数学分析_第三版_复旦大学欧阳光中

214 2.E K 1. e (1) u = f(x,y)x = rcos,y = rsin u r , 2u r2 (2) u = f(x,y)x = a,y = b u , 2u 2 , 2u , u , 2u 2 (3) u = f(x2+ y2+ z2) u x , 2u x2 , 2 xy , u y , u z (4) u = f x, x y ! u x , 2u x2 , u y . ) (1) u r = fxcos + fysin 2u r2 = fx2cos2 + fxysin2 + fy2sin2 (2) u = afx, 2u 2 = a2fx2, 2u = abfxy, u = bfy, 2u 2 = b2fy2 (3) u x = 2xf 0(x2+y2+z2), 2u x2 = 2f 0(x2+y2+z2)+4x2f00(x2+y2+z2), 2 xy = 4xyf 00(x2+y2+z2), u y = 2yf 0(x2 + y2+ z2), u z = 2zf 0(x2 + y2+ z2) (4) u x = f1+ 1 y f2, 2u x2 = f11+ 2 y f12+ 1 y2 f22, u y = x y2 f2 2. = (x,y,z),x = u + v,y = u v,z = uvu,v. )u= x+ y+ vz,v= x y+ uz 3. e() (1) u = f(x + y) (2) u = f(x + y,x y) (3) u = f(ax2+ by2+ cz2) ) (1) du = f 0(x + y)(dx + dy) (2) du = (f1+ f2)dx + (f1 f2)dy (3) du = 2f 0(ax2 + by2+ cz2)(axdx + by dy + cz dz) 4. ?ye (1) z = (x2+ y2)Ky z x x z y = 0 (2) u = y(x2 y2)Ky u x + x u y = xu y (3) u = x(x + y) + y(x + y)K 2u x2 2 2u xy + 2u y2 = 0. y (1) z x = 2x 0(x2 + y2), z y = 2y 0(x2 + y2) Ky z x x z y = 0. (2) u x = 2xy 0(x2 y2), u y = (x2 y2) 2y2 0(x2 y2) Ky u x + x u y = x(x2 y2) = xu y . 215 (3) u x = (x + y) + x 0(x + y) + y0(x + y), u y = x 0(x + y) + (x + y) + y0(x + y) K 2u x2 = 2 0(x+y)+x00(x+y)+y00(x+y), 2u xy = 0(x+y)+0(x+y)+x00(x+y)+y00(x+y) 2u y2 = 2 0(x + y) + x00(x + y) + y00(x + y) u 2u x2 2 2u xy + 2u y2 = 0. 5. u = 2u x2 + 2u y2 + 2u z2 ,u = f(x + y + z,x2+ y2+ z2). ) u x = f1+ 2xf2K 2u x2 = f11+ 4xf12+ 4x2f22+ 2f2 5 2u y2 = f11+ 4yf12+ 4y2f22+ 2f2, 2u z2 = f11+ 4zf12+ 4z2f22+ 2f2 uu = 2u x2 + 2u y2 + 2u z2 = 3f11+ 4(x + y + z)f12+ 4(x2+ y2+ z2)f22+ 6f2. 6. eu = f(r),r = px2 + y2f(r)gy 2u x2 + 2u y2 = d2u dr2 + 1 r du dr y u x = x px2 + y2 f 0(r)K 2u x2 = y2 (x2+ y2) 3 2 f 0(r) + x2 x2+ y2 f 00(r) 5 2u y2 = x2 (x2+ y2) 3 2 f 0(r) + y2 x2+ y2 f 00(r) u 2u x2 + 2u y2 = d2u dr2 + 1 r du dr 7. eu,vx,yx = rcos,y = rsind u x = v y , u y = v x y u r = 1 r v , v r = 1 r u . yu,vx,yx = rcos,y = rsin K u r = cos u x + sin u y , v r = cos v x + sin v y , u = rsin u x + rcos u y , v = rsin v x + rcos v y q u x = v y , u y = v x K u r = 1 r v , v r = 1 r u . 8. f(tx,ty) = tnf(x,y)Kk x f x + y f y = nf k5ngg.|(Jz = px2 + y2x z x + y z y . yf(tx,ty) = tnf(x,y)Kt f1(tx,ty)x + f2(tx,ty)y = ntn1f(x,y) -t = 1Kf1(x,y)x + f2(x,y)y = nf(x,y)=x f x + y f y = nf z(x,y) = px2 + y2Kz(tx,ty) = tpx2+ y2(t 0) ux z x + y z y = z = px2 + y2. 9. ?y z = x y x ! + y x ! 216 vx2 2z x2 + 2xy 2z xy + y2 2z y2 = 0 y z x = y x ! y x 0 y x ! y x2 0 y x ! , z y = 0 y x ! + 1 x 0 y x ! K 2z x2 = y2 x3 00 + 2y x3 0 + y2 x4 00, 2z xy = y x2 00 1 x2 0 y x3 00, 2z y2 = 1 x 00 + 1 x2 00 ux2 2z x2 + 2xy 2z xy + y2 2z y2 = 0 10. u = (x + at) + (x at),?gy 2u t2 = a2 2u x2 . yu = (x + at) + (x at),?g K u t = a( 0 0), u x = 0 + 0u 2u t2 = a2( 00 + 00), 2u x2 = 00 + 00 l 2u t2 = a2 2u x2. 217 3.d(|)( 1. de(z = f(x,y) (1) x + y + z = ez (2) xyz = x + y + z ) (1) ux1 + zx= zxezKzx= 1 ez 1 uzx2= ez (1 ez)3 zy= 1 ez 1 ,zy2= ez (1 ez)3 ,zxy= zyx= ez (1 ez)3 (2) uxyz + xyzx= 1 + zx()Kzx= yz 1 1 xy ()ux2yzx+ xyzx2= zx2Kzx2= 2yzx 1 xy = 2y(yz 1) (xy 1)2 zy= xz 1 1 xy ,zy2= 2x(xz 1) (xy 1)2 ,zxy= zyx= 2z (xy 1)2 2. de( (1) f(x + y,y + z,z + x) = 0 z x , z y (2) z = f(xz,z y)dz (3) F(x y,y z,z x) = 0 z x , z y (4) F(x,x + y,x + y + z) = 0 z x , z y , 2z x2 . ) (1) ux z = z(x,y)f1+ f2zx+ f3(zx+ 1) = 0Kzx= f1+ f3 f2+ f3 zy= f1+ f2 f2+ f3 (2) dz = (xdz + z dx)f1+ (dz dy)f2Kdz = zf1dx f2dy 1 xf1 f2 (3) ux z = z(x,y)F1 F2zx+ F3(zx 1) = 0Kzx= F1 F3 F2 F3 zy= F2 F1 F2 F3 (4) ux z = z(x,y)F1+ F2+ F3(1 + zx) = 0 ()Kzx= F1+ F2+ F3 F3 3()2ux F11+ F12+ F13(1 + zx) + F21+ F22+ F23(1 + zx) + zx2F3+ (1 + zx)F13+ F23+ F33(1 + zx) = 0 Kzx2= 1 F3 3 F2 3(F11+ 2F12+ F22) 2F3(F1+ F2)(F13+ F23) + F33(F1+ F2) 2 zy= F2+ F3 F3 3. dz = x + y (z)(z = z(x,y)1 y 0(z) 6= 0y z y = (z) z x y z = z(x,y)dz = dx + (z)dy + y 0(z)dz q1 y 0(z) 6= 0Kdz = dx + (z)dy 1 y0(z) u z y = (z) 1 y0(z) , z x = 1 1 y0(z)l z y = (z) z x 218 4. ydax + by + cz = (x2+ y2+ z2)z = z(x,y)v(cy bz) z x + (az cx) z y = bx ay(u)ua,b,c. y z = z(x,y)(u)u Kadx + bdy + cdz = 2(xdx + y dy + z dz) 0 u z x = 2x 0 a c 2z0 , z y = 2y 0 b c 2z0 l (cy bz) z x + (az cx) z y = bx ay 5. ?yd(cx az,cy bz) = 0z = z(x,y)va z x + b z y = c. yOux,y z = z(x,y) c1 a1zx b2zx= 0,a1zy+ c2 b2zy= 0 u z x = c1 a1+ b2 , z y = c2 a1+ b2 l a z x + b z y = c. 6. ydF(x + zy1,y + zx1) = 0(z = z(x,y)vx z x + y z y = z xy. yOux,y z = z(x,y) F1 1 + zx y ! + F2 zx x z x2 ! = 0,F1 zy y z y2 ! + F2 1 + zy x ! = 0 u z x = yzF2 x2yF1 x(xF1+ yF2) , z y = xzF1 xy2F2 y(xF1+ yF2) l x z x + y z y = z xy. 7. e|( (1) ? x + y + z = 0, x y z = 1, dy dx , dz dx , d2y dx2 (2) x + y = u + v, x y = sinu sinv , du, dv (3) ? xu + yv = 0, yu + xv = 1, u x , u y , v x , v y , 2u xy (4) x = coscos, y = cossin, z = sin, z x , z y (5) ? u = f(u,x,v + y), v = g(u x,u2 y), u x , v x . ) (1) x 1 + dy dx + dz dx = 0 yz + xz dy dx + xy dz dx = 0 () ) dy dx = y(z x) x(y z) , dz dx = z(x y) x(y z) ()2x d2y dx2 + d2z dx2 = 0 z dy dx + y dz dx + z dy dx + x dy dx dz dx + xz d2y dx2 + y dz dx + x dy dx dz dx + xy d2z dx2 = 0 d2y dx2 = 2z dy dx + 2y dz dx + 2x dy dx dz dx x(y z) dy dx , dz dx d2y dx2 = yz(x y)2+ (x z)2+ (y z)2 x2(z y)3 219 (2) ?U ? u + v = x + y y sinu = xsinv ? du + dv = dx + dy sinudy + y cosudu = sinv dx + xcosv dv Kdu = 1 xcosv + y cosu(sinv + xcosv)dx (sinu xcosv)dy dv = 1 xcosv + y cosu(sinv y cosu)dx + (sinu + y cosu)dy (3) ? xdu + y dv = udx v dy y du + xdv = v dx udy udu = 1 x2 y2(yv xu)dx + (yu xv)dy, dv = 1 x2 y2(yu xv)dx + (yv xu)dy K u x = yv xu x2 y2 , u y = yu xv x2 y2 , v x = yu xv x2 y2 , u x = yv xu x2 y2 u 2u xy = (yux v xvx)(x2 y2) 2x(yu xv) (x2 y2)2 u x , v x 2u xy = 2(x2v + y2v 2xyu) (x2 y2)2 (4) dx,yx 1 = sin cos x cos sin x 0 = sin sin x + cos cos x K x = cos sin , x = sin cos u z x = z x = cotcos = x z n z y = y z (5) x u x = f1 u x + f2+ f3 v x v x = g1 u x 1 ! + 2vyg2 v x K u x = f2(1 2vyg2) g1f3 (f1 1)(2vyg2 1) g1f3 , v x = g1(f1+ f2 1) (f1 1)(2vyg2 1) g1f3 . 8. x = u + v,y = u2+ v2,z = u3+ v3zx,y z x , z y . )x2 y = 2uvKz = (u + v)(u2 uv + v2) = x 2 (3y x2) u z x = 3 2 (y x2) , z y = 3 2 x. 9. x = rcos,y = rsinC dx dt = y + kx(x2+ y2) dy dt = x + ky(x2+ y2) 4I. )dx,ytl4ICr,tx = rcos,y = rsin t dx dt = cos dr dt rsin d dt dy dt = sin dr dt + rcos d dt x,y, dx dt , dy dt ?| cos dr dt rsin d dt = rsin + krcos r2 sin dr dt + rcos d dt = r cos + kr sin r2 u dr dt = kr3, d dt = 1. 10. x = eucos,y = eusin C 2z x2 + 2z y2 = 0. )x = eucos,y = eusinKu = ln(x2+ y2), = arctan y x 220 K u x = x x2+ y2 , u y = y x2+ y2 ; x = y x2+ y2 , y = x x2+ y2 u u x = y , u y = x q z x = z u u x + z x , z y = z u u y + z y K 2z x2 = 2z u2 u x !2 + 2 2z u u x x + 2z 2 x !2 + z u 2u x2 + z 2 x2 2z y2 = 2z u2 u y !2 + 2 2z u u y y + 2z 2 y !2 + z u 2u y2 + z 2 y2 q 2u x2 = x y ! = y x ! = y u y ! = 2u y2 2 x2 = 2 y2 K 2 x2 + 2 y2 = 2u x2 + 2u y2 = 0 q u x !2 + u y !2 = x !2 + y !2 , u x x = u y y K 2z x2 + 2z y2 = e2u 2z u2 + 2z 2 ! = 0 = 2z u2 + 2z 2 = 0. 11. x = rcos,y = rsinKf(x,y) = (r,)ur, 5L 2f x2 + 2f y2 . )f(x,y) = (r,)ur, f x x r + f y y r = r = f x cos + f y sin = r f x x + f y y = = f x rsin + f y rcos = K 2 r2 = 2f x2 cos2 + 2f xy sin2 + 2f y2 sin2 2 2 = r2 2f x2 sin2 2f xy sin2 + 2f y2 cos2 ! f x rcos f y rsin u 2 r2 + 1 r2 2 2 = 2f x2 + 2f y2 1 r r l 2f x2 + 2f y2 = 2 r2 + 1 r2 2 2 + 1 r r 12. x = e,y = eCax2 2z x2 + 2bxy 2z xy + cy2 2z y2 = 0(a,b,c). )x = e,y = eK = lnx, = lnyu d dx = 1 x , d dy = 1 y K z x = 1 x z , z y = 1 y z u 2z x2 = 1 x2 2z 2 z ! , 2z y2 = 1 y2 2z 2 z ! , 2z xy = 1 xy 2z ?zna 2z 2 z ! + 2b 2z + c 2z 2 z ! = 0. 13. = x, = x2+ y2Cy z x x z y = 0 . )dzx,y ,qx,yl zwLmC,ux,yE u z x = z + 2x z , z y = 2y z y z x x z y = y z 221 y 6 0Kdy z x x z y = 0 z = 0. 14. = x, = y x, = z xC u x + u y + u z = 0. )dux,y,z ,qx,y,zl uwLmC,ux,y.zE u u x = u u u , u y = u , u z = u Kd u x + u y + u z = 0 z = 0 15. 5C = x + 1y, = x + 2yy3rA 2u x2 + 2B 2u xy + C 2u y2 = 0(A,B,C AC B2 0)?:3Iua. y3?:P0(x0,y0,z0) K3T: 1 2x0 (x x0) + 1 2y0 (y y0) + 1 2z0 (z z0) = 0 =y0z0(x x0) + x 0z0(y y0) + x 0y0(z z0) = 0 u3Iax0, ay 0, az 0a(x0+ y 0+ z 0) = a. 4. x2+ y2= a2,bz = xy?. )?:M0(x0,y0,z0) d3M0:n1= 2x0,2y0,0,n2= y0,x0,b u?vcos = n1 n2 |n1|n2| = 2bz0 |a|a2+ b2 . 226 6.F 1. u = x2 xy + y23(1,1)?l = (cos,sin).? (1) 3=k (2) 3=k? (3) 3=0 (4) uF. ) ux= 2x y,uy= x + 2yKux(1,1) = 1,uy(1,1) = 1 q u l = ux(1,1)cos + uy(1,1)sinK u l = cos + sin = 2sin + 4 ! u (1) ? = 4 3l = 2 2 , 2 2 ! k2 (2) ? = 3 4 3l = 2 2 , 2 2 ! k?2 (3) ? = 4 , 3 4 3l = 2 2 , 2 2 ! l = 2 2 , 2 2 ! 0 (4) gradu = ux(1,1)i + uy(1,1)j = i + j. 2. u = xyz3:M(1,1,1)l = (2,1,3)9F. )ux= yz,uy= xz,uz= xyK3(1,1,1):ux= uy= uz= 1 qlucos = 2 14,sin = 1 14,cos = 3 14 K u l = ux(1,1,1)cos + uy(1,1,1)cos + uz(1,1,1)cos = 2 7 14 gradu = i + j + k. 3. u = x2+ 2y2+ 3z2+ xy + 3x 2y 6z3O(0,0,0)9A(1,1,1)F9?. )ux= 2x + y + 3,uy= 4y + x 2,uz= 6z 6 K3O(0,0,0):ux= 3,uy= 2,uz= 6ugradu = 3i 2j + 6k,|gradu| = 7 3A(1,1,1):ux= 6,uy= 3,uz= 0ugradu = 6i + 3j,|gradu| = 35. 4. y (1) grad(u + v) = gradu + gradv, (2) grad(uv) = ugradv + vgradu (3) gradF(u) = F 0(u)gradu y?5y.-u = u(x,y),v = v(x,y) (1) (u + v) x = u x + v x , (u + v) y = u y + v y Kgrad(u + v) = (u + v) x , (u + v) y ! = u x , v y ! + u x , v y ! = gradu + gradv. (2) (uv) x = v u x + u v x , (uv) y = v u y + u v y Kgrad(uv) = (uv) x , (uv) y ! = v u x , u y ! + u v x , v y ! = ugradv + vgradu. (3) gradF(u) = F x , F y ! = F 0(u)u x ,F 0(u)u y ! = F 0(u) u x , u y ! = F 0(u)gradu ?y. 5. ygrad 1 r = r r3 r = px2 + y2+ z2,r = xi + yj + zk. y r x = x r , r y = y r , r z = z r Kgrad 1 r = d dr 1 r ! gradr = 1 r2 r x i + r y j + r z k ! = 1 r2 1 r(xi + yj + zk) = r r3 . 232 2.4 1. e3e4 (1) f = x + y,ex2+ y2= 1 (2) f = x 2y + 2z,ex2+ y2+ z2= 1 (3) f = xyz,e 1 x + 1 y + 1 z = 1 a (x 0,y 0,z 0,a 0) (4) f = 1 x + 1 y ,ex + y = 2 (5) f = xyz,ex2+ y2+ z2= 1,x + y + z = 0. ) (1) L = x + y + (x2+ y2 1) )| Lx= 1 + 2x = 0 Ly= 1 + 2y = 0 L= x2+ y2 1 = 0 x1= 2 2 y1= 2 2 1= 2 2 x2= 2 2 y2= 2 2 2= 2 2 qLx2= 2,Lxy= 0,Ly2= 2 Kd2L 2 2 , 2 2 ! = 2(dx2+ dy2) 0u3 1 3 , 2 3 , 2 3 ! ?4?3 n3 1 3 , 2 3 , 2 3 ! ?43. (3) L = xyz + 1 x + 1 y + 1 z 1 a ! )| Lx= yz x2 = 0 Ly= xz y2 = 0 Lz= xy z2 = 0 L= 1 x + 1 y + 1 z 1 a = 0 x = y = z = 3a, = 81a4 qLx2(3a,3a,3a) = Ly2(3a,3a,3a) = Lz2(3a,3a,3a) = 6a, Lxy(3a,3a,3a) = Lxz(3a,3a,3a) = Lyz(3a,3a,3a) = 3a Kd2L(3a,3a,3a) = 3a(dx + dy + dz)2+ dx2+ dy2+ dz2 0u3(3a,3a,3a)?4? 27a3. (4) L = 1 x + 1 y + (x + y 2) 233 )| Lx= 1 x2 + = 0 Ly= 1 y2 + = 0 L= x + y 2 = 0 x = y = = 1 qLx2(1,1) = Ly2(1,1) = 2,Lxy(1,1) = 0 Kd2L(1,1) = 2(dx2+ dy2) 0u3(1,1)?4?2. (5) L = xyz + u(x2+ y2+ z2 1) + v(x + y + z) )| Lx= yz + 2ux + v = 0 Ly= xz + 2uy + v = 0 Lz= xy + 2uz + v = 0 Lu= x2+ y2+ z2 1 = 0 Lv= x + y + z = 0 x1= 6 6 y1= 6 6 z1= 6 3 u1= 6 12 v1= 1 6 x2= 6 6 y2= 6 6 z2= 6 3 u2= 6 12 v2= 1 6 x3= 6 3 y3= 6 6 z3= 6 6 u3= 6 12 v3= 1 6 x4= 6 3 y4= 6 6 z4= 6 6 u4= 6 12 v4= 1 6 x5= 6 6 y5= 6 3 z5= 6 6 u5= 6 12 v5= 1 6 x6= 6 6 y6= 6 3 z6= 6 6 u6= 6 12 v6= 1 6 qd2L = 2u(dx2+ dy2+ dz2) + 2(z dxdy + y dxdz + xdy dz) K3:(x1,y1,z1)?d2L = 6 6 (dx2+ dy2+ dz2 4dxdy + 2dxdz + 2dy dz) dx2+ y2+ z2= 12xdx + 2y dy + 2z dz = 0K3:(x1,y1,z1)?kdx + dy = 2dz qdx+y+z = 0dx+ dy+ dz = 0Kdx = dy, dz = 0ud2L(x1,y1,z1) = 6 dx2 0 K3 6 6 , 6 6 , 6 3 ! ?4? 6 18 n3(x3,y3,z3),(x5,y5,z5)?4? 6 18 3(x2,y2,z2),(x4,y4,z4),(x6,y6,z6)?4 6 18 . 2. f = xmynzp3x + y + z = a,a 0,m 0,n 0,p 0,x 0,y 0,z 0e. )x 0,y 0,z 0Kf = xmynzplnf = mlnx + nlny + plnz,I lnf4:f4: -L = mlnx + nlny + plnz + (x + y + z a) K) Lx= m x + = 0 Ly= n y + = 0 Lz= p z + = 0 L= x + y + z a = 0 x = ma m + n + p y = na m + n + p z = pa m + n + p = m + n + p a K ma m + n + p , na m + n + p , pa m + n + p ! U4: qLx2= m x2 ,Lxy= Lyz= Lxz= 0,Ly2= n y2 ,Lz2= p z2 , d2L = m x2 dx2 n y2 dy2 p z2 dz2 ! . ? x + y = a z = 0 ? x + z = a y = 0 ? y + z = a x = 0 f 0f4: :. 3. x2+ 3y2= 12Sn/.1u . )duKn/Su x2 (23)2 + y2 4 = 1n/ .1u 234 .:7 :(0,2) n/,:I(x,y) (x,y 0)KSn/.2xpy + 2 n/n:IA(0,2),B(x,y),C(x,y)d 5A(0,2),B(x,y),C(x,y) : KS = x(y + 2):(x,y)3 x2+ 3y2= 12 qdKS = x(y + 2)3x2+ 3y2= 12(x,y 0) L = x(y + 2) + (x2+ 3y2 12) K) Lx= y + 2 + 2x = 0 Ly= x + 6y = 0 L= x2+ 3y2 12 = 0 x = 3 y = 1 = 1 2 u:IA(0,2),B(3,1),C(3,1)A(0,2),B(3,1),C(3,1) dKSK73K3(0,2),(3,1),(3,1)(0,2),(3,1),(3,1)? 9. 4. y2= 4x:x y + 4 = 0C. ):I(x,y)Kld = 1 2|x y + 4|y2= 4x x y + 4 = 0!mx y + 4 0 y2= 4x3m :(x,y)ld = 1 2(x