考研数学公式手册随身看

(一)函数、极限、连 (二)一元函数微分 (四)向量代数和空间解析几 (一)行列 (三)向 这是一套真正能用的资料，如想打包以上资料，请通过参与活动，加入会员，免费领 公共课297分学长免费带你全程实战[了解详情 (一)yf1幂函数yxRyaxa0a1对数函数ylogax(a0a1三角函数:ysinxycosxytanx等yarcsinxyarccosxyarctanx等及其性limf(x)Af(x)f(x) limf(xAf(x0Aa(x其中lima(x)1设limf(x)A,又A0(或A0),则一个0当x(x0x0),且xx0时，f(x0(或f(x若

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准线为fx,y0,//z 准线为xz0,y fx,y,z准线为: ,母线的方向矢量为l,m,首先,在准线上任取一点x,yz,则过点x,yz的母线方程XxYyZz X,YZ为母线上任一点的流动坐标消去方程组fx,y,zgx,y,z0 X Y Z x2y22y2y方程x22py,pxzyx2y2z2 a,bc均为正数zc yx a,bc均为正数 xyz a,bc均为正数x2y2 2a,b,p为正数面x2y2 2a,b,p均为正数x2y2z2 a,bc为正数z xzf(xy连续，可导（两偏导存在）可导可微zf'(x,y)xf'(x,lim 是否为0 Th1(求偏导与次序无关定理设zf(xy)的两个混合偏导数fxyfx z,z必存在,且有dzzdxzx Th3(偏导存在与可微的关系定理若zf(xy)zz在P(xx上的某领域内存在,且在P(x,y)则zf(xy)在P(xy)阶偏导zzuz设zf(u,v),u(x,y),v(x,y),则 u v z z u v设zf(uvu(xv则dzzduzdv称之为zdxu v设zf(xuvu(xyv(xzffuf则 u vz0fuf u v其中间变量用数字1，2，3……表示更简洁.设F(xy)0,则dyF'x(x F'y(x,F(xyz)0,则zF'x(xyz,zF'y(xy F'z(x,y,z) F'z(x,y,设由方程组 确定的隐函数yy(x),z则dydzdydzdx dxF'F'dyF'dz F'dyF'dzF' y z y z G'G G G G G y z y z Th1zf(xy)M0x0y0)处可微，则f(xy)在点M0(x0y0沿任意方向lcoscos)f(x0,y0)f(x0,y0)cosf(x0,y0)cos lcos,sin是l的极角，[0,2数的计 为f(x0,y0)f(x0,y0)cosf(x0,y0) Th2设三元函数uf(xyzM0x0y0z0uf(xyzM0x0y0z0f(x0,y0,z0)f(x0,y0,z0)cosf(x0,y0,z0)cos f(x0,y0)(f(x0,y0),f(x0,y0))(cos,cos grad(f(x0,y0))lgradf(x0,y0)cosgrad(f(x0,y0gradf(xyf(x0y0f(x0y0) 0 zf(xyM0的梯度(向量f(x0y0随llgradf(x0y0) grad(f(x0,y0gradf(x0y0曲线yy(t)在(xyztz 0 xx0yy0z： 法平面方程：x'(t0xx0y'(t0yy0z'(t0zz0G(x,y,z)（x y _z(F, (F, (F,(y, (z, (x, (F,G)（xx)(F, (yy)(F, (zz)(y, (z, (x, z(xxzyyzz x yxx0yy0z： x p切平面方程：F'x(xx0Fyyy0Fz(zz00 xx0yy0z：F'x F'y F'zP(x0y0)点的任一点Q(x,y)f(xyf(x0y0或f(x0y0则称f(x0y0为f(xy)的极小值(极大值1取极值的必要条件设zf(xy)在P(x0y0f'(x,y)P(x,y)是zf(x,y)的极值点，则 fy(x0,y0)Th（设zf(xy)在P(x0y0连续的二阶偏导数,且f'x(x0y00,f'y(x0y0[f"(x,y)]2f"2(x,y)f"2(x,y)xy 则P(x0y0是zf(xy)若f"2xy0(或f"2xy0),则P(xy) 若f2xy0(或f"2xy0),则P(xy) 求出zf(xy)的驻点（x0y0zf(xy3条件极值（日乘数法解题程序令F(x,y)=f(x,解方程组f'(x,y)'(x,y) 求驻点(x,y (x,y)令F（xyz（xyz);f'y(x,y,z)'y(x,y,z)解方程组fxyz'(xyz ）2）分的概nI=f(x,y)d=limf(i,i)i,其中dmaxdid i di为i的直径（i1, nI=F(xyz)dvlimf(i,i,i)vi其中dmaxdid i di为vi的直径（i1,2, [f(x,y)g(x,y)]df(x,y)dg(x, nf(xy)df(xy)d其中DiD i1 ,D若在D上恒有f(xyg(xy),则f(xy)dg(x A为DmAf(xy)dMADD,则二重积分f(xD0,f关于y为奇函数，即f(xyf(x2f(xy)df关于y为偶函数，即f(xyf(x则二重积分f(x,y)dD0,f关于x的奇函数，即f(xyf(x2f(xy)df关于x为偶函数，即f(xyf(x则二重积分f(xD0,f关于x,y的奇函数，即f(xyf(x2f(xy)df关于x，y为偶函数，即f(xyf(x4）如果D关于直线yx对称，则f(xy)df(x 注意到二重积分积分域Df(xy的Df(xy数，则LPdxQdyQP,(x,y) LPdxQdy0,L存在函数u(x,y),(x,yDdu(x,y)PdxQdy(x,u(x,y)(x,y)Pdx(QP)dxdyPdxD 或者或者(QP)dxdyPdxD ，1(Gauss)设(PQR)dVPdydzQdzdxRdxdy (PQR)dV（PcosQcosRcos 这里S()coscoscosS上点(xyz处的外法向量的方向余弦.2设为分段光滑的又向闭曲线，S是以为边界的分块光滑)(RQ)dydz(PR)dzdx(QP)dxdyPdxS dydzdzdx( PdxQdyRdzS (RQ)cosPR)cosQP)cos S PdxQdyijkPQR 若unsvn则(unvn)s 1级数un 若unvn均发散,则(unvn)敛散性不定 p级数的收敛正项级数un（un0）比较判敛法：设0unvn, 且limunA(vn n 若0A,且vn收敛，则un 若0A,且vn发散，则un a,|r|等比级数arn11 p级数np发散，p1 （设un0,n1, n 与1(1)n1uu0 若交错级数(1)n1uu0) (1)unun1,(n1, );(2)limunnnnn|u(x)lim|un1(x|(x)(或limn|u(x|n收敛半径，若liman1,则R1nnaxn a 1aaxax2) () 设axnf(xbxng(x R1R2Rmin(R1R2则对x(RR), axnbxn(ab)xnf(xg(x且在（- f设b00,则在x0f(x)a0a1x anxn CCxCx2 Cxn bbx bxn Ca0,Ca1b0a0b1 nn axn可逐项微分，且f'=(axn (anxn)'nanxn1 axn可逐项积分，且xft）dt=xatn (xatndt) an n0 ff(n)(x f"(x级数 0(xx)nf(x)f'(x)(xx) 0(xx)2 f(n)(x0)(xx)n ff(n) f xnf(0)f'(0)x 2!x2 (0)xnf(n)(x n!0(xx0收敛于f(x)的充分条件limRn(x其中R(x) f(n1)[x(xx)](xx)n1,00 (n 11uu2 un 1u 1uu2 (1)nun 1u eu1uu2 un n0 n nsinuu (2n (2n n cosu1 (1) n n , n n(1u)a1aua(a1)u2 a(a 与[la=1f(x)cosnxdx12f(x)cosnxdx,(n0,1,2,n b1f(x)sinnxdx12f(x)sinnxdx,(n1,2, 2 10 2a (acosnxbsina1lf(xcosnxdx,(n0,1,2) l b1lf(x)sinnxdx,(n0,1,2 l 为系数的三角级数2a0 (ancoslxbnsinl 2a0 (ancoslxbnsinl3狄里赫莱收敛定理：设函数f(x)在[-,]只有有限个极值点，则f(x)的级数在[-，]上收敛，且 f(x),x为f(x)0(acosxbsinnx1f(x0f(x0x为fx）[ 1[f(0)f(0)],x [0,l]f(x为[0,lF(xf(x0x f(x~a0acosnx（f(x),lx 数a2lf(xcosnxdx l f(x为[0,lF(xf(x0x F(x)除x=0外在区间[,]f(x),lxf(x~bsinnx（正弦级数 b2lf(x)sinnxdx l 常微分方程含有自变量、未知函数及未知函数的某些导f1x)g1y)dxf2x)g2y)dygyf(x0，得f1(x)dxg2ydy f g( f1(x)dxg2(y)dyCf2(x) g1(y)程yfyx解法：令uyyuxyuxdu uxduf(u) du dx lnxC f(u)u f(u)u可化为齐次型的方程dyfa1xb1yc1 axby 2解法：(1)当c1c20abydyfa1xb1yf 1xgy属于 axby y 2 a2 x(2)a1b10a1b1 方 2dyf(a2xb2y)c1g(axby) axby 2axbyuduabf(u属于 (3)．a1b10,c,c不全为 解方程组a1xb1yc1 1 axbyc 求交点( xXyY则原方程dyX属于 3y'p(xyy'p(xy0yCepyC(x)epp( p(C q(x)C(x) dx y[q(x)ep(x)dxdxC]ep(4y'p(xyq(xynn解法：令Zy1n，则方程 dzp(x)zq(x)1ndz(1np(x)z(1n)q(x5M(xy)dxN(xy)dy0MN.通解为xM(x,y)dxyN(x,y)dy yp(xyq(xyf (8.1)p(xq(x),f(x)1y1xy2xyp(xyq(xy0(8.2)y(xy(x线性无关(y1(x)(常数，则(8.2) y2),)y(x)yxCy(xCy(x)，其中C,C为任意常数1 2 1二阶常系数线性齐次方程y''py'qy (1)其中p,2pqy(x)CexCe 当y(xCC 当i（复根）y(x)ex(CcosxCsin 2n阶常系数齐次线性方程y(n)py(n1)py(n2) py0(＊) np(n1)p(n2) p 若1,2 y(x)CexCex Ce 0k(k重实根，则(＊)的通解中含有：(CCx Cxk1)e ex[(CCxCxk1)cosx(DDxDxk1)sin 程，1二阶常系数线性非齐次方程y''py'qyf(x)(2)(1)．求对应齐次方程的通解Y(x)(2)．求出(2)的特解y*(x)(3)．方程(2)yY(xy)待定系数法2形如xny(n)axn1y(n1) xyay0的方程成为 (一)设A(a ,则aAaA aAA,iij i1 i2j in 0,i或aAaA aAA,i1i1 2i2 ni AA*A*AAE,其 A n1 A(A)(A n2 A nnABnABABBA ABAB|kA|kn|A|A为n设A为n阶方阵，则|AT||A|;|A1||A|1(若A可逆）|A*||A|n1（nA A |A||B|，A,B为方阵，OB OB (6) 行列式D (xx 1jxn1 设A是n阶方阵，i(i1, ,n)是A的n个特征值，n|A|i a1n 矩阵：mn个数a排成m行n列的表格 2n mn矩阵CcijaijbijABABms矩阵Ccijn ainbnjaikbkjA与B的乘积k积，记为C1)(AT)TA,(AB)TBTAT,(kA)TkAT,(AB)TAT2)(A1)1AAB)1B1A1kA)11A1k

(AB)1A1B13)(A*)*|A|n2A(n3)，(AB)*B*(kA)*kn1A*(n2)但(AB)*A*B*4)(A1)T(AT)1,(A1)*(A*)1,(A*)T(ATAA*A*A|A||A*||A|n1(n2),(kA)*kn1A*,(A*)*|A|n2A(nAA*|A|A1A*)*1|AAnrA*)1,rAn0,r(A)nA可逆ABE|A|0rA2）r(Amn)min(m,n);3）A0r(A)14）r(AB)r(A)r(Ar(Bnr(ABmin(r(Ar(B)),ABr(Ar(B)若A1存在r(AB) r(AB)r(Amnnr(ABr(Amsnr(ABr(AmsnAx0AO OOB B1 AC OB AO O B B1 A BO O (三)1,2, ,s线性相关至少有一个向量可以用其余向 ,s，线性相关可以由1,2 ,s惟一线性表示可以由1,2 ,s线性表r(1,2 ,s)r(1,2 ,s,nn n线性无关|[1，2n|n个n维向量 n线性相 ,n]|n+1n维向量线性相关③若1 S线性无关，则添加分量后仍线性无关 ,s线性相关 ,s，线性相关可以由1,2 ,s惟一线性表示(3)可以由 ,s线性表r(1,2 ,s)r(1,2 ,s,n若1,2 ,n与1,2,,n是向量空间V的两组基，则c11 c1n c(,,,)(,,,) 2n(,,,1 1 n 1 nn其中C是可逆矩阵，称为由基1,2 ,2若向量在基1,2 ,n的坐标分别Xxx x)TYyy y)T1 1 x11x22xnny11y22 n，则向量坐标变换为XCY或Y其中C是从基1,2 ,n的过渡矩(abababT1 2 n若1,2,,s线性无关，则可构造1,2,,s使其两两正交，且i仅是1,2, 的线性组合(i1,2,,n)，再把i单位化，记i，则, ,是规范正交向量组.其 1 i11 (2,1) (,) (3,1)(3,2) (,) (,)1 (s,1)(s,2) (s,s1) (,) (,) (, )1 s sjjDnD2D1DA0xD1xD2 annxn22n21 22axax ax a1nxn1 nA可逆Ax0只有零解bAxb总有唯r(AmnnAx0只有零解r(Ar(AbmAxb有解 xs为Axb的解，则k1x1k2x2 ksxsk1k2 ks1时仍为Axb的解但当k1k2 ks时，则为Ax0的解.x1x2Axb的解；2xxx Ax0的解非齐次线性方程组Axb无解r(A)1r(A)b不能由A的列向量1,2, ,n线性表示.Ax0恒有解(必有零解).当有非零解时，由于Ax0的全体解向量构成一个向量空间，称为该方程组的解nr(A，解空间的一组基称为齐次 ,tAx0 ,tAx0 ,tAx0的任一解都可以由1,2 ,t线性表出k11k22kttAx0k1k2kt是任意,kab2mf(1|A|且对应特征向量相同（ ,n为A的n个特征值则iaii,i|Ai i i从而|A|0A没有特征值3设1,2 ,s为A的s个特征值，对应特征向量 ,sk11k22 kssAnkAnkAn kAnknkn k 11 22 ss1若 B，(1) BT, B1,A*B (2)|A||B|,Aiibii,r(A)i i(3)|EA||EB|对AnA可对角化对每个kiA可对角化，则由P1AP,APP1，从而AnPn若 D，则A OOC 若 B，则f f(B),f f(B)，其中f(A)为计算)＝秩(A)阵P使得BP1AP成立则称矩阵A与B相似记为 B如果 B则 ( (4)EAEB,从而AB(6秩(A秩(B),EAEBA、B nf(x1,x2 ,xn)aijxiyj，其中aijaji(i,j1, ,n)i1jn元二次型，简称二次型.x1 a11 a1nx ax2A 2nfx aan nn向量形式fxTAx其中A称为二次型矩阵，因为aijaji(i,j1 )A的秩称为二次型二次型f(x,x ,x)xTAx经过合同变换xCy化1 rfxTAxyTCTACydy2iif(rn)的标准形.在一般的数域内，二次型的标准形不是个数由rA的秩)唯一确定.3任一实二次型f都可经过合同变换化为规范形fz2z2 z2z2 1A正定kA(k0ATA1A*|A|0,Aaii0，且|Aii|A，B正定A+BAB，BAA正定f(xxTAx0,xPAPT 存在正交矩阵Q，使QTAQQ1AQ n其中0,i1, ,n.正定kA(k0),AT,A1,A*正定i|A|0,Aaii0，且|Aii|随机间，相等：A=B，即AB，且BA A+B，A差：A-B，A发生但B不发生 B（或AB，A与B同时发生 ： B=AB,且AB,记AB或B交换律：AB A，AB 结合律：（AB）CA（B（AB）CA（B分配律：（AB）C（AC）（B 律：ABAB BAn件，即 Aj，i 1（ ,若AiAj(ij),则 Ai)P(i iP(A)1P(P(AB)P(A) B)P(AP(BP(ABBAP(AB)P(AP(BP(B)P(AP(ABC)P(A)P(B)P(C)P(AB)P(AC) Ai)(P(Ai 古典型概率: 几何型概率:样本空间为欧氏空间中的一个区域，PAA的度量（长度、面积、体积本1概率的基本（A）， P（A）P(A|Bi)P(Bi),BiBj,i Bii i ：P(B|A)P(A|Bj)P(Bj),j1, , iP(A1A2)P(A1)P(A2|A1)P(A2)P(A1|A2 An)P(A1)P(A2|A1)P(A3|A1A2 P(An|A1 2的独立AB相互独立P(AB)P(AB)P(A)P(B); P(BC)P(B)P(C);P(AC)P(A)P(C);P(AB) P(BC)P(AC) P(ABC)3独立重复试验:n次，若每次实验中事P(Xk)Ckpk(1p)nknP(A)1P( B)P(A)P(B) C)P(A)P(B)P(C)P(AB)P(AC)P(AB)P(A)P(AB)P(A)P(AB),P(A)P(AB) B)P(A)P(AB)P(AB)P(AB)满足概率的所有性质，例如：.P(A1|1P(A1| A2|B)P(A1|B)P(A2|B)P(A1A2|B)P(A1A2|B)P(A1|B)P(A2|A1B) i i i iAB互逆AB互斥，但反之不成立，AB互斥（或互逆）且均非零概率A与B不独立. 随随量及概率分布:取值带有随机性的变量，严格地说性质：(1)0F(x) (2)F(x)单调不(3)右连续F(x0) (4)F()0,F()P(Xxi)pi,i1, , pi0,piif(xf(x)f(x)dxxx为f(x)的连续点，则f(x)F'(x)分布函数F(x)f(1)0-1P(Xkpk(1p)1kkP(Xk)Ckpk(1p)nk,k ,nkP(Xk) e,0,k0,1,k1,axU（a，b：(x(x) ,0,xe2E():f(x几何分布G(p):P(Xk)(1p)k1p,0p1,k1, H(N,M,n):P(Xk)CkCnk,k MNNP(Xx1pi,Yg(XP(Yyj)P(Xxig(x)X~fX(x),Yg(xFy(y)P(Yy)P(g(X)y) fx(x)dxg(x)fY(y)F'Y(X~N(0,1)(0)1,(0)1 (a)P(Xa) X~N(,2X~N(0,1)且PXaa X~E()P(Xst|Xs)P(XX~G(p)P(Xmk|Xm)P(X离散型随量的分布函数为阶梯间断函数；连续型随随联合分布为F(x,y)P(Xx,Yy) P{Xxi,Yyj}pij;i,j1,2, pipij,i1,jpjpij,j1,iP{Xx|Yy} jP{Yy|Xx} 1f(xyf(x,y)f(x,y)dxdy 2F(xyf(u fX(x)f(x, fY(y)f(x, 4条件概率密度 (x|y)f(x, (y|x)f(x, f( f 1,(x,y) U(D),f(x,y) f(x,y) 11 (x (x)(y (y)2 2 22(12 1 布XY的相互独立F(xy)FX(x)FYypijpipj(离散型）f(xyfX(xfYy)(连续型XYXY0XY不相关，否则称X和Y相关P(Xxi,YyipijZg(X,Y) P(Xxi,Yyjg(x,y) f(xyZg(X,Y)Fz(z)Pg(X,Y)z f(x,y)dxdy，fz(z)F'zg(x,y) fX(x)f(x, fY(y)f(x,P(X,Y)Df(x,D ①X~N(,2),Y~N(,2 ②XY相互独立0XY不相关③CXCY~N(CC,C22C222CC 1 2 1 2 121N(1(y),2(12 2⑤Y关于X=xN(2(x),2(12 1XYN(,2),N(,2 则(X,Y)~N(,2,2 CXCY~N(CC,C22C22 1 2 1 2XYf(x)和g(x为连续函数，则f(X)与g(Y)也相互独立.（PXxipiE(XxipiiX~f(x),E(X)xf(1)E(C)C,E[E(X)]E(X(2)(2)E(C1XC2Y)C1E(X)C2E(YXYE(XYE(X)E(YE(XY)2E(X2)E(Y22D(X)EXE(X)2E(X2E(X标准差：DX)DXxEX)2 i 5DXxEX)fD(C)0,D[E(X)]0,D[D(X)]XYD(XYD(XD(YD(CXC)C2D(X D(XY)D(X)D(Y)2Cov(X,Y)D(X)D(Y)2D(X)D(X)E(XC)2,CE(XD(X)0PXC随(1)对于函数YXP{XxipiE(Yg(xipiXiX~f(xE(Yg(xf(2)Zg(X,Y);(X,Y)~P{Xxi,Yyj}pijE(Z)g(xi,yj) (X,Y)~f(x,y);E(Z)g(x,y)f(x,协方差Cov(X,Y)E(XE(X)(YE(Y相关系数 Cov(X,Y ,k阶原点矩E(Xk) D(X)D(Yk阶中心矩E[XE(X)]kCov(X,Y)Cov(Y,XCov(aX,bY)abCov(Y,XCov(X1X2,Y)Cov(X1,Y)Cov(X2,Y(X,Y)(X,Y1P(YaXb1,其中a(X,Y1P(YaXb1,其中a（1）D(X)E(X2)E2(XCov(X,Y)E(XY)E(X)E(YX,Y)1, (X,Y1P(YaXb1,其中a(X,Y1P(YaXb1,其中a(X,Y)Cov(X,Y)E(X,Y)E(X)E(YD(XY)D(X)D(YD(XY)D(X)D(Y注：XY5个条件中任何一个成立的充分条切1 不等式：P|XE(X)|D(X)P|XE(X)|1D(X2切大数定律：设X1, ,Xn E(X)D(X)2(i1,2,),则对于任意正数 1 limP Xinni 设设X1, ,Xn 1正数， 1 limP nni 2大数定设X1,X2 ,Xn, 相互独立同分布，EXi,i1,2,则对于任意正数，有limP1 X inni 设n~B(n,p),（即X1, ,Xn,相互独立且同服从0-1分nnXi）inp limP x e2nnp(1 设X1, ,Xn E(Xi),D(Xi)2(0)i1, t则limP x e2n XXn个相互独立且与总体同分布的随量X1,X2,Xn,称为容量为n的X1X2Xn,是来自总体X的一个样本，g(X1, Xn）g含任何知参数，则称g(X1, ,Xn)为统计1XnXi S2XXn i1kAXkk1 n i1 (XX)k,k1, i2分布：2X2X2 X2~2(n)，其中X, ,X t分布：T ~t(n)其中X~N(0,1),Y~2(n),且Y/F分布：FXn1~F(nnX~2(n),Y~2(nY/ 1 2P(XxxX的nnXN(,2 1X1 X XS2 XX)2 i n1 i i XX~N ~N n/n 1 (XX)2~2(n i1(X)2~2 iX~t(nSn（1）2~2(nE(2(nnD(2(n（2）对于T~t(n)，有E(T)0,D(T) n(n2)n（3）F~F(mn)1~F(n,m), (m,n) (n,1a/E(X)E(X),E(S2)D(X),D(X)D(Xn概法矩估计E(XE(XE(S2D(XXS2EX),DX)的无偏估计量由大数定律X，S2也分别是E(X),D(X)的一致估量1 是1念(X,X P{}UXnN(XtS,XtS P{Tt}TX t(n W'1(X i2 (X)2(X (i1 ,i 2 2 P{W'2(n)}2P{W' (n)}1 2n1SW 2nn1S2n1S2 2n1,2n1