成考专升本高等数学公式笔记

简单运算

(a±b)2=a2+b2±2ab(a \pm b)^2 = a^2 + b^2 \pm 2ab(a±b)2=a2+b2±2ab
(a±b)3=a3±3a2b+3ab2±b3(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3(a±b)3=a3±3a2b+3ab2±b3
a2−b2=(a+b)∗(a−b)a^2 - b^2 = (a + b) * (a - b)a2−b2=(a+b)∗(a−b)
x2+(a+b)x+a∗b=(x+a)∗(x−a)x^2 + (a + b)x + a * b = (x + a) * (x - a)x2+(a+b)x+a∗b=(x+a)∗(x−a)
a3+b3=(a+b)∗(a2+b2−ab)a^3 + b^3 = (a + b) * (a^2 + b^2 - ab)a3+b3=(a+b)∗(a2+b2−ab)
a3−b3=(a−b)∗(a2+b2+ab)a^3 - b^3 = (a - b) * (a^2 + b^2 + ab)a3−b3=(a−b)∗(a2+b2+ab)
a3+b3+c3=(a+b+c)∗(a2+b2+c2−ab−bc−ac)−3abca^3 + b^3 + c^3 = (a + b + c) * (a^2 + b^2 + c^2 - ab - bc - ac) - 3abca3+b3+c3=(a+b+c)∗(a2+b2+c2−ab−bc−ac)−3abc
奇函数：f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x)
偶函数：f(−x)=f(x)f(-x) = f(x)f(−x)=f(x)

对数指数

ax∗ay=ax+ya^x * a^y = a^{x+y}ax∗ay=ax+y
ax∗bx=(a∗b)xa^x * b^x = (a * b)^xax∗bx=(a∗b)x
(ax)y=ax∗y(a^x)^y = a^{x * y}(ax)y=ax∗y
αx=y⟺yx=α⟺log⁡αy=x\alpha^x = y \iff \sqrt[x]y = \alpha \iff \log_\alpha y = xαx=y⟺xy=α⟺logαy=x
αx=α1x\sqrt[x]\alpha = \alpha^{\frac 1x}xα=αx1
α−x=1αx\alpha^{-x} = {1 \over \alpha^x}α−x=αx1
log⁡a(x∗y)=log⁡ax+log⁡ay\log_a(x * y) = \log_ax + \log_ayloga(x∗y)=logax+logay
log⁡axy=log⁡ax−log⁡ay\log_a\frac xy = \log_ax - \log_aylogayx=logax−logay
log⁡axy=ylog⁡ax\log_ax^y = y\log_axlogaxy=ylogax
log⁡ax=log⁡bxlog⁡ba\log_ax = {\log_bx \over \log_ba}logax=logbalogbx
alog⁡ax=xa^{\log_ax} = xalogax=x
ln⁡x=log⁡ex\ln x = \log_exlnx=logex
lg⁡x=log⁡10x\lg x = \log_{10}xlgx=log10x

三角函数

sin⁡0=0\sin 0 = 0sin0=0
sin⁡π=0\sin \pi = 0sinπ=0
sin⁡π2=1\sin \frac \pi 2 = 1sin2π=1
sin⁡−α=−sin⁡α\sin -\alpha = -\sin \alphasin−α=−sinα
sin⁡(α±β)=sin⁡α∗cos⁡β±cos⁡α∗sin⁡β\sin(\alpha \pm \beta) = \sin \alpha * \cos \beta \pm \cos \alpha * \sin \betasin(α±β)=sinα∗cosβ±cosα∗sinβ
cos⁡0=1\cos 0 = 1cos0=1
cos⁡π=−1\cos \pi = -1cosπ=−1
cos⁡π2=0\cos \frac \pi 2 = 0cos2π=0
cos⁡−α=cos⁡α\cos -\alpha = \cos \alphacos−α=cosα
cos⁡(α±β)=cos⁡α∗cos⁡β∓sin⁡α∗sin⁡β\cos(\alpha \pm \beta) = \cos \alpha * \cos \beta \mp \sin \alpha * \sin \betacos(α±β)=cosα∗cosβ∓sinα∗sinβ
tan⁡x=sin⁡xcos⁡x\tan x = {\sin x \over \cos x}tanx=cosxsinx
sin⁡2x+cos⁡2x=1\sin^2x + \cos^2x = 1sin2x+cos2x=1
sin⁡x∗csc⁡x=1\sin x * \csc x = 1sinx∗cscx=1
cos⁡x∗sec⁡x=1\cos x * \sec x = 1cosx∗secx=1
sin⁡2x=1−cos⁡2x2\sin^2x = {1 - \cos 2x \over 2}sin2x=21−cos2x
cos⁡2x=1+cos⁡2x2\cos^2x = {1 + \cos 2x \over 2}cos2x=21+cos2x
sin⁡(arcsin⁡x)=cos⁡(arccos⁡x)=tan⁡(arctan⁡x)=x\sin(\arcsin x) = \cos(\arccos x) = \tan(\arctan x) = xsin(arcsinx)=cos(arccosx)=tan(arctanx)=x
y=arctan⁡x⟺x=tan⁡yy = \arctan x \iff x = \tan yy=arctanx⟺x=tany

极限运算法则

lim⁡x→x0f(x)=lim⁡x→x0±αf(x∓α)\lim\limits_{x \to x_0}f(x) = \lim\limits_{x \to x_0 \pm \alpha}f(x \mp \alpha)x→x0limf(x)=x→x0±αlimf(x∓α)
lim⁡x→x0[f(x)±g(x)]=lim⁡x→x0f(x)+lim⁡x→x0g(x)\lim\limits_{x \to x_0}[f(x) \pm g(x)] = \lim\limits_{x \to x_0}f(x) + \lim\limits_{x \to x_0}g(x)x→x0lim[f(x)±g(x)]=x→x0limf(x)+x→x0limg(x)
lim⁡x→x0[f(x)∗g(x)]=lim⁡x→x0f(x)∗lim⁡x→x0g(x)\lim\limits_{x \to x_0}[f(x) * g(x)] = \lim\limits_{x \to x_0}f(x) * \lim\limits_{x \to x_0}g(x)x→x0lim[f(x)∗g(x)]=x→x0limf(x)∗x→x0limg(x)
lim⁡x→x0f(x)g(x)=lim⁡x→x0f(x)lim⁡x→x0g(x)\lim\limits_{x \to x_0}{f(x) \over g(x)} = {\lim\limits_{x \to x_0}f(x) \over \lim\limits_{x \to x_0}g(x)}x→x0limg(x)f(x)=x→x0limg(x)x→x0limf(x)
lim⁡x→∞(1+1x)x=e⟺lim⁡x→0(1+x)1x=e⟺1∞\lim\limits_{x \to \infty}(1+\frac 1x)^x = e \iff \lim\limits_{x \to 0}(1+x)^{\frac 1x} = e \iff 1^\inftyx→∞lim(1+x1)x=e⟺x→0lim(1+x)x1=e⟺1∞

极限的无穷

有限的无穷小量的和、差、积为无穷小量
无穷小量和有界函数相加时无穷小量
已知lim⁡x→x0f(x)=A\lim\limits_{x \to x_0}f(x) = Ax→x0limf(x)=A则f(x)=A+α且lim⁡x→x0α=0f(x)=A+\alpha且\lim\limits_{x \to x_0}\alpha = 0f(x)=A+α且x→x0limα=0
已知α∼α′，β∼β′\alpha \sim \alpha\prime，\beta \sim \beta\primeα∼α′，β∼β′则lim⁡x→x0ab=lim⁡x→x0a′b′\lim\limits_{x \to x_0}{\frac ab} = \lim\limits_{x \to x_0}{\frac {a\prime}{b\prime}}x→x0limba=x→x0limb′a′
已知x→0x \to 0x→0则x∼sin⁡x∼tan⁡x∼ln⁡(1+x)∼ex−1,1−cos⁡x∼12x2x \sim \sin x \sim \tan x \sim \ln (1+x) \sim e^x - 1,1 - \cos x \sim \frac 12x^2x∼sinx∼tanx∼ln(1+x)∼ex−1,1−cosx∼21x2
高阶无穷小记做o(x)o(x)o(x)
极限连续中无意义的点是间断点
lim⁡x→x0f[g(x)]=f[lim⁡x→x0g(x)]=f[g(x0)]\lim\limits_{x \to x_0}f[g(x)] = f[\lim\limits_{x \to x_0}g(x)] = f[g(x_0)]x→x0limf[g(x)]=f[x→x0limg(x)]=f[g(x0)]

导数运算

f(n)(x0)=lim⁡x→x0f(n−1)(x)−f(n−1)(xo)x−x0⟹f′(x0)=lim⁡x→x0f(x)−f(xo)x−x0f^{(n)}(x_0) = \lim\limits_{x \to x_0} {f^{(n-1)}(x)-f^{(n-1)}(x_o) \over x - x_0} \implies f'(x_0) = \lim\limits_{x \to x_0} {f(x)-f(x_o) \over x - x_0}f(n)(x0)=x→x0limx−x0f(n−1)(x)−f(n−1)(xo)⟹f′(x0)=x→x0limx−x0f(x)−f(xo)
$(c * \mu)^{(n)} = c * \mu^{(n)} $
(μ±ν)(n)=μ(n)±ν(n)(\mu \pm \nu)^{(n)} = \mu^{(n)} \pm \nu^{(n)}(μ±ν)(n)=μ(n)±ν(n)
$(\mu * \nu)’ = \mu’ * \nu + \mu * \nu’ $
(μν)′=μ′∗ν−μ∗ν′ν′({\mu \over \nu})' = {\mu' * \nu - \mu * \nu' \over \nu'}(νμ)′=ν′μ′∗ν−μ∗ν′
(xμ)′=μxμ−1(x^{\mu})' = \mu x^{\mu-1}(xμ)′=μxμ−1
(f[g(x)])′=f′[g(x)]g′(x)(f[g(x)])' = f'[g(x)]g'(x)(f[g(x)])′=f′[g(x)]g′(x)
隐函数求导：dydx=−Fx(x,y)Fy(x,y)\frac {dy}{dx} = - {\frac {F_x(x,y)}{F_y(x,y)}}dxdy=−Fy(x,y)Fx(x,y)

常用导数运算

(ex)(n)=ex(e^x)^{(n)} = e^x(ex)(n)=ex
$ (\ln x)’ = \frac 1x$
(sin⁡x)′=cos⁡x(\sin x)' = \cos x(sinx)′=cosx
(cos⁡x)′=−sin⁡x(\cos x)' = -\sin x(cosx)′=−sinx
(arcsin⁡x)′=11−x2(\arcsin x)' = {1 \over \sqrt{1 - x^2}}(arcsinx)′=1−x21
(arctan⁡x)′=11+x2(\arctan x)' = {1 \over 1 + x^2}(arctanx)′=1+x21
(ax)(n)=axln⁡na(a^x)^{(n)} = a^x\ln^na(ax)(n)=axlnna
(sin⁡kx)(n)=kn∗sin⁡(k∗x+n∗π2)(\sin kx)^{(n)} = k^n * \sin(k*x + n * \frac\pi2)(sinkx)(n)=kn∗sin(k∗x+n∗2π)
[ln⁡(x+b)](n)=(−1)(n−1)(n−1)!(x+b)(n)[\ln(x+b)]^{(n)} = (-1)^{(n-1)}{(n-1)! \over (x+b)^{(n)}}[ln(x+b)](n)=(−1)(n−1)(x+b)(n)(n−1)!

微分基本运算

dy=f′(x)dx⟺df(x)=f′(x)dxdy=f'(x)dx \iff df(x) = f'(x) dxdy=f′(x)dx⟺df(x)=f′(x)dx
dydx=dydu∗dudx=dydudxdu{dy \over dx} = {dy \over du} * {du \over dx} = {{dy \over du} \over {dx \over du}}dxdy=dudy∗dxdu=dudxdudy
dy=f′(u)∗dudy = f'(u) * dudy=f′(u)∗du
d(μ+c)=dμd(\mu+c) = d\mud(μ+c)=dμ
d(μ∗c)=c∗dμd(\mu * c) = c * d\mud(μ∗c)=c∗dμ

洛必达法则

00\frac 0000、∞∞\frac \infty\infty∞∞、0∗∞0 * \infty0∗∞、∞−∞\infty - \infty∞−∞
进行反复求导，直到不满足不定式
0∗∞⟺01∞⟺∞10⟺∞∗00 * \infty \iff {0 \over {1 \over \infty}} \iff {\infty \over {1 \over 0}} \iff \infty * 00∗∞⟺∞10⟺01∞⟺∞∗0
f(x)−g(x)=f(x)−g(x)f(x)∗g(x)∗f(x)∗g(x)=1g(x)−1f(x)1f(x)∗1g(x)f(x) - g(x) = {f(x) - g(x) \over f(x) * g(x)} * f(x) * g(x) = {{1 \over g(x)} - {1 \over f(x)} \over {{1 \over f(x)} *{1 \over g(x)}}}f(x)−g(x)=f(x)∗g(x)f(x)−g(x)∗f(x)∗g(x)=f(x)1∗g(x)1g(x)1−f(x)1
f(x)g(x)=eln⁡f(x)g(x)=eg(x)∗ln⁡f(x)f(x)^{g(x)} = e^{\ln f(x)^{g(x)}} = e^{g(x) * \ln f(x)}f(x)g(x)=elnf(x)g(x)=eg(x)∗lnf(x)

导数的平面几何

切线与法线关系：k′=−1kk' = -\frac 1kk′=−k1
切线公式：y−f(x0)=f′(x0)∗(x−x0)y - f(x_0) = f'(x_0) * (x - x_0)y−f(x0)=f′(x0)∗(x−x0)
法线公式：y−f(x0)=−1f′(x0)∗(x−x0)y - f(x_0) = - {1 \over f'(x_0)} * (x - x_0)y−f(x0)=−f′(x0)1∗(x−x0)
导数为0的点为驻点
二阶导数在驻点处的值大于0为极大值，小于0为极小值
已知lim⁡x→∞f(x)=A\lim\limits_{x \to \infty}f(x) = Ax→∞limf(x)=A则y=Ay = Ay=A是水平渐近线
已知lim⁡x→af(x)=∞\lim\limits_{x \to a}f(x) = \inftyx→alimf(x)=∞则x=ax = ax=a是垂直渐近线
在区间内导数大于零是单调增加，小于零是单调减少，等于零则判断两侧符号
驻点区间内导数符号判断单调性
比较所有驻点、不可导点的值大小可以得到最值
二阶导数大于零为凹，小于零为凸
二阶导数等于零时的点和不存在的点，将点的两侧代入二阶导函数结果异号为拐点

拉格朗日割线

f(b)−f(a)=f′(c)∗(b−a)⟺f′(c)=f(b)−f(a)b−af(b) - f(a) = f'(c) * (b - a) \iff f'(c) = {f(b)- f(a) \over b - a}f(b)−f(a)=f′(c)∗(b−a)⟺f′(c)=b−af(b)−f(a)
f(a)=f(b)⟺f′(c)=0f(a) = f(b) \iff f'(c) = 0f(a)=f(b)⟺f′(c)=0

不定积分

∫f(x)dx=F(x)+C\int f(x)dx = F(x) + C∫f(x)dx=F(x)+C

∫kf(x)dx=k∫f(x)dx\int kf(x)dx = k\int f(x)dx∫kf(x)dx=k∫f(x)dx

∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx\int [f(x) \pm g(x)]dx = \int f(x)dx \pm \int g(x)dx∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx

[∫f(x)dx]′=f(x)⟺d[∫f(x)dx]=f(x)dx[\int f(x)dx]' = f(x) \iff d[\int f(x)dx] = f(x)dx[∫f(x)dx]′=f(x)⟺d[∫f(x)dx]=f(x)dx

∫f′(x)dx=f(x)+c⟺∫df(x)=f(x)+c\int f'(x)dx = f(x) + c \iff \int df(x) = f(x) + c∫f′(x)dx=f(x)+c⟺∫df(x)=f(x)+c

∫xadx=xa+1a+1+C\int x^adx = {x^{a+1} \over a+1} + C∫xadx=a+1xa+1+C

∫dxx=ln⁡∣x∣+C\int {dx \over x} = \ln|x| + C∫xdx=ln∣x∣+C

∫axdx=axln⁡a+C\int a^xdx = {a^x \over \ln a} + C∫axdx=lnaax+C

∫sin⁡xdx=−cos⁡x+C\int\sin xdx = -\cos x + C∫sinxdx=−cosx+C

∫cos⁡xdx=sin⁡x+C\int\cos xdx = \sin x + C∫cosxdx=sinx+C

∫11−x2dx=arcsin⁡x+C\int {1 \over \sqrt {1 - x^2}}dx = \arcsin x + C∫1−x21dx=arcsinx+C

∫11+x2dx=arcsin⁡x+C\int {1 \over {1 + x^2}}dx = \arcsin x + C∫1+x21dx=arcsinx+C

∫f[ϕ(x)]ϕ′(x)dx=∫f(μ)du=F(μ)∣μ=ϕ(x)+C=F[ϕ(x)]+C\int f[\phi(x)]\phi'(x)dx = \int f(\mu)du = F(\mu)|_{\mu=\phi(x)} + C = F[\phi(x)] + C∫f[ϕ(x)]ϕ′(x)dx=∫f(μ)du=F(μ)∣μ=ϕ(x)+C=F[ϕ(x)]+C

∫f(x)dx=∫f[ϕ(t)]ϕ′(t)dt=Φ(t)+C=Φ[ϕ−1(x)]+C\int f(x)dx = \int f[\phi(t)]\phi'(t)dt = \Phi(t) + C = \Phi[\phi^{-1}(x)] + C∫f(x)dx=∫f[ϕ(t)]ϕ′(t)dt=Φ(t)+C=Φ[ϕ−1(x)]+C

∫uv′dx=uv−∫u′vdx⟺∫udv=uv−∫vdu\int uv'dx = uv - \int u'vdx \iff \int udv = uv - \int vdu∫uv′dx=uv−∫u′vdx⟺∫udv=uv−∫vdu

定积分

∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)∫abf(x)dx=F(b)−F(a)

∫abkf(x)dx=k∫abf(x)dx\int_a^b kf(x)dx = k\int_a^b f(x)dx∫abkf(x)dx=k∫abf(x)dx

∫ab[f(x)±g(x)]dx=∫abf(x)dx±∫abg(x)dx\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx∫ab[f(x)±g(x)]dx=∫abf(x)dx±∫abg(x)dx

∫abf(x)dx=∫acf(x)dx±∫cbf(x)dx\int_a^b f(x)dx = \int_a^c f(x)dx \pm \int_c^b f(x)dx∫abf(x)dx=∫acf(x)dx±∫cbf(x)dx

∫ab1dx=b−a\int_a^b 1dx = b - a∫ab1dx=b−a

f(x)⩽g(x)⟹∫abf(x)dx⩽∫abg(x)dxf(x) \leqslant g(x) \implies \int_a^bf(x)dx \leqslant \int_a^bg(x)dxf(x)⩽g(x)⟹∫abf(x)dx⩽∫abg(x)dx

∣∫abf(x)dx∣⩽∫ab∣f(x)∣dx|\int_a^b f(x)dx| \leqslant \int_a^b |f(x)|dx∣∫abf(x)dx∣⩽∫ab∣f(x)∣dx

∫abf(x)dx=f(ξ)(b−a)\int_a^b f(x)dx = f(\xi)(b-a)∫abf(x)dx=f(ξ)(b−a)

ddx∫axf(t)dt=f(x)\frac d{dx}\int_a^xf(t)dt = f(x)dxd∫axf(t)dt=f(x)

f(x)f(x)f(x)是在对称区间是奇函数：∫−aaf(x)dx=0\int_{-a}^af(x)dx = 0∫−aaf(x)dx=0

f(x)f(x)f(x)是在对称区间是偶函数：∫−aaf(x)dx=2∗∫−aaf(x)dx\int_{-a}^af(x)dx = 2 * \int_{-a}^af(x)dx∫−aaf(x)dx=2∗∫−aaf(x)dx

∫abf(x)dx=∫αβf[ϕ(x)]ϕ′(x)dx\int_a^b f(x)dx = \int_\alpha^\beta f[\phi(x)]\phi'(x)dx∫abf(x)dx=∫αβf[ϕ(x)]ϕ′(x)dx

∫abuv′dx=uv∣ab−∫abu′vdx⟺∫abudv=uv∣ab−∫abvdu\int_a^b uv'dx = uv|_a^b - \int_a^b u'vdx \iff \int_a^b udv = uv|_a^b - \int_a^b vdu∫abuv′dx=uv∣ab−∫abu′vdx⟺∫abudv=uv∣ab−∫abvdu

积分几何

空间平面一般方程Ax+By+Cz+D=0Ax + By + Cz + D = 0Ax+By+Cz+D=0

空间曲面标准方程A(x−x0)+B(y−y0)+C(z−z0)+D=0A(x - x_0) + B(y - y_0) + C(z - z_0) + D = 0A(x−x0)+B(y−y0)+C(z−z0)+D=0的法向量为(A,B,C)

空间直线标准方程x−x0A=y−y0B=z−z0C{x - x_0 \over A} = {y - y_0 \over B} = {z - z_0 \over C}Ax−x0=By−y0=Cz−z0的方向向量为(A,B,C)

垂直 A1A2+B1B2+C1C2=0A_1A_2 + B_1B_2 + C_1C_2 = 0A1A2+B1B2+C1C2=0

平行 A1A2=B1B2=C1C2\frac {A_1}{A_2} = \frac {B_1}{B_2} = \frac {C_1}{C_2}A2A1=B2B1=C2C1

曲线区间面积：S=∫ab[f(x)−g(x)]dxS =\int_a^b [f(x) - g(x)]dxS=∫ab[f(x)−g(x)]dx

旋转体体积：dV=π∫abf2(x)dxdV =\pi\int_a^b f^2(x)dxdV=π∫abf2(x)dx

偏导数、全微分、二元积分

lim⁡x→x0f(x0+x,y0)−f(x0,y0)x=σzσx∣x=x0y=y0=zx∣x=x0y=y0=fx(x0,y0)\lim\limits_{x \to x_0} {{f(x_0 + x,y_0) - f(x_0,y_0)} \over x} = \frac {\sigma z}{\sigma x}|_{x = x_0 \atop y = y_0} = z_x|_{x = x_0 \atop y = y_0} = f_x(x_0,y_0)x→x0limxf(x0+x,y0)−f(x0,y0)=σxσz∣y=y0x=x0=zx∣y=y0x=x0=fx(x0,y0)

dz=σzσx∗dx+σzσy∗dydz = \frac {\sigma z}{\sigma x} * dx + \frac {\sigma z}{\sigma y} * dydz=σxσz∗dx+σyσz∗dy

σσx(σzσy)=σ2zσx∗σy=fxy(x,y)\frac \sigma {\sigma x}(\frac {\sigma z}{\sigma y}) = \frac {\sigma ^2z}{\sigma x * \sigma y} = f_{xy}(x,y)σxσ(σyσz)=σx∗σyσ2z=fxy(x,y)

σzσx=σzσu∗σuσx+σzσv∗σvσx\frac {\sigma z}{\sigma x} = \frac {\sigma z}{\sigma u} * \frac {\sigma u}{\sigma x} + \frac {\sigma z}{\sigma v} * \frac {\sigma v}{\sigma x}σxσz=σuσz∗σxσu+σvσz∗σxσv

σyσx=−Fx(x,y,z)Fy(x,y,z)\frac {\sigma y}{\sigma x} = -{F_x(x,y,z) \over F_y(x,y,z)}σxσy=−Fy(x,y,z)Fx(x,y,z)

二重积分∬Df(x,y)dσ=lim⁡λ→0∑i=0nf(ξi,ηi)σi\iint\limits_D f(x,y)d\sigma = \lim\limits_{\lambda \to 0}\displaystyle\sum_ {i=0}^n f(\xi_i,\eta_i)\sigma_iD∬f(x,y)dσ=λ→0limi=0∑nf(ξi,ηi)σi

∬Dkf(x,y)dσ=k∬Df(x,y)dσ\iint\limits_D kf(x,y)d\sigma = k\iint\limits_D f(x,y)d\sigmaD∬kf(x,y)dσ=kD∬f(x,y)dσ

∬D[f(x,y)±g(x,y)]dσ=∬Df(x,y)dσ±∬Dg(x,y)dσ\iint\limits_D [f(x,y) \pm g(x,y)]d\sigma = \iint\limits_D f(x,y)d\sigma \pm \iint\limits_D g(x,y)d\sigmaD∬[f(x,y)±g(x,y)]dσ=D∬f(x,y)dσ±D∬g(x,y)dσ

∬Ddσ=σ\iint\limits_D d\sigma = \sigmaD∬dσ=σ

∬Df(x,y)dσ=∬D1f(x,y)dσ+∬D2f(x,y)dσ\iint\limits_D f(x,y)d\sigma = \iint\limits_{D_1} f(x,y)d\sigma + \iint\limits_{D_2} f(x,y)d\sigmaD∬f(x,y)dσ=D1∬f(x,y)dσ+D2∬f(x,y)dσ

二元函数极值

驻点fx(x0,y0)=0,fy(x0,y0)=0f_x(x_0,y_0) = 0,f_y(x_0,y_0) = 0fx(x0,y0)=0,fy(x0,y0)=0

极值存在fxx(x0,y0)∗fyy(xo,y0)>fxy(xo,y0)∗fxy(xo,y0)f_{xx}(x_0,y_0) * f_{yy}(x_o,y_0) > f_{xy}(x_o,y_0) * f_{xy}(x_o,y_0)fxx(x0,y0)∗fyy(xo,y0)>fxy(xo,y0)∗fxy(xo,y0)

存在的极之是极小值fxx(x0,y0)>0f_{xx}(x_0,y_0) > 0fxx(x0,y0)>0

存在的极之是极大值fxx(x0,y0)<0f_{xx}(x_0,y_0) < 0fxx(x0,y0)<0

拉格朗日函数 L(x,y)=f(x,y)+λϕ(x,y)⟺{fx(x,y)+λϕx(x,y)=0fy(x,y)+λϕy(x,y)=0ϕ(x,y)=0L(x,y) = f(x,y) + \lambda\phi(x,y) \iff \begin{cases} {f_x(x,y) + \lambda\phi_x(x,y) = 0} \\ {f_y(x,y) + \lambda\phi_y(x,y) = 0} \\ {\phi(x,y) = 0}\end{cases}L(x,y)=f(x,y)+λϕ(x,y)⟺⎩⎪⎨⎪⎧fx(x,y)+λϕx(x,y)=0fy(x,y)+λϕy(x,y)=0ϕ(x,y)=0

二重积分几何

二重积分在平面上的面积：σ\sigmaσ

∬Df(x,y)dσ=∫abf1(x)dx∗∫cdf2(y)dy\iint\limits_D f(x,y)d\sigma =\int_a^bf_1(x)dx * \int_c^df_2(y)dyD∬f(x,y)dσ=∫abf1(x)dx∗∫cdf2(y)dy

dσ=dxdy⟺∬Df(x,y)dxdy=∫abdx∫ϕ1(x)ϕ2(x)f(x,y)dyd\sigma = dxdy \iff \iint\limits_Df(x,y)dxdy = \int_a^bdx\int_{\phi_1(x)}^{\phi_2(x)}f(x,y)dydσ=dxdy⟺D∬f(x,y)dxdy=∫abdx∫ϕ1(x)ϕ2(x)f(x,y)dy

dσ=rdrdθ⟺∬Df(x,y)dσ=∫αβdθ∫r1(θ)r2(θ)f(rcos⁡θ,rsin⁡θ)rdrd\sigma = rdrd\theta \iff \iint\limits_Df(x,y)d\sigma = \int_{\alpha}^{\beta}d\theta\int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta,r\sin\theta)rdrdσ=rdrdθ⟺D∬f(x,y)dσ=∫αβdθ∫r1(θ)r2(θ)f(rcosθ,rsinθ)rdr

级数

∑n=1∞un=u1+u2+u3+...+un+...\displaystyle\sum_ {n=1}^\infty u_n= u_1+u_2+u_3+...+u_n+...n=1∑∞un=u1+u2+u3+...+un+...

数列极限存在则收敛，不存在则发散

调和级数是发散的∑n=1∞1n\displaystyle\sum_ {n=1}^\infty \frac 1nn=1∑∞n1

等比级数∣q∣<1|q|<1∣q∣<1收敛，∣q∣⩾1|q|\geqslant1∣q∣⩾1发散，∑n=1∞aqn−1⟺a∗1−qn1−q\displaystyle\sum_ {n=1}^\infty aq^{n-1} \iff a * {1-q^n \over 1-q}n=1∑∞aqn−1⟺a∗1−q1−qn

级数乘以常数，两个相同敛散性的级数加减，基数前面改变有限项，敛散性都不变

正项级数每一项都非负

p级数：∑n=1∞1np\displaystyle\sum_ {n=1}^\infty \frac 1{n^p}n=1∑∞np1在p>1p>1p>1时收敛，在0<p⩽10 < p \leqslant 10<p⩽1时发散

正项级数大的收敛小的也收敛，小的发散大的也发散

正项级数∑n=1∞un\displaystyle\sum_ {n=1}^\infty u_nn=1∑∞un和∑n=1∞vn\displaystyle\sum_ {n=1}^\infty v_nn=1∑∞vn敛散性相同：0<lim⁡m→∞unvn<+∞0 < \lim\limits_{m \to \infty} \frac {u_n}{v_n} < +\infty0<m→∞limvnun<+∞

正项级数lim⁡m→∞un+1un<1\lim\limits_{m \to \infty} {u_{n+1} \over u_n} < 1m→∞limunun+1<1 收敛，lim⁡m→∞un+1un>1\lim\limits_{m \to \infty} {u_{n+1} \over u_n} > 1m→∞limunun+1>1 发散，等于1无法判断

交错级数：∑n=1∞(−1)nun\displaystyle\sum_ {n=1}^\infty (-1)^nu_nn=1∑∞(−1)nun

绝对收敛：∑n=1∞∣un∣\displaystyle\sum_ {n=1}^\infty |u_n|n=1∑∞∣un∣收敛，∑n=1∞un\displaystyle\sum_ {n=1}^\infty u_nn=1∑∞un收敛

条件收敛：∑n=1∞un\displaystyle\sum_ {n=1}^\infty u_nn=1∑∞un收敛，∑n=1∞∣un∣\displaystyle\sum_ {n=1}^\infty |u_n|n=1∑∞∣un∣发散

若un>un+1u_n > u_{n+1}un>un+1且lim⁡n→∞un=0\lim\limits_{n \to \infty}u_n = 0n→∞limun=0，则∑n=1∞(−1)n−1un\displaystyle\sum_ {n=1}^\infty (-1)^{n-1}u_nn=1∑∞(−1)n−1un收敛，且S⩽u1S \leqslant u_1S⩽u1

幂级数：∑n=0∞an(x−x0)n\displaystyle\sum_ {n=0}^\infty a_n(x-x_0)^nn=0∑∞an(x−x0)n

∑n=0∞anxn\displaystyle\sum_ {n=0}^\infty a_nx^nn=0∑∞anxn在∣x∣<R|x| < R∣x∣<R时绝对收敛，在∣x∣>R|x| > R∣x∣>R时发散，在x=0x=0x=0时收敛

∑n=0∞anxn\displaystyle\sum_ {n=0}^\infty a_nx^nn=0∑∞anxn在x=x0x = x_0x=x0收敛，则−∣x0∣<x<∣x0∣-|x_0| < x < |x_0|−∣x0∣<x<∣x0∣时绝对收敛

∑n=0∞anxn\displaystyle\sum_ {n=0}^\infty a_nx^nn=0∑∞anxn在x=x0x = x_0x=x0发散，则x<−∣x0∣x < -|x_0|x<−∣x0∣和x>∣x0∣x > |x_0|x>∣x0∣时发散

收敛半径：R=lim⁡n→∞∣anan+1∣R = {\lim\limits_{n \to \infty} |{a_n \over {a_{n+1}}}|}R=n→∞lim∣an+1an∣

泰勒级数：f(x)=∑n=0∞f(n)(x0)n!(x−x0)nf(x) = \displaystyle\sum_ {n=0}^\infty {f^{(n)}(x_0) \over n!}(x - x_0)^nf(x)=n=0∑∞n!f(n)(x0)(x−x0)n

麦克劳林级数：f(x)=∑n=0∞f(n)(0)n!(x)nf(x) = \displaystyle\sum_ {n=0}^\infty {f^{(n)}(0) \over n!}(x)^nf(x)=n=0∑∞n!f(n)(0)(x)n

S′(x)=∑n=0∞nanxx−1S'(x) = \displaystyle\sum_ {n=0}^\infty na_nx^{x-1}S′(x)=n=0∑∞nanxx−1

∫0xS(x)dx=∑n=0∞ann+1xn+1\int_0^xS(x)dx = \displaystyle\sum_ {n=0}^\infty {a_n \over n+1}x^{n+1}∫0xS(x)dx=n=0∑∞n+1anxn+1

常见幂级数展开

11−x=∑n=0∞xn，x∈(−1,1){1 \over 1 - x} = \displaystyle\sum_ {n=0}^\infty x^n，x \in (-1,1)1−x1=n=0∑∞xn，x∈(−1,1)

ex=∑n=0∞xnn!，x∈(−∞,+∞)e^x = \displaystyle\sum_ {n=0}^\infty {x^n \over n!}，x \in (-\infty,+\infty)ex=n=0∑∞n!xn，x∈(−∞,+∞)

sin⁡x=∑n=0∞(−1)n(2n+1)!x2n+1，x∈(−∞,+∞)\sin x = \displaystyle\sum_ {n=0}^\infty {(-1)^n \over (2n+1)!}x^{2n+1}，x \in (-\infty,+\infty)sinx=n=0∑∞(2n+1)!(−1)nx2n+1，x∈(−∞,+∞)

cos⁡x=∑n=0∞(−1)n(2n)!x2n，x∈(−∞,+∞)\cos x = \displaystyle\sum_ {n=0}^\infty {(-1)^n \over (2n)!}x^{2n}，x \in (-\infty,+\infty)cosx=n=0∑∞(2n)!(−1)nx2n，x∈(−∞,+∞)

ln(1+x)=∑n=1∞(−1)n−1xnn，x∈(−1,1]ln(1 + x) = \displaystyle\sum_ {n=1}^\infty (-1)^{n-1}{x^n\over n}，x \in (-1,1]ln(1+x)=n=1∑∞(−1)n−1nxn，x∈(−1,1]

ln(1−x)=−∑n=1∞xnn，x∈[−1,1)ln(1 - x) = - \displaystyle\sum_ {n=1}^\infty {x^n\over n}，x \in [-1,1)ln(1−x)=−n=1∑∞nxn，x∈[−1,1)

一阶微分方程

可分离变量微分方程：dydx=f(x)g(y)⟺∫g(y)dy=∫f(x)dx\frac {dy}{dx} = f(x)g(y) \iff \int g(y)dy = \int f(x)dxdxdy=f(x)g(y)⟺∫g(y)dy=∫f(x)dx
一阶线性微分方程：dydx+P(x)y=Q(x)\frac {dy}{dx} + P(x)y = Q(x)dxdy+P(x)y=Q(x)，等于零为齐次，否则非齐次
一阶线性齐次微分方程通解：y=Ce−∫P(x)dxy = Ce^{-\int P(x)dx}y=Ce−∫P(x)dx
一阶线性非齐次微分方程通解：y=Ce−∫P(x)dx[∫Q(x)e∫P(x)dxdx+C]y = Ce^{-\int P(x)dx}[\int Q(x)e^{\int P(x)dx}dx+ C]y=Ce−∫P(x)dx[∫Q(x)e∫P(x)dxdx+C]

二阶常系数线性微分方程

y′′+py′+qy=f(x)y''+ py' + qy = f(x)y′′+py′+qy=f(x)，等于零为齐次，否则非齐次
若y1=y1(x)y_1=y_1(x)y1=y1(x)和y2=y2(x)y_2=y_2(x)y2=y2(x)线性无关，则有通解y=C1y1+C2y2y = C_1y_1 + C_2y_2y=C1y1+C2y2

齐次方程的特征函数：r2+pr+q=0r^2 + pr + q = 0r2+pr+q=0
特征函数有两个不相等的实根，则y‾=C1er1x+C2er2x\overline y = C_1e^{r_1x} + C_2e^{r_2x}y=C1er1x+C2er2x
特征函数有两个相等的实根，则y‾=(C1+C2x)erx\overline y = (C_1 + C_2x)e^{rx}y=(C1+C2x)erx
特征函数无实根，则y‾=(C1cos⁡βx+C2sin⁡βx)eax\overline y = (C_1\cos \beta x + C_2\sin \beta x)e^{ax}y=(C1cosβx+C2sinβx)eax

非齐次方程通解：齐次方程通解加非齐次方程特解：y∗=xkQn(x)eaxy^* = x^kQ_n(x)e^{ax}y∗=xkQn(x)eax，a不是特征根，则k=0，a是某一特征根，则k=1，a是唯一特征根，则k=2