自动化学院学习分数分工科数学总复习下

:yy(x):x(t),y(t),z(tzz(:yf(z切线方程为'(txx0yy0zz0G(t (t{'(t0),(t0),(t0G000:F(x,y,z)GG(x,y,z)'(t0)(xx0)(t0)(yy0)(t0)(zz0)):F(x,y,z) x y z .Fx(x0,y0,z0 Fy(x0,y0,z0 Fz(x0,y0,z0S:zf(x,Gn (1)定义设zfx,y)在M0x0y0的某个邻域内有定义若当x,yUM0有fx,y)fx0y0称zfx,y)在M0x0y0有极大值fx0,y0点Mxy)为fxy)小00(2)小定义满足方程组fxx,y)0的点x,yf(x,y)yzfx,y) 设f(x,y)在点P0的邻域UP0)内具有二阶连续偏导数,且P0为fHP为不定矩阵fP不是极值 0H(P)f(P fxy (P f(P)fyy∵detHf(P0)detf fxyfABP CAC记＝fxyx0y0))2fxxx0,y0fyyx0,y(2)当0时x0y0) (3)当0时不能确定xy)0zzf(xy第一步解方程组fx(x,y) fy(x,y)x0y0第三步定出ACB2的符号，再判定是否有极值其中：Bfxyx0y0Afxxx0y0Cfyyx0y0求zfx,y)在附加条件x,y)0Fxfx(x,y)λx(x,y)从方程Fyfyx,y)λyx,y)0x,y,,两类曲线积分的关系和两类曲面积分的关系⑩重积分、曲线积分及曲面积分求一些几何量与物理量（形的面积、体积、曲面面积、弧长、质量、质心、转动惯量、引力、功及流量等）. f(x,y)d ①（D）关于x轴对称（x轴上方部分为 fxy)d当fxyfxD当fxyfx②（D）关于y轴对称（y轴右边部分为（D则f(x,y)d 2f(xy)d当f(xyf(x D4f(x,则fxD当fx，yfxf(x,当f(xyfxy f(x,,2fxy，z)dV,当fxyz)是z(或y;或x)的偶其中V1是V在z0（或y0;或x0)f(x,y)dxdyf(y,如：fx,yC(Df(x,y)dxdyf(y,axdxdy ydxdy2212x (axby)dxdy(ab)xdxdy(ab)x2dxdydzy2dxdydzVVV13Vxyz yyy2b f(x,y)day(xy1(xDyabf(x,y)ddd f(x,y)dx(yDcx(yyxx2(y)Dcr2(Dxx1(y) (r(1 d f(rcos,rsin)rdoff(x,y)dxdyf[x(u,v),y(u,v (x,(u,v选择坐标化为三次定积分(累次积分(1)利用直角坐标V"先后"其平面区域Sz(截面积也记Sz),在xoy面上的投影区域记为Dz,则f(z)dxdydz f(z dxdy f(z)Sddccz利用柱坐标利用柱坐标dvrdrddzxrcos,yrsinzzxrsinzyrsinsinzrcosx2y2z2rrx2y2zrxoyM'(x,f(x,y,z)dVVz(x,ydxdyf(x,y,z(x,yDdcdzf(x,y, r( z(r,dr()rdrz(r,)f(rcos,rsin, ( r(, dsind (4)利用变量代换(4)利用变量代换*dvJxx(u,v,w),yy(u,v,w),zz(u,v,w)(x,y,z)(u,v,w)' J f(x,y,z)dvf(x(u,v,w),y(u,v,w),z(u,v,w))J设fxy)在平面曲线弧L,Lf(x,y)dsf((t),(t))'2(t)'2(t)dt,L:x(ty(taf(x,y(x))1y'(2L:yy(f(x(y),y)1x'2(L:xx(f(rcos,rsin)rr'd L:rr(设f设fxyz)在空间,1.:x(t),y(t),z(t).(tf(x,y,f[(t),(t),(t)]2(t)2(t)22.F(x,y,z)G(x,y,z)():zz(x,y),(x,y)（Df(x,y,z)dS f[x,y,z(x,y)]1 zD11求曲线弧 2求平面区域D的面积D3求曲顶曲底立体Ω的体积V[f2(x,y)f1(x,y(vf(xy)d(顶f(x,yDDV4.SdS1z2xz.W nJGF F(Mi)0iP(x,y,z)dxQ(x,y,z)dyR(x,y,6.6.vndSlimviniGnG0i[P(x,y,z)cosQ(x,y,z)cosR(x,y,z)cosP(x,y,z)dydzQ(x,y,z)dzdxR(x,y, z n{cos,cos,cos}为点x,yz)处指定侧的法向量G∑OyxdydzdzdxcosdSdxdycos平面曲线：1.化为定积分2.用空间曲线：1.化为定积分2.用 G(x,y,z)=0参数化的方法 GGLP(x,y)dxQ(x,y)dyLF(终点(1)Lx(ty(t)t终点{P[x(t),y(t)]x'(t)Q[x(t),y(t)]y'(tyy((2)y(t),t起点终点x(tz(t或xx或rr(GJGFdsLP(x,y,z)dxQ(x,y,z)dyR(x,y,z)dzFLP(x,y)dxQ(x,y)dyLP(x,y)dxQ(x,LPdxQdyL(PcosQcost{cos,cost{cos,cos,cosL上点xy)DD )dxdy Pdx P L 与(I)L的正向;L③p、QDL)上有连续的 ①若PQ且QP简单 当L开添辅助线成闭路,② 则 当L开:选一特殊路径计算L③若PQ,但被积函数在点x,y) 0PQRPQR注意 ②用 (2)记(2)记FPxyzQxyzRxyzGGGG(3) GotFdSFGGGPQD内具有一阶连续偏导数LPdxQdyD内与路径无关DL有LPdxQdy 在D内 DduPdxQd向量场Axy（Pxy)，Qxy)）在DG向量场Axy（Pxy)，Qxy)）在D G注意:注意:i)以上D(ii)u(x,y)(x,y(x0,y0P(x,y)dxQ(x,y)d P(x,y)dx Q(x,y)dxx00yy0或u(x,y) Q(x,y)dy P(x,yx0x设G是 单连通域,向量函AMPM),QMRM在G内具有连续一阶偏导数,则下列五个条件相互等价:GvCPdxQdyRdzGC,CPdxQdyRdG内存在某一函数u, G(4)在G内是处处 rotAAduPdxQdyRd(PQ,QR,RP 用R(x,y,z)dxdyR[x,y,z(x, P(x,y,z)dydzP[x(y,z),y,z](dydz DcoscosQ(x,y,z)dzdxQ[x,y(x,z), cos:yy(z,x),(z,x):xx(y,z),(y,z)coscos:zz(x,y),(x,y)PdydzQdzdx(zxzy[P(x,y,z(x,y))(∓zx)Q(x,y,z(x,y))(∓zyR(x,y,z(x,对应下侧(zxzy:zz(z,x),(x,y)PdydzQdzdx(PcosQcosRcos)dsF(PcosQcosRcos(QQR 的正向③P、Q、R在闭区域()上有连续的一阶偏导数; (2)记FPx,yzQx,yzRx,yzGGG FndSG1010 graduuGuGuGu{,, y zxyGdivA GPQ GA{P,Q,GrotA(yz)i(zx)j(xyRQ PR QPGiAPG 无旋场向量场沿任意简单闭曲线C的环量都等于0GGdivAA0A为无源场GGGGAA为调和场AGG理G是单连通域,数,则下列五个条件相互等价G线C的环量为 PdxQdyRdzGAM)在G内是保守场(对GCPdxQdyRdz与路径无关GduPdxQdyRd(4)AM)G,Gu(M),使得AM)G(5)在G内是处处 rotAA(PQ,QR,RP Icosx2y2dD1Icos(xy)d,Icos(x2y2)2d 23DD其中Dxyx2y21}，则（（A(A)I3I2 （B）I1I2(C)I2I1 (D)I3I1I20(x2y2)2x2y2x2y2120cosx2y2cos(x2y2)cos(x2y2解：解：f(u,v)dudv:af(xyxyDaf(x,y)dxdy(xy xydya dxdy aa 11 1218(D)xy8(C)xy1(A)xyf(x,y)xyf(u,D 设连续函数f(x,y)满3f(xy)二次积分 f(x,y)dy等于 2 arcsinf(x, f(x, arcsin0f(x,y)dx f(x,022提示：确定y的取值范围时应注意：当x2y=sinx=sin(x0x,于是xarcsiny2而xarcsin例4.计算例4.计算 edx e y 1yx1y 2解D14y22x 1yD:1y1,yx2D D:1x1,x2y2y原式dxe x1x2 x(ee)dxe 1x 12 例1f(x0dx dx1f(x01I dx f(x 1f(xf(x 1f(y00100 f(x)f(yD:0x1,0yex1 x(,)(仅当x0时等号成立Ief(x)f(y)dxdy(1f(x)f(yDDdxdyf(x)dxdyf(y)dxdy 即， dx1f(x 1f(x00 6x2y2z)dv是由曲线y2xz轴旋转一周而成的曲面与平面z4所围的立体解法1xyz)x2y22z,0zD{(x,y)xy224(x2y2z)dvdxdy (x2y2x 2{(x2y2)(4x2242(x2y2)2/2Dd(4r28 )r83008(xyz)dv (r2242r2 )r2 r4D28 (4r8 )rdr2 83解法2“先二后一4 (xy xy2 (rz)rdr 2 03Dz{(x,y)xy22A(t)xf(t dnxdtdn1x(3') dnxdta1(tdn1x "an(t)xdxA(t)10"001"0 0"0x1(t) 00 G ""xG(t)x2(t)f# 1#a(t)a(t)a(t)"a xn(t)0f(t) 1 x(t)(x(t),x(t"x(t x(tx1(t)就是方程（3）的解若求得了（3）的解x(t)x(tx(t)x'(t",x(n1t))T(3')dnx dn1x dt a1(t)dtn1"an(t) f(t a(t "a(t)xdt A(t)xf(t A(t) ①G(tG(t"G(t)是(2)的m个解则 若GG②若G(t),G(t"G(t)是（2）的n个线性无关的特解这里c1,c2"是任常数 x(tc1x1(tc2x2(tcnxn(t)=X(t)C是（2）的通解 GGGGGX(tx(t)x2txn(t1dX(t)A(t)X(t),atb;且t[ab]使detX(t00dxA(t)GGA(t)GG(t ①方程组(2)的nG(t),G(t"G(t 线性相关t0[ab使得W(t0)②若X(t)是(2)的基解矩阵B是n阶非奇异常数矩阵X(t)B也是在区间也是(2)的基解矩阵.③如果X(tX*(t)是(2)的两个基解矩阵非奇异n阶常数矩阵B,使得有X*(tX(t④齐次线性微分方程(4)的n个解x1t),x2t"xn(t线性相关t0[a,b]Wronsky行列式W(t0 x2x1x2##""#xn(t)xn(t)# x(n1)(t)2 A(t)xf(t 4.常系数齐次dx(t))或det(EA若属于i的线性无关特征向量个数=ni,x(x(t)X(t)X(t)f(GtC11,2",nr1r2",rnG X(t[er1,er2",rn tGtt(ii)1,2,",kn1n2nkr",rr",r",r",r G 1n X(t) "GG ,GGGG," tx(t) GG(kt t2 1! 2!(k (k "rG(nn(k))ei ①求代数方程AE)r=0的n个线性无关nGiir，r"， （1） G 0②对每个r，（k1,",n0i应 其中 j1,2,...,niG(kG(k(AEj(kiG)"， ..Ax(t)GGG当X(0)