数学分析讲义第四版刘玉琏傅沛仁著高等教育出版社课后答案第十三单元.pdf

1?n? ?SK13.1(?) 56e?1317? 1.U?,? ZZ R xydxdy, ?R0 6 x 6 1;0 6 y 6 1. )?f(x,y) = xy34?/?RY,l?f(x,y) = xy3R?.?AA? ?:Pik?A?. ?|?x = i n,y = k n(n N,i,k = 1,2,.,n 1)?Rn 2?/?Rik,? 4Rik= 1 n2,?Pik(i,k) = ? i n, k n ? (i,k = 1,2,.,n).? n X i,k=1 f(i,k)4Rik= n X i,k=1 i n k n 1 n2 = 1 n4 n X i=1 i n X k=1 k = 1 n4 ?n(n + 1) 2 ?2 = 1 4 ? 1 + 1 n ?2 . u,RRRxydxdy = lim n Pn i,k=1f(i,k)4Rik = lim n 1 4 ? 1 + 1 n ?2 . 3.?R0 6 x 6 1;0 6 y 6 1.(x,y) R, f(x,y) = ? 1,x = y 0,x 6= y. y:f(x,y)3R?,?RRRf(x,y)dxdy = 0. yyy?f(x,y)3R?Y:?3?y = x,0 6 x 6 1?.?n3,f(x,y)3R? .d,?AA?:Pk?A?. ?T?/?Rn?:R1,R2,.,Rn.?O4R1,4R2,.,4Rn.3 Rk?:Pk(k,k).k6= k(o?U?),l?f(k,k) = 0,k = 1,2,.,n.? ? n X k=1 f(k,k)4Rk= 0 u,RRRf(x,y)dxdy =lim kTk0 n P k=1 f(k,k)4Rk= 0. 5.y:ef(x,y)3k.4?RY,?f(x,y) 0,K ZZ R f(x,y)dxdy 0. yyy fi ?f(x,y)3k.4?RY,?10.2n6,f(x,y)3R?m.?(x,y) R,kf(x,y) 0,f(x,y) m 0.l?,f(x,y)3R? n X k=1 f(k,k)4k n X k=1 m4k= m n X k=1 4k= m R, 1 ?RR?,?.u ZZ R f(x,y)dxdy =lim kTk0 n X k=1 f(k,k)4k R 0. 6.y:ef(x,y)?g(x,y)3k.4?RY,?g(x,y) 0,K(,) R, ZZ R f(x,y)g(x,y)dxdy = f(,) ZZ R g(x,y)dxdy. yyy fi ?f(x,y)3RY,?10.2n6,f(x,y)3R?m?M,=(x,y) R,k m 6 f(x,y) 6 M, ?(x,y) R,kg(x,y) 0, mg(x,y) 6 f(x,y)g(x,y) 6 Mg(x,y). l?,k m ZZ R g(x,y)dxdy 6 ZZ R f(x,y)g(x,y)dxdy 6 M ZZ R g(x,y)dxdy. ?n8,RRRg(x,y)dxdy 0. eRRRg(x,y)dxdy = 0,KRRRf(x,y)g(x,y)dxdy = 0.u,(,) R,k ZZ R f(x,y)g(x,y)dxdy = f(,) ZZ R g(x,y)dxdy; eRRRg(x,y)dxdy 0,k m 6 RR Rf(x,y)g(x,y)dxdy RR Rg(x,y)dxdy 6 M. ?10.2n7(Y?0?5),(,) R, RR Rf(x,y)g(x,y)dxdy RR Rg(x,y)dxdy = f(,), =RRRf(x,y)g(x,y)dxdy = f(,) RR Rg(x,y)dxdy. 7.y:ef(x,y)3?RY,?k.4?D R?k ZZ D f(x,y)dxdy = 0 K(x,y) R,kf(x,y) = 0. yyy?y.b?P(x0,y0) R,?f(x0,y0) 6= 0,?f(x0,y0) 0.?Y ?5,r 0,=?3:P(x0,y0)?%r?U(P,r),?(x,y) G = U(P,r) R,k f(x,y) f(x0,y0) 2 ( 0). l?, ZZ G f(x,y)dxdy f(x0,y0) 2 ZZ D dxdy = f(x0,y0) 2 G 0, 2 ? G?G?,?fi ?g.u,(x,y) R,k f(x,y) = 0. 8.y:ef(x,y)?g(x,y)3k.4?R?,Kf(x,y)g(x,y)3R?. yyy?T?Rn?:R1,R2,.,Rn.?4kRk?,q?k(f),k(g),k(fg) Of(x),g(x),f(x)g(x)31k?Rk ?.qfi ?f(x,y)?g(x,y)3Rk .(?),= M 0,(x,y) R,k|f(x,y)| 6 M?|g(x,y)| 6 M. ?:P0 k,P 00 k Rk,k |f(P0 k)g(P 0 k) f(P 00 k)g(P 00 k)| 6|f(P0 k)g(P 0 k) f(P 00 k)g(P 0 k)| + |f(P 00 k)g(P 0 k) f(P 00 k)g(P 00 k)| 6|g(P0 k)|f(P 0 k) f(P 00 k)| + |f(P 00 k)|g(P 0 k) g(P 00 k)| 6Mk(f) + k(g). u,k(fg) 6 Mk(f) + k(g),k = 1,2,.,n. n X k=1 k(fg)4k6 M ? n X k=1 k(f)4k+ n X k=1 k(g)4k ? . fi ?f(x,y)?g(x,y)3R?,= 0, 0,T :k T kYn,k lim d(G)0 1 G ZZ G f(x,y)dxdy = f(x0,y0) . 3 10.y:eY?fn(x,y)3k.4?R?uf(x,y),K lim n ZZ R fn(x,y)dxdy = ZZ R f(x,y)dxdy. yyy fi ?f(x,y)3RY,l?.qfi ?fn(x,y)3k.4?R?uf(x,y),= 0,N N,n N,(x,y) R,k |fn(x,y) f(x,y)| 0)?.X13.h(I). ?C?,?u = xy,v = y x. ?C?xy?I?RC?uv?I?o?:u = 1,u = 2,v = 1,v = 4 ?/?R, X13.h(II). (u,v) (x,y) = ? ? ? ? ? ? yx y x2 1 x ? ? ? ? ? ? = 2y x = 2v (x,y) (u,v) = 1 (u,v) (x,y) = 1 2v . u, V = ZZ R x2y2dxdy = ZZ R0 u2 1 2vdudv = Z 4 1 dv Z 2 1 u2 2vdu = 7 2 ln2 . (2)f(x,y) = px2 + y2,Ra26 x2+ y26 b2,a ? ? ? ?1 3 ?s2 3 ?2? ? ? ? = 1. l?, n P k=1 k4 k P R2 k4 k P R2 4k= 1 s 2 3(R2?),= lim kTk0 n X k=1 k4 k6= 0. u,f(x,y)3R?. (2)gR 1 0 dx R1 0 f(x,y)dy?3. yyyx 0,1. ?xkn,f(x,y) = 3y230,1(uy)?,? Z 1 0 f(x,y)dy = Z 1 0 3y2dy = 1; ?xn,f(x,y) = 130,1(uy)?,? Z 1 0 f(x,y)dy = Z 1 0 dy = 1; l?,x 0,1,f(x,y)30,1(uy)?,? Z 1 0 f(x,y)dy = 1, u,gR 1 0 dx R1 0 f(x,y)dy?3. (3)kx?y ?g?3. yyyy0 0,1,?y06= 1 3,?f(x,y0)30,1?. fl ?,0,1?T.31k?mxk1,xk? k= |1 3y2 0|(?),k = 1,2,.,n. 13 l? n P k=1 k4 xk= |1 3y2 0| n P k=1 4xk= |1 3y2 0|, =lim l(T)0 n P k=1 k4 xk6= 0. u,f(x,y0)30,1?,l?kx?y ?g?3. 13.?f(x,y)Y,F(t) = RR R1 f(x,y)dxdy,?Rt: x2+ y26 t2,F0(t). )?x = rcos,y = rsin,|J| = r.k F(t) = ZZ R1 f(x,y)dxdy = Z r 0 dr Z 2 0 rf(rcos,rsin)d, ?12.3n1,?R 2 0 rf(rcos,rsin)dr?Y.?8.4n1,k F0(t) = Z 2 0 tf(tcos,tsin)d. 14.y:ef(x,y)3Ra16 x 6 b1;a26 y 6 b2Y,(,) R,-Ra16 x 6 ;a26 y 6 ,K 2 ZZ R f(x,y)dxdy = f(,). yyy fi ?f(x,y)3RY,(,) R,? F(,) = ZZ R f(x,y)dxdy = Z a1 dx Z a2 f(x,y)dy. fi ?(x) = R a2 f(x,y)dy3a1,Y.?8.4n1,k F = Z a2 f(,y)dy. qfi ?f(,y)3a2,b2Y,2?8.4n1,k 2F = f(,), = 2 ZZ R f(x,y)dxdy = f(,). ?SK13.2 56e?1358? 1.e?n: (1)RRRVxy2z3dxdydz,?Vz = xy?y = x,x = 1,z = 0?. )NVV?z = xy?n?y = x,x = 1,z = 0?,X13.q.NV3xy? I?Kn?y = 0,x = 1,x = y?n?/?D,X13.q.kz,?g 3?K?D?,= 14 ZZZ V xy2z3dxdydz= ZZ D xy2dxdy Z xy 0 z3dz = Z 1 0 xdx Z x 0 y2dy Z xy 0 z3dz = 1 4 Z 1 0 x5dx Z x 0 y5dy = 1 28 Z 1 0 x12dx = 1 364. (3)RRRVxyzdxdydz. ?V = (x,y,z)|x2+ y2+ z26 1,x 0,y 0,z 0. )NV?N?o?,X13.r.NV3xy?I?K?D(x2+ y26 1,x 0,y 0).kz,?g3?K?D?,= ZZZ V xy2z3dxdydz= ZZ D xydxdy Z 1x2y2 0 zdz = 1 2 Z 1 0 xdx Z 1x2 0 y(1 x2 y2)dy = Z 1 0 x(1 x2)2dx = 1 48. 2.?e?nU?gSS?: (1) R1 0 dx R1x 0 dy Rx+y 0 f(x,y,z)dz. )NV:0 6 z 6 x + y,0 6 y 6 1 x,0 6 x 6 1,=NV?:x = 0,y = 0,z = 0,x + y = z,x + y = 1?,X13.s. ky.?NV?K?xy?I?,?K?/?D(0 6 x 6 1;0 6 z 6 1). z = x?NV?N:V1?V2,X13.s.u 15 Z 1 0 dx Z 1x 0 dy Z x+y 0 f(x,y,z)dz = ZZZ V1 f(x,y,z)dxdydz + ZZZ V2 f(x,y,z)dxdydz = Z 1 0 dx Z x 0 dz Z 1x 0 f(x,y,z)dy + Z 1 0 dx Z 1 x dz Z 1x zx f(x,y,z)dy. kx,?NV?K?yz?I?.?K?/?G(0 6 y 6 1;0 6 z 6 1). z = y?NV?N:V1?V2.u,?k Z 1 0 dx Z 1x 0 dy Z x+y 0 f(x,y,z)dz = Z 1 0 dz Z 1 z dy Z 1y 0 f(x,y,z)dx + Z 1 0 dz Z z 0 dy Z 1y zx f(x,y,z)dx. (2)R 1 1dx R 1x2 1x2 dy R1 x2+y2 f(x,y,z)dz. )NV:px2+ y26 z 6 1,1 x26 y 6 1 x2,1 6 x 6 1,=NVI z2= x2+ y2?z = 1?,X13.t. ky,?NV?K?xz?I?.?Kn?/?D(x = z,x = z,z = 1)?.u ,ky,?gx,?z,k Z 1 1 dx Z 1x2 1x2 dy Z 1 x2+y2 f(x,y,z)dz = Z 1 0 dz Z z z dx Z z2x2 z2x2 f(x,y,z)dy. ky,?gz,?x.x=0(yz?I)?NV?N:V1?V2,X 13.t.u, Z 1 1 dx Z 1x2 1x2 dy Z 1 x2+y2 f(x,y,z)dz = ZZZ V1 f(x,y,z)dxdydz + ZZZ V2 f(x,y,z)dxdydz = Z 1 0 dx Z 1 x dz Z z2x2 z2x2 f(x,y,z)dy + Z 0 1 dx Z 1 x dz Z z2x2 z2x2 f(x,y,z)dy. 16 kx,)?,l?. 3.?C?e?n? (1)RRRV(x2+ y2)dxdydz,?Vx2+ y2= 2z?z = 2?. )NV=?x2+ y2= 2z?z = 2?,X13.u. ?IO?: x = rcos,y = rsin,z = z. |J| = r.?NVC?NV: r2 2 6 z 6 2,0 6 r 6 2,0 6 6 2. u,RRRV(x2+ y2)dxdydz = RRR V0 r2 r drddz = Z 2 0 d Z 2 0 r3dr Z 2 r2 2 dz = 16 3 . (2)RRRvzpx2+ y2dxdydz,?Vy = 2x x2?z = 0,z = a,y = 0?(a0). )NV?e.?(x 1)2+ y26 1(y 0)p?a1z?N,X13.v ?IO?: x = rcos,y = rsin,z = z,|J| = r. ?NVC?NV: 0 6 z 6 a,0 6 6 2 ,0 6 r 6 2cos. u,RRRvzpx2+ y2dxdydz = Ra 0 dz R 2 0 d R2cos 0 zr rdr = 4a2 3 Z 2 0 cos3d = 8 9a 2 (3)RRRv s 1 x2 a2 y2 b2 z2 c2dxdydz,?V ? x2 a2 + y2 b2 + z2 c2 6 1. )2?IO?: x = arsincos,y = brsinsin,z = crcos. ? ? ? ? ? ? asincosarcoscosarsinsin bsinsinbrcossinbrsincos ccoscrsin0 ? ? ? ? ? ? = abcr2sin. ?NVC?NV:0 6 r 6 1,0 6 6 ,0 6 6 2.u ZZZ v s 1 x2 a2 y2 b2 z2 c2dxdydz 17 = abc Z 2 0 d Z 0 sind Z 1 0 p 1 r2r2dr = 4abc Z 1 0 r2 p 1 r2dr = 2 4 abc. (4)RRRV(x2+ y2)dxdydz,?Vz = pb2 x2 y2?z = pa2 x2 y2(b a 0)9 z = 0?. )NV%3?:?a?b?xy?I?. ?IO?: x = rsincos, y = rsinsin, z = rcos. (x,y,z) (r,) = r2sin. ?NVC?NV:a 6 r 6 b,0 6 6 2,0 6 6 2.u ZZZ V (x2+ y2)dxdydz = Z 2 0 d Z 2 0 sin3d Z b a r2 r2dr = 2 5 (b5 a5) Z 2 0 sin3d = 2 5 (b5 a5) 2 3 = 4 15(b 5 a5). 4.e?N?N: (1)z = xy,x2+ y2= x,z = 0. )NV3xy?I?K?D : x2+ y26 x.x?D?D1: x2+y26 x,y 0?D2: x2+y26 x,y 6 .0X13.w.3D1?V?z = xy 03xy? I?, 3D2?V?z = xy 6 03xy?I?e?,?NVux. d,NV?N1?%?N?2?.u,NV ?N I = 2 ZZ D1 dxdy Z xy 0 dz = 2 Z 1 0 dx Z xx2 0 dy Z xy 0 dz = 2 Z 1 0 xdx Z xx2 0 ydy = Z 1 0 x(x x2)dx = 1 12. (2)z = x2+ y2,z = 2(x2+ y2),y = x,y = x2. )NV3xy?I?K?D : y = x?y = x2?.NV3xy?I?,e ?=?z = x2+ y2,?=?z = 2(x2+ y2).u,NV?N I = ZZZ V dxdydz = Z 1 0 dx Z x x2 dy Z 2(x2+y2) x2+y2 dz = Z 1 0 dx Z x x2 (x2+ y2)dy = 3 35. (3)x2+ y2+ z2= a2,x2+ y2+ z2= b2,x2+ y2= z2(z 0,0 0). ?IO?: x = rcos, y = rsin, z = z. |J| = r. NV 3?IXNV : 0 6 6 2,0 6 r 6 1,r 6 z 6 p 2 r2. 20 duz?=.?“,k Jz= ZZZ V (x2+ y2)dxdydz = Z 2 0 d Z 1 0 dr Z 2r2 r r2 rdz =2 Z 1 0 r3( p 2 r2 r)dr = 4 15(4 2 5). 7.y: Z x 0 dv Z v 0 du Z u 0 f(t)dt = 1 2 Z x 0 (x t)2f(t)dt. yyy?“?z?n,?NV 0 : 0 6 t 6 u,0 6 u 6 v,0 6 v 6 x,= NVtuv?m?IXo?:t = 0,t = u,u = v?v = x?,X13.y. ?“?g?gS:kv,?gu,?t.?NV? K?tu?I?,n?t = u,u = x,t = 0?n?/?D,X13.y.u Z x 0 dv Z v 0 du Z u 0 f(t)dt = Z x 0 dt Z x t du Z x u f(t)dv = Z x 0 dt Z x t f(t)(x u)du = Z x 0 f(t)dt Z x t (x u)d(x u) = 1 2 Z x 0 (x t)2f(t)dt. 8.?F(t) = RRR V f(x2+ y2+ z2)dxdydz,?V : x2+ y2+ z26 t2,f?,F0(t). ) ?IO?: x = rsincos, y = rsinsin, z = rcos. |J| = r2sin. L?IO?,NV3?IX?NV: 0 6 r 6 t,0 6 6 ,0 6 6 2. l?,k F(t) = Z 02d Z 0 d Z t 0 f(r2)r2sindr =4 Z t 0 r2f(r2)dr u,F0(t) = 4t2f(t2). 21 9.?f(x,y,z)3NV : x2+ y2+ z26 1Y,Vr: x2+ y2+ z26 r2,(0 0,31?%?,? 4 6 6 2.l?1?%?NV1: 0 6 6 2, 4 6 6 2,0 6 r 6 a pcos2. u,NV?N I= ZZZ V dxdydz = 8 ZZZ V1 r2sindrdd =8 Z 2 0 d Z 2 4 d Z acos2 0 r2sindr =8 2 a3 3 Z 2 4 (cos2) 3 2sind = 4 3a 3 Z 2 4 (1 2cos2) 3 2sind (?u = cos) = 4 3a 3 Z 2 2 0 (1 2u2) 3 2du (?u = 1 2sint) = 4 3 1 2a3 Z 2 0 cos4tdt = 2a3 42. 22 (2) ?x2 a2 + y2 b2 + z2 c2 ?2 = x h (h 0). )w,x 0,=?NV?uyz?Ix?. XJ:(x,y,z)(? ?K?,x 0)3?,Kux?:(x,y,z)?3?.d, ? NVux,l?NV?N1?%?NV1?N?4?.?O?1?%? ?NV1?N,?2?IO?: x = arsincos, y = brsinsin, z = crcos. |J| = abcr2sin. 3?IX? r4= a hrsincos r = 3 s a hrsincos. 1?%?NV1: 0 6 6 2,0 6 6 2,0 6 r 6 3 s a hrsincos. u,NV?N I= ZZZ V dxdydz = 4 ZZZ V1 dxdydz =4 Z 2 0 d Z 2 0 d Z 3 v u u ta h r sincos 0 abcr2sindr = 4a2bc 3h Z 2 0 cosd Z 2 0 sin2d = a2bc 3h . 11.n ZZZ V dxdydz px2 + y2+ (z a)2 , ?VNx2+ y2+ z26 1,?a 1. )?IO?: x = rsincos, y =