\forall \delta \gt 0, B(x_0,\delta) \bigcap \Omega \ne \varnothing \text{ and } B(x_0,\delta) \bigcap \Omega \ne B(x_0,\delta), 则x0x_0为边界点.

聚点: δ>0,B(x0,δ)Ω\forall \delta \gt 0, B(x_0,\delta) \bigcap \Omega \ne \varnothing, 则x0Ωx_0 \in \Omega'.

闭包: Cl(Ω)=ΩΩ\text{Cl}(\Omega) = \Omega' \bigcup \Omega.

xx为非聚点, x∉Ωx \not \in \Omega, 则xx为外点.
xx为非聚点, xΩx \in \Omega, 则xx为孤立点.

开集: Ω\Omega s.t.: Ω=Ω0\Omega = \Omega_0.

闭集: Ω\Omega s.t.: Ω=Ω\Omega = \overline \Omega.

有界集: Ω\Omega s.t.: r>0,ΩB(0,r)\exists r \gt 0, \Omega \subseteq B(0,r).

Rn\mathbb R^n\varnothing既开又闭.
开集的并是开集, 有限个开集的交是开集.
闭集的交是闭集, 有限个闭集的并是闭集.


点列收敛: {xi}\lbrace x_i \rbraceRn\mathbb R^n中点列, 若limixix0=0\lim_{i\to\infty} \| x_i-x_0 \| = 0, 则limixi=x0\lim_{i\to\infty} x_i = x_0.

收敛点列极限唯一.

Cauchy列(基本列): ε>0,N>0,l,kN+,l,k>0\forall \varepsilon \gt 0, \exists N \gt 0, \forall l,k \in \mathbb N_+, l,k \gt 0, xkxl<ε\| x_k-x_l \| \lt \varepsilon.

基本列 \Leftrightarrow 收敛.

Ω\Omega为闭集, {xi}Ω\lbrace x_i \rbrace \subseteq \Omegalimixi\lim_{i\to\infty} x_i存在, 则limixiΩ\lim_{i\to\infty} x_i \in \Omega. 特别地, 因为Rn\mathbb R^n为闭集, 所以Rn\mathbb R^n具有完备性.

连通集: x,yΩ\forall x,y \in \Omega, φiC[a,b]\exists \varphi_i \in \mathscr C[a,b], φi(a)=x(i),φi(b)=y(i)\varphi_i(a) = x^{(i)}, \varphi_i(b) = y^{(i)}. 连通非空开集为开区域, 开区域闭包为闭区域.

二元函数: f:ΩRf:\Omega \to \mathbb R s.t.: ΩR2,(x,y)Ω,!zR,z=f(x,y)\Omega \subseteq \mathbb R^2, \forall (x,y) \in \Omega, \exists! z \in \mathbb R, z = f(x,y).


二元函数极限与连续

二重极限: p0ΩR2p_0 \in \Omega \subseteq \mathbb R^2为聚点, 二元函数ff定义在B0(p0,δ0)ΩB_0(p_0,\delta_0) \bigcap \Omega上, 若AR,ε>0,0<δ<δ0,pB0(p0,δ0)Ω\exists A \in \mathbb R, \forall \varepsilon \gt 0, \exists 0 \lt \delta \lt \delta_0, p \in B_0(p_0,\delta_0) \bigcap \Omega, 都有: f(p)A<ε\vert f(p)-A \vert \lt \varepsilon, 则limpp0pΩ=A\lim_{p\to p_0 \atop p\in\Omega} = A. 若p0p_0Ω\Omega内点, 则可简记为limpp0f(p)=limpp00f(p)=limxx0yy0f(x,y)=A\lim_{p \to p_0} f(p) = \lim_{\| p-p_0 \| \to 0} f(p) = \lim_{x\to x_0 \atop y\to y_0} f(x,y) = A.

若二重极限存在, 则任意路径趋近于p0p_0, 极限值相同.

累次极限: limxx0limyy0f(x,y)\lim_{x \to x_0} \lim_{y \to y_0} f(x,y)limyy0limxx0f(x,y)\lim_{y \to y_0} \lim_{x \to x_0} f(x,y).

若二重极限和累次极限均存在, 则相等.
若累次极限不相等, 则二重极限不存在.


连续: ε>0,δ>0,pB(p0,δ)D(f)\forall \varepsilon \gt 0, \exists \delta \gt 0, \forall p \in B(p_0,\delta) \bigcap D(f), 都有f(p)f(p0)<ε\vert f(p)-f(p_0) \vert \lt \varepsilon, 则ffp0p_0处连续.
否则ffp0p_0处间断, p0p_0ff间断点. limpp0f(p)\lim_{p \to p_0}f(p)存在但不等于f(p0)f(p_0)则为可去间断点, 不存在则为本性间断点.

ffp0p_0二重极限存在, 则ffp0p_0处连续.
p0p_0D(f)D(f)聚点, ffp0p_0处连续 \Leftrightarrow ffp0p_0二重极限为f(p0)f(p_0).
p0p_0D(f)D(f)孤立点, ffp0p_0处必连续.
p0p_0ff间断点, p0p_0必为D(f)D(f)聚点.
p0p_0ff连续, 则一元函数f(x,y0)f(x,y_0)f(x0,y)f(x_0,y)(x0,y0)(x_0,y_0)处连续.

ff在开集DD上连续或在闭集DD内部和边界点上连续, 则fC(D)f \in \mathscr C(D).

ff在连通集DD上连续, 则ffDD上最值存在.


二元函数导数

偏导数:

fx(p0)=fxp0=limΔx0f(x0+Δx,y0)f(x0,y0)Δxfy(p0)=fyp0=limΔy0f(x0,y0+Δy)f(x0,y0)Δy\begin{aligned} f'_x(p_0) &= \left.\frac{\partial f}{\partial x}\right\vert_{p_0} = \lim_{\Delta x \to 0} \frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x} \\ f'_y(p_0) &= \left.\frac{\partial f}{\partial y}\right\vert_{p_0} = \lim_{\Delta y \to 0} \frac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y} \\ \end{aligned}

全微分:

Δf(p0)=fx(p0)Δx+fy(p0)Δy+o(Δx2+Δy2)o(ρ)df(p0)=fx(p0)  dx+fy(p0)  dy\begin{aligned} \Delta f(p_0) &= f'_x(p_0)\Delta x + f'_y(p_0)\Delta y + \underbrace{o\left(\sqrt{\Delta x^2+\Delta y^2}\right)}_{o(\rho)} \\ \text{d}f(p_0) &= f'_x(p_0)\;\text{d}x + f'_y(p_0)\;\text{d}y \\ \end{aligned}

可微 \Rightarrow 连续, 偏导存在
偏导连续 \Rightarrow 可微

可微的充要条件:

limpp0f(x+Δx,y+Δy)f(x,y)fx(p0)Δxfy(p0)ΔyΔx2+Δy2=0\lim_{p \to p_0} \frac{f(x+\Delta x,y+\Delta y)-f(x,y)-f'_x(p_0)\Delta x-f'_y(p_0)\Delta y}{\sqrt{\Delta x^2 + \Delta y^2}} = 0

复合函数求导:
z=f(x,y)=f(x(t,s),y(t,s))z = f(x,y) = f(x(t,s),y(t,s)):

zt=fxxt+fyytzs=fxxs+fyys\begin{aligned} \frac{\partial z}{\partial t} &= \frac{\partial f}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial t} \\ \frac{\partial z}{\partial s} &= \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s} \\ \end{aligned}


高阶偏导数:

fxx(p0)=2fx2p0=limΔx0fx(x0+Δx,y0)fx(x0,y0)Δxfxy(p0)=2fyxp0=limΔy0fx(x0,y0+Δy)fx(x0,y0)Δyfyx(p0)=2fxyp0=limΔx0fy(x0+Δx,y0)fy(x0,y0)Δxfyy(p0)=2fy2p0=limΔy0fy(x0,y0+Δy)fy(x0,y0)Δy\begin{aligned} f''_{xx}(p_0) &= \left.\frac{\partial^2 f}{\partial x^2}\right\vert_{p_0} = \lim_{\Delta x \to 0} \frac{f'_x(x_0+\Delta x,y_0)-f'_x(x_0,y_0)}{\Delta x} \\ f''_{xy}(p_0) &= \left.\frac{\partial^2 f}{\partial y \partial x}\right\vert_{p_0} = \lim_{\Delta y \to 0} \frac{f'_x(x_0,y_0+\Delta y)-f'_x(x_0,y_0)}{\Delta y} \\ f''_{yx}(p_0) &= \left.\frac{\partial^2 f}{\partial x \partial y}\right\vert_{p_0} = \lim_{\Delta x \to 0} \frac{f'_y(x_0+\Delta x,y_0)-f'_y(x_0,y_0)}{\Delta x} \\ f''_{yy}(p_0) &= \left.\frac{\partial^2 f}{\partial y^2}\right\vert_{p_0} = \lim_{\Delta y \to 0} \frac{f'_y(x_0,y_0+\Delta y)-f'_y(x_0,y_0)}{\Delta y} \\ \end{aligned}

fxy,fyxf''_{xy},f''_{yx}p0p_0处连续, 则fxy(p0)=fyx(p0)f''_{xy}(p_0)=f''_{yx}(p_0).


方向导数: 对于eR2,e=1e \in \mathbb R^2, \| e \| = 1, 可定义:

fe(p0)=fep0=limt0+f(p0+te)f(p0)t=d[f(p0+te)]dtf'_e(p_0) = \left.\frac{\partial f}{\partial e}\right\vert_{p_0} = \lim_{t\to0^+} \frac{f(p_0+te)-f(p_0)}{t} = \frac{\text{d} [f(p_0+te)]}{\text{d} t}

注意到上式中ee仅指代射线方向, 即λ>0\forall \lambda \gt 0:

f(λe)p0=fep0\left.\frac{\partial f}{\partial(\lambda e)}\right\vert_{p_0} = \left.\frac{\partial f}{\partial e}\right\vert_{p_0}

可微 \Rightarrow 方向导数存在.
方向导数存在且同一点处反方向导数为正方向导数相反数 \Rightarrow 偏导数存在.

f(e)p0=fep0fip0=fxp0                 if fip0=f(i)p0fjp0=fyp0                 if fjp0=f(j)p0\begin{aligned} \left.\frac{\partial f}{\partial(-e)}\right\vert_{p_0} &= - \left.\frac{\partial f}{\partial e}\right\vert_{p_0} \\ \left.\frac{\partial f}{\partial \vec i}\right\vert_{p_0} &= \left.\frac{\partial f}{\partial x}\right\vert_{p_0} \;\;\;\;\;\;\;\; \text{ if } \left.\frac{\partial f}{\partial \vec i}\right\vert_{p_0} = -\left.\frac{\partial f}{\partial(-\vec i)}\right\vert_{p_0} \\ \left.\frac{\partial f}{\partial \vec j}\right\vert_{p_0} &= \left.\frac{\partial f}{\partial y}\right\vert_{p_0} \;\;\;\;\;\;\;\; \text{ if } \left.\frac{\partial f}{\partial \vec j}\right\vert_{p_0} = -\left.\frac{\partial f}{\partial(-\vec j)}\right\vert_{p_0} \\ \end{aligned}


梯度:

f(p0)=gradf(p0)=(fx(p0),fy(p0))\nabla f(p_0) = \text{grad} f(p_0) = (f'_x(p_0),f'_y(p_0))

fep0=f(p0)e=f(p0)cosf(p0),e\left.\frac{\partial f}{\partial e}\right\vert_{p_0} = \nabla f(p_0) \cdot e = \| \nabla f(p_0) \| \cos \langle \nabla f(p_0), e \rangle


向量值函数与隐函数

向量值函数: f:ΩRmf:\Omega \to \mathbb R^m s.t.: ΩRn,xΩ,!yRm,y=f(x)\Omega \subseteq \mathbb R^n, \forall \vec x \in \Omega, \exists! \vec y \in \mathbb R^m, \vec y = f(\vec x).

连续: fC(Ω)1in,fiC(Ωi)f \in \mathscr C(\Omega) \Leftrightarrow \forall 1 \le i \le n, f_i \in \mathscr C(\Omega_i).

映射微分:

Δf(x)=AΔx+o(Δx)o(ρ)df(x)=Adx\begin{aligned} \Delta f(\vec x) &= A\vec{\Delta x} + \underbrace{o\left(\|\vec{\Delta x} \|\right)}_{o(\rho)} \\ \text{d}f(\vec x) &= A\text{d} \vec x \\ \end{aligned}

其中AMm×n(R)A \in M_{m \times n}(\mathbb R), 满足:

Aij=yixjp0A_{ij} = \left.\frac{\partial y_i}{\partial x_j}\right\vert_{p_0}

AAff的Jacobi矩阵, 记作Jf(x0)=(y1,,ym)(x1,,xn)x0=AJ f(\vec x_0) = \left.\frac{\partial (y_1,\cdots,y_m)}{\partial (x_1,\cdots,x_n)}\right\vert_{\vec x_0} = A.
Φ(x0)(x)=Jf(x0)×x\Phi(\vec x_0)(\vec x) = J f(\vec x_0) \times \vec xffx0\vec x_0的微分映射.
df(x)=Jf(x0)×dx\text{d} f(\vec x) = J f(\vec x_0) \times \text{d} \vec xffx0\vec x_0的微分.

[dy1dy2dym]=(y1,,ym)(x1,,xn)×[dx1dx2dxn]\begin{bmatrix} \text{d}y_1 \\ \text{d}y_2 \\ \cdots \\ \text{d}y_m \\ \end{bmatrix} = \frac{\partial (y_1,\cdots,y_m)}{\partial (x_1,\cdots,x_n)} \times \begin{bmatrix} \text{d}x_1 \\ \text{d}x_2 \\ \cdots \\ \text{d}x_n \\ \end{bmatrix}

复合映射微分:
若向量值函数ffx0\vec x_0处可微, ggu0=f(x0)\vec u_0 = f(\vec x_0)处可微, Im(f)D(g)\text{Im}(f) \subseteq D(g). 则:

J(gf)(x0)=Jg(u0)×Jf(x0)J(g \circ f)(\vec x_0) = J g(\vec u_0) \times J f(\vec x_0)

逆映射微分:

J(ff1)=In×nJf1(y)=(Jf(x))1\begin{aligned} J (f \circ f^{-1}) &= I_{n \times n} \\ J f^{-1}(\vec y) &= \left( J f(\vec x) \right)^{-1} \\ \end{aligned}


隐函数: D(F)=W×ERn×RmD(F) = W \times E \subseteq \mathbb R^n \times \mathbb R^m, xW,!yE\forall \vec x \in W, \exists! y \in E s.t.: F(x,y)=0F(\vec x,y) = 0. 则F(x,y)=0F(\vec x,y)=0确定隐函数y=f(x)y = f(\vec x).

隐函数存在性:
FFWW内有定义, 且:

  1. FC(q)F \in \mathscr C^{(q)}, q1q \ge 1.
  2. p0W×E,F(p0)=0\exists \vec p_0 \in W \times E, F(\vec p_0) = 0.
  3. Fy(p0)0F'_y(\vec p_0) \ne 0.

则: I×JW×E\exists I \times J \subseteq W \times E s.t.: x0I,y0J\vec x_0 \in I, y_0 \in J (i.e.: p0\vec p_0的邻域):

  1. xI\forall \vec x \in I, !y=f(x)J\exists ! y = f(\vec x) \in J. (隐函数存在唯一性)
  2. y0=f(x0)y_0 = f(\vec x_0). (初值条件)
  3. fC(q)(I)f \in \mathscr C^{(q)}(I). (隐函数连续性)
  4. xI\forall \vec x \in I,
f'_j(\vec x) = -\frac{F'_j(\vec p)}{F'_y(\vec p)} $$ (隐函数导数)

若二元函数F(x,y)=0F(x,y)=0确定隐函数f(x)f(x), f1=gf^{-1} = g存在 \Leftrightarrow F(x,y)=0F(x,y)=0确定隐函数g(y)g(y).

隐函数方程组: 1im,Fi(x1,,xn,y1,,ym)=0\forall 1 \le i \le m, F_i(x_1,\cdots,x_n,y_1,\cdots,y_m) = 0, 其中D(Fi)=Wx×WyD(F_i) = W_x \times W_y.

隐函数方程组解存在性:
FFWxW_x内有定义, 且: 1im\forall 1 \le i \le m,

  1. FC(q)F \in \mathscr C^{(q)}, q1q \ge 1.
  2. p0Wx×Wy,Fi(p0)=0\exists p_0 \in W_x \times W_y, F_i(\vec p_0) = 0.
  3. (F1,,Fm)(y1,,ym)p0\left.\frac{\partial (F_1,\cdots,F_m)}{\partial (y_1,\cdots,y_m)}\right\vert_{\vec p_0}可逆.

则: p0\exists \vec p_0的邻域Ix×IyI_x \times I_y: 1im\forall 1 \le i \le m,

  1. xIx,!y\forall \vec x \in I_x, \exists! \vec y s.t.: Fi(x,y)=0F_i(\vec x,\vec y) = 0. 可以相应定义定义yi=(y)i=fi(x)y_i = (\vec y)_i = f_i(\vec x).
  2. (y0)i=fi(x0)(\vec y_0)_i = f_i(\vec x_0).
  3. fiC(q)(Ix)f_i \in \mathscr C^{(q)}(I_x).
  4. xIx\forall \vec x \in I_x,

(y1,,ym)(x1,,xn)x=((F1,,Fm)(y1,,ym)p)1×(F1,,Fm)(x1,,xn)x\left.\frac{\partial(y_1,\cdots,y_m)}{\partial(x_1,\cdots,x_n)}\right\vert_{\vec x} = -\left( \left.\frac{\partial (F_1,\cdots,F_m)}{\partial (y_1,\cdots,y_m)}\right\vert_{\vec p} \right)^{-1} \times \left.\frac{\partial(F_1,\cdots,F_m)}{\partial(x_1,\cdots,x_n)}\right\vert_{\vec x}


空间曲线和曲面

空间曲线切向量: p=tτ\vec p = t \vec \tau, 其中切向量τ=(xt,yt,zt)(p0)\vec \tau = (x'_t,y'_t,z'_t)(\vec p_0). 若空间曲线处处有非零、连续的切向量, 则称空间曲线光滑.

空间曲面切向量与法向量: 曲面S:F(p)=0S:F(\vec p) = 0p0\vec p_0处可微, 且F(p0)0\nabla F(\vec p_0) \ne 0, FF在邻域内连续, 则:

τ,F(t0)=nτ=0\forall \vec \tau, F'(t_0) = \vec n \cdot \vec \tau = 0

其中法向量n=F(p0)=(Fx,Fy,Fz)(p0)\vec n = \nabla F(\vec p_0) = (F'_x,F'_y,F'_z)(\vec p_0), 切向量τ=(xt,yt,zt)(p0)\vec \tau = (x'_t,y'_t,z'_t)(\vec p_0)对于任意曲线l:x=x(t),y=y(t),z=z(t)l:x=x(t),y=y(t),z=z(t) s.t: p0lS\vec p_0 \in l \subseteq S.

切向量求切线方程:

xx0τx=yy0τy=zz0τz\frac{x-x_0}{\tau_x} = \frac{y-y_0}{\tau_y} = \frac{z-z_0}{\tau_z}

法向量求法线方程:

xx0nx=yy0ny=zz0nz\frac{x-x_0}{n_x} = \frac{y-y_0}{n_y} = \frac{z-z_0}{n_z}

切向量求法平面方程:

τx(xx0)+τy(yy0)+τz(zz0)=0\tau_x(x-x_0) + \tau_y(y-y_0) + \tau_z(z-z_0) = 0

法向量求切平面方程:

nx(xx0)+ny(yy0)+nz(zz0)=0n_x(x-x_0) + n_y(y-y_0) + n_z(z-z_0) = 0

切向量求法向量(法向量求切向量类似):

n=τ1×τ2=ijkτ1xτ1yτ1zτ2xτ2yτ2z=(det(y,z)(u,v),det(z,x)(u,v),det(x,y)(u,v))\vec n = \vec \tau_1 \times \vec \tau_2 = \left \vert \begin{matrix} \vec i & \vec j & \vec k \\ \tau_{1x} & \tau_{1y} & \tau_{1z} \\ \tau_{2x} & \tau_{2y} & \tau_{2z} \\ \end{matrix} \right \vert = \left( \det \frac{\partial (y,z)}{\partial (u,v)}, \det \frac{\partial (z,x)}{\partial (u,v)}, \det \frac{\partial (x,y)}{\partial (u,v)} \right)

显函数法向量: n=(fx,fy,1)(p0)\vec n = (f'_x,f'_y,-1)(p_0).


二元函数泰勒展开与极值

高阶全微分: 若fC(n)f \in \mathscr C^{(n)},

dnf=(dxx+dyy)nf(x,y)=i=0nCninfxiyni(x,y)dxidyni\text{d}^n f = \left( \text{d}x \cdot \frac{\partial}{\partial x} + \text{d}y \cdot \frac{\partial}{\partial y} \right)^nf(x,y) = \sum_{i=0}^n C_n^i \frac{\partial^n f}{\partial x^i \partial y^{n-i}}(x,y) \cdot \text{d}x^i \cdot \text{d}y^{n-i}

二元函数泰勒展开: 若fC(n)f \in \mathscr C^{(n)},

Tf(x,y)=k=0n1k!((xx0)x+(yy0)y)kf(x0,y0)f(x,y)=Tf(x,y)+Rn\begin{aligned} T_f(x,y) &= \sum_{k=0}^n \frac{1}{k!} \left( (x-x_0) \cdot \frac{\partial}{\partial x} + (y-y_0) \cdot \frac{\partial}{\partial y} \right)^kf(x_0,y_0) \\ f(x,y) &= T_f(x,y) + R_n \\ \end{aligned}

皮亚诺余项: fC(n)f \in \mathscr C^{(n)}, Rn=o((xx0)2+(yy0)2n)R_n = o \left( \sqrt{(x-x_0)^2+(y-y_0)^2}^n \right).
拉格朗日余项: fC(n+1)f \in \mathscr C^{(n+1)}, Rn=1(n+1)!(Δxx+Δyy)n+1f(x0+θΔx,y0+θΔy)R_n = \frac{1}{(n+1)!}\left( \Delta x \cdot \frac{\partial}{\partial x} + \Delta y \cdot \frac{\partial}{\partial y} \right)^{n+1}f(x_0+\theta \Delta x,y_0+\theta \Delta y), θ[0,1]\theta \in [0,1].

e.g.: n=2n=2

f(x,y)=f(x0,y0)+(xx0)fx(x0,y0)+(yy0)fy(x0,y0)+12(xx0)2fxx(x0,y0)+(xx0)(yy0)fxy(x0,y0)+12(yy0)2fyy(x0,y0)+R2\begin{aligned} f(x,y) &= f(x_0,y_0) \\ &+ (x-x_0)f'_x(x_0,y_0) + (y-y_0)f'_y(x_0,y_0) \\ &+ \frac{1}{2}(x-x_0)^2f''_{xx}(x_0,y_0) + (x-x_0)(y-y_0)f''_{xy}(x_0,y_0) + \frac{1}{2}(y-y_0)^2f''_{yy}(x_0,y_0) \\ &+ R_2 \end{aligned}

二元函数微分中值定理: fCf \in \mathscr C, θ[0,1]\theta \in [0,1],

f(x,y)f(x0,y0)=(Δxfx+Δyfy)(x0+θΔx,y0+θΔy)f(x,y)-f(x_0,y_0) = \Big(\Delta xf'_x+\Delta yf'_y\Big)(x_0+\theta\Delta x,y_0+\theta\Delta y)


二元函数极值: B(p0,δ)D(f)B(p_0,\delta) \subseteq D(f), 若pB(p0,δ),f(p)f(p0)\forall p \in B(p_0,\delta), f(p) \le f(p_0), 则p0p_0为极大值点, f(p0)f(p_0)为极大值. 极小值点和极小值同理.

极值点 \Rightarrow 内点.
极值点, 偏导存在 \Rightarrow 驻点(临界点).

p0p_0ff驻点, fC(2)(B(p0,δ0))f \in \mathscr C^{(2)}(B(p_0,\delta_0)):

  • fxxfyyfxy2>0f''_{xx}f''_{yy} - {f''_{xy}}^2 \gt 0, p0p_0为极值点
    • fxx>0,fyy>0f''_{xx} \gt 0, f''_{yy} \gt 0, p0p_0为极小值点.
    • fxx<0,fyy<0f''_{xx} \lt 0, f''_{yy} \lt 0, p0p_0为极大值点.
  • fxxfyyfxy2<0f''_{xx}f''_{yy} - {f''_{xy}}^2 \lt 0, p0p_0不为极值点
  • fxxfyyfxy2=0f''_{xx}f''_{yy} - {f''_{xy}}^2 = 0, 无法确定
    • 0<δ<δ0,pB(p0,δ)\exists 0 \lt \delta \lt \delta_0, \forall p \in B(p_0,\delta), fxxfyyfxy20f''_{xx}f''_{yy} - {f''_{xy}}^2 \ge 0, 则为极值点, 按照第一条方式判断.

一般地, 对于nn原函数, 判断Hessian矩阵Hf(p0)H_f(p_0):

[fx1x1fx1x2fx1x3fx1xnfx2x1fx2x2fx2x3fx2xnfx3x1fx3x2fx3x3fx3xnfxnx1fxnx2fxnx3fxnxn]\begin{bmatrix} f''_{x_1x_1} & f''_{x_1x_2} & f''_{x_1x_3} & \cdots & f''_{x_1x_n} \\ f''_{x_2x_1} & f''_{x_2x_2} & f''_{x_2x_3} & \cdots & f''_{x_2x_n} \\ f''_{x_3x_1} & f''_{x_3x_2} & f''_{x_3x_3} & \cdots & f''_{x_3x_n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f''_{x_nx_1} & f''_{x_nx_2} & f''_{x_nx_3} & \cdots & f''_{x_nx_n} \\ \end{bmatrix}

若正定或在邻域内连续半正定, 则为极小值点; 若负定或在邻域内连续半负定, 则为极大值点.


最小二乘法求回归直线:

[kb]=[i=1nxi2i=1nxii=1nxin]1[i=1nxiyii=1nyi]=1Var[x][1E[x]E[x]E[x2]][i=1nxiyii=1nyi]\begin{bmatrix} k \\ b \\ \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^n x_i^2 & \sum_{i=1}^n x_i \\ \sum_{i=1}^n x_i & n \\ \end{bmatrix}^{-1} \begin{bmatrix} \sum_{i=1}^n x_iy_i \\ \sum_{i=1}^n y_i \\ \end{bmatrix} = \frac{1}{\text{Var}[x]} \begin{bmatrix} 1 & -\mathbb E[x] \\ -\mathbb E[x] & \mathbb E[x^2] \\ \end{bmatrix} \begin{bmatrix} \sum_{i=1}^n x_iy_i \\ \sum_{i=1}^n y_i \\ \end{bmatrix}


拉格朗日乘子法求条件极值: 求f(x,y)f(x,y)φ(x,y)=0\varphi(x,y)=0条件下的极值, 可构造拉格朗日函数并求解其驻点:

L(x,y,λ)=f(x,y)+λφ(x,y)L(x,y,\lambda) = f(x,y) + \lambda \varphi(x,y)


含参定积分

一致连续: ε>0,δ>0,p1,p2Ω,p1p2<δ\forall \varepsilon \gt 0, \exists \delta \gt 0, \forall p_1,p_2 \in \Omega, \| p_1-p_2 \| \lt \delta, 都有f(p1)f(p2)<ε\vert f(p_1)-f(p_2) \vert \lt \varepsilon, 则ffΩ\Omega上一致连续.

有界闭集上连续则一致连续.

含参定积分: II为任意区间, D=[a,b]×ID(f)D = [a,b] \times I \subseteq D(f), 若uI\forall u \in I, φ(u)=abf(x,u)  dx\varphi(u) = \int_a^b f(x,u)\;\text{d}x存在, 则称其为ff含参uu的定积分.

连续性: fC(D)f \in \mathscr C(D) \Rightarrow φC(I)\varphi \in \mathscr C(I), 即:

limuu0φ(u)=limuu0abf(x,u)  dx=ablimuu0f(x,u)  dx=φ(u0)\lim_{u\to u_0}\varphi(u) = \lim_{u\to u_0} \int_a^b f(x,u) \;\text{d}x = \int_a^b \lim_{u\to u_0} f(x,u) \;\text{d}x = \varphi(u_0)

可导性: fuC(D)f'_u \in \mathscr C(D) \Rightarrow φ\varphi'存在且:

φ(u)=abfu(x,u)  dx\varphi'(u) = \int_a^b f'_u(x,u) \;\text{d}x

可积性: fC(D)f \in \mathscr C(D) \Rightarrow φR(I)\varphi \in \mathscr R(I)且:

αβφ(u)  du=αβabf(x,u)  dx  du=abαβf(x,u)  du  dx\int_\alpha^\beta\varphi(u)\;\text{d}u = \int_\alpha^\beta \int_a^b f(x,u) \;\text{d}x \;\text{d}u = \int_a^b \int_\alpha^\beta f(x,u) \;\text{d}u \;\text{d}x

变限积分: fuC(D)f'_u \in \mathscr C(D), a(u),b(u)C[a,b]a(u),b(u) \in \mathscr C[a,b][α,β][\alpha,\beta]上可导 \Rightarrow φ\varphi'[α,β][\alpha,\beta]上存在且:

φ(u)=a(u)b(u)fu(x,u)  dx+b(u)f(b(u),u)a(u)f(a(u),u)\varphi'(u) = \int_{a(u)}^{b(u)} f'_u(x,u) \;\text{d}x + b'(u) \cdot f(b(u),u) - a'(u) \cdot f(a(u),u)


含参广义积分: II为任意区间, D=[a,+)×ID(f)D = [a,+\infty) \times I \subseteq D(f), 若uI\forall u \in I, φ(u)=a+f(x,u)  dx\varphi(u) = \int_a^{+\infty} f(x,u)\;\text{d}x