微分几何课后答案.pdf

? 1.1?p5. 3.?t = 0? (1)?r = (acosht,asinht,at): s(t) = 2asinht. (3)?r = (acost,aln(sect + tant) asint,0): s(t) = alnsect. 4.?c : r = r(t)?r(t0)? r(t0) 6= 0.?r(t0)?r(t0). ?r2(t0)? 0 = d dt ? ? ? ? t=t0 r2= 2r(t0)r(t0). ? 6.?c?l?l?c? p.?l?c?p? ?l?x?c : r(t) = (x(t),y(t)?p = r(0) ?y(t)?0?y(0) = 0, r(0) = (x(0),0) k l. ? 1.2?p11. 2.? (2)r = (cos3t,sin3t,cos2t). ? k(t) = 6 25|sin2t|, (t) = 8 25sin2t. (3)r = a(3t t3,3t2,3t + t3)(a 0). ? k(t) = (t) = 1 3a(t2+ 1)2 . 1.3frenet?frenet?p15. 5.?kr? (4)?r = a(t sint,1 cost) ?z-?z = (0,0,1)?r = (x(t),y(t),0)? ? kr= (r,r,z) |r|3 = xy xy (x2+ y2)3/2 . 1 ? kr= 1 4a|sin t 2| . (5)?r = ?t,acosh t a ? ? kr= 1 acosh 2 t a . 6.?kr? ?kr= 0?frenet? n r= krt ? ? r + 1 kr nr ? = t t = 0, ? r + 1 kr nr=?:= a, |r a| = 1 |kr|. ? 1 |kr| ? 7.?(1)? ?(1)?r(s)?a?(r a) k t, 0 = (r a)n ? 0 = (r a)(kt + b) = k(r a)t. ?k = 0? 1.5?p15. 4.? bertrand? ?c1,c2?bertrand ? ?c1: r1= r1(s)?s? ?s? c2: r2= r2(s)? (1)? r2(s) = r1(s) + (s)n1(s) 2 ?(s)?n1(s)?c1? ? r 2= (1 k1)t1+ n1 + 1b1. ?n1k n2?n1?(s) = 0? (2)? s?c2? ? (t1(s)t2(s)= k1n1(s)t2(s) + t1(s)k2n2(s)d s ds = 0, ? 5.?(2)?r = r(s)?k 6= 0?r(s)?r,r ?bertrand?(6= 0)?,?k + = 1. ? ”=”?r? r(s) = r(s) + n(s), ? r(s) = (1 k)t(s) + b(s). ?t,tr?.?n = nr? tr= t cos + b sin. ? 1 k cos = sin , ? = , = cot? ”=”?r(s) = r(s) + n(s)? 6.? ? ? ? ?r1= r1(s)?s? ? r2(s) = r1(s) + (s)b1(s). ? r 2(s) = t1(s) + (s)b1(s) (s)k1(s)n1(s). ?b1= b2,?b1? (s) = 0,(s) . 3 ? r2(s) = t1(s) k1(s)n1(s). ? t2+ n2= r 2(s) = k 2 1t1(s) + (k1 k 1)n1 k11b1(s), ?b1?k11= 0.? = 0? 6= 0? k11= 0,? ? 1.2?p61. 3.?xyz = a3? ? ? 0 = d(xyz) = yzdx + zxdy + xydz, ?(x0,y0,z0)? y0z0(x x0) + z0 x0(y y0) + x0y0(z z0) = 0, ? y0z0 x + z0 x0y + x0y0z = 3a3. ? v = 1 6 ? ? ? ? 3a3 y0z0 3a3 z0 x0 3a3 x0y0 ? ? ? ? = 9 2|a| 3. ? 1.3?p62. 1.?c : r = (s)?n,b? ?a?s? r = (s) + an(s)cosv + b(s)sinv ? ? ?frenet? rs=(1 ak cosv)t a sinvn + a cosvb, rv=a(n sinv + b cosv), 4 ?k,t?c? ? rs= (1 ak cosv)t + rv, ? rs rv=(1 akcosv)t rv =a(1 akcosv)t (n sinv + b cosv) =a(1 akcosv)(b sinv n cosv), ?s? n = n cosv b sinv ? n = n cosv + b sinv.? 3.? x2 a + y2 b + z2 c = 1 ?a b c? c?c b? ?b a? ? ? ? ?f(x,y,z) := x2 a + y2 b + z2 c 1. ?(x0,y0,z0),?f(x0,y0,z0) = 0?(x0,y0,z0)?x0y0z06= 0,?f(x0,y0,z0) = 0 ? ():=x2 0(b )(c ) + y 2 0(c )(a ) + z 2 0(a )(b ) (a )(b )(c ). ? () 0,(b) 0, ?() = 0? ? 1 c,c 2 b,b 3 a. ? ?fi(x,y,z) = 0, i = 1,2,3. 5 ? ?fi?fi(x,y,z) = 0?fi fj= 0, i 6= j? fi= ? 2x a i , 2y b i , 2z c i ? , ? fi fj= 4x2 (a i)(a j) + 4y2 (b i)(b j) + 4z2 (c i)(c j). ?fi= fj? 0=fi(x,y,z) fj(x,y,z) =(i j) ? x2 (a i)(a j) + y2 (b i)(b j) + z2 (c i)(c j) ? = i j 4 ?f i fj ? , ?i 6= j,?fi fj= 0,? 1.4?p63. 1.?m obius? r = (cos, sin, 0) + v ? sin 2 cos, sin 2 sin, cos 2 ? ,1 2 v 1 2 ?n. ? ? rv= ? sin 2 cos, sin 2 sin, cos 2 ? r=(sin, cos, 0) + v 2 ? cos 2 cos, cos 2 sin, sin 2 ? +v ? sin 2 sin, sin 2 cos, 0 ? r rv= ? cos 2 cos, cos 2 sin, sin 2 ? +1 2v ? sin + sincos, sin2 cos, 2sin2 2 ? . ? 6 1.6? ?p73. 2.?r = (cosv (u+v)sinv, sinv +(u+v)cosv, u+2v)? ?r = a(v) = (cosv, sinv, v)? ? ? r = (cosv, sinv, v) + (u + v)(sinv, cosv, 1), ? ( x = v y = u + v, ? r = (cosx, sinx, x) + y(sinx, cosx, 1) = a(x) + ya(x), ? 3.?r = (acos, asin, b) (ab 6= 0)? ?r = (ucosv, usinv, bv)? ? ?|r|2= a2+ b2=? ? ? n() = (cos, sin, 0), ? r = (acos, asin, b) + (cos, sin, 0), ? ( u = a v = , ? r=(ucosv, usinv, bv) =(0, 0, bv) + u(cosv, sinv, 0) :=(v) + ul(v). ? (,l,l) = ? ? ? ? ? ? ? ? 00b cosvsinv0 sinvcosv0 ? ? ? ? ? ? ? ? = b 6= 0, 7 ? 5.?a2x + 2ay + 2z = 2a? ? ? ? ( a2x + 2ay + 2z = 2a ax + y = 1, ?a?2xz = (y 1)2? x = (x + z)/2 z = (x z)/2 y = y 1, ? x2= y2+ z2.? 6.? ? ? ?c : r = r(s)? ? ?c? r(s,t) = r(s) + tn(s), (r,n,n) = (t,n,kt + b) = ; ?c? r(s,t) = r(s) + tb(s), (r,b,b) = (t,b,n) = . ? 6= 0? 7.?c?s?l?c?l? ? ? ?c? ?r = r(s)?s? r(s,t) = r(s) + tt(s). ? rs= t(s) + tk(s)n(s), rt= t(s), rs rt= tk(s)b(s) 8 ?s?r(s,t)?c?r(s)?b(s)? 9.?m obius? r = (cos, sin, 0) + v ? sin 2 cos, sin 2 sin, cos 2 ? ,1 2 v 0? lab= z 1 0 ? ? ? ? ds dv ? ? ? ?dv = z 1 0 q a2v2+ (av2/2)2+ a2dv = a z 1 0 (1 + v2/2)dv = 7 6a; lac= z 1 0 ? ? ? ? ds dv ? ? ? ?dv = z 1 0 q a2v2+ (av2/2)2+ a2dv = 7 6a; lbc= z a/2 a/2 ? ? ? ? ds du ? ? ? ?du = z a/2 a/2 1du = a. 12 ?(du,dv)?(u,v)? cos = duu + (u2+ a2)dvv pdu2 + (u2+ a2)dv2pu2+ (u2+ a2)v2 , ? cosa= a2v2+ u2+ a2 a2v2+ u2+ a2 ? ? ? ? a(0,0) = 1, cosb= av a2v2 + u2+ a2 ? ? ? ? b(a/2,1) = 2 3, cos( c)= av a2v2 + u2+ a2 ? ? ? ? c(a/2,1) = 2 3, a = 0,b = c = arccos 2 3. ? 2.2?p81. 1.?u = ucos + v sin, v = usin + v cos? ? ? ?(2-29)? e=e cos2 f sin2 + gsin2 f=(e g)sincos + f cos2 g=e sin2 + f sin2 + gcos2.? 2.? a = z z d p eg f2dudv ? ? (2-29)? e f f g ! = u uu v v uv v !t ef fg ! u uu v v uv v ! . ? eg f2= (eg f2) ? ? ? ? (u,v) ( u, v) ? ? ? ? 2 , 13 ?(2-9)? eg f2= |r u r v|2= |ru rv|2 ? ? ? ? (u,v) ( u, v) ? ? ? ? 2 = (eg f2) ? ? ? ? (u,v) ( u, v) ? ? ? ? 2 . ? a= z z d p eg f2d ud v = z z d p eg f2 ? ? ? ? (u,v) ( u, v) ? ? ? ?d ud v = z z d p eg f2dudv = a.? 2.3 2.4?p86. 3.?r = (ucosv, usinv, u+v)?r = (cos, sin, p2 1) ? = arctanu + v, = p u2+ 1. ? ?ru= (cosv,sinv,1),rv= (usinv,ucosv,1), ds2= 2du2+ 2dudv + (u2+ 1)dv2. ?r= (cos,sin, 21), r= (sin,cos,0), d s2= 22 1 2 1 d2+ 2d2. ? = arctanu + v, = u2 + 1? d s2= 2u2+ 1 u2 ? udu u2 + 1 ?2 + (u2+ 1) ? du u2+ 1 + dv ?2 = ds2.? 4.?r = (a(cosu+cosv), a(sinu+sinv), b(u+v)? ? ? ? r = ? 2acos u + v 2 cos u v 2 , 2asin u + v 2 cos u v 2 , b(u + v) ? . ?x = (u + v)/2,y = 2acos(u v)/2,?r? r = (ycosx, y sinx, 2bx), 14 ? ds2= (y2+ 4b2)dx2+ dy2. ? r = (f(y)cosx, f(y)sinx, g(y), ? d s2= f2(y)dx2+ (f2+ g2)dy2 ? f(y) = p y2+ 4b2,g(y) = z y 0 2b py2 + 4b2 dy, ?ds2= d s2,?r? r? 5.? ? ? ?ds2= 2ds2? e= 2e,f= 2f,g= 2g. ? p eg f2= p eg f2. ? = 1? 6.?f(z) = u(x,y) + iv (x,y)?z = x + iy? (1)?z = u(x,y),z = v (x,y)? (2) xy? = u(x,y), = v (x,y)? ? (1).? ?z = u(x,y)?d = q 1 + u2 x+ u2ydxdy, ?z = v (x,y)?d= q 1 + v 2 x + v2 ydxdy. ?f?ux= vy, uy= vx,?d = d? (2). ? d2+ d2=(uxdx + uydy)2+ (vxdx + vydy)2 =(vydx vxdy)2+ (vxdx + vydy)2 =(v 2 x + v 2 y)(dx 2 + dy2), 15 ? 9.? ?2.1?3? 2.5?p89. 1.? (2)?r = (acoshucosv,bcoshusinv,csinhu); (4)?r = (u,v, 1 2( u2 a2 + v2 b2). ? (2).? ru= (asinhucosv,bsinhusinv,ccoshu), rv= coshu(asinv,bcosv,0), ru rv= coshu(bccoshucosv,cacoshusinv,absinhu); ruu= (acoshucosv,bcoshusinv,csinhu), ruv= sinhu(asinv,bcosv,0), rvv= coshu(acosv,bsinv,0). ? ii = abc (du2+ cosh2udv2) = a2b2sinh2u + b2c2cosh2ucos2v + c2a2cosh2usin2v. (4).? ru rv= ? u a2 , v b2 ,1 ? , ruu= ? 0,0, 1 a2 ? ,ruv= 0,rvv= ? 0,0, 1 b2 ? . ii = ? 1 + u2 a4 + v2 b4 ?1 2 ?du2 a2 + dv2 b2 ? .? 2.?z = f(x,y)? ? r = (x,y,f(x,y), 16 rx= (1,0,fx),ry= (0,1,fy), ? i = (1 + f2 x)dx 2 + 2fxfydxdy + (1 + f2 y)dy 2. ? rxx= (0,0,fxx), rxy= (0,0,fxy), ryy= (0,0,fyy), n = 1 p1 + |f|2(fx,fy,1),|f|2= f2 x+ f 2 y. ? ii = fxxdx2+ 2fxydxdy + fyydy2 p1 + |f|2.? 4.? ln m2= 1 g (r11,r1,r2)(r22,r1,r2) (r12,r1,r2)2. ?(2-39).? 5.? r11r1= e1/2,r12r1= e2/2, r22r2= g2/2,r12r2= g1/2, r11r2= f1 e2/2,r12r1= f2 g1/2. ? e1= (r1r1)1= 2r11r1, e2= (r1r1)2= 2r12r1; g1= (r2r2)1= 2r21r2, g2= (r2r2)2= 2r22r2; f1= (r1r2)1= r11r2+ r1r21, f2= (r1r2)2= r12r2+ r1r22. ? 3.2?p96. 1.? ln g uk = 1 1k+ 2 2k, g = det(gij). ? ? dg = d(det(gij) = dgij (gij?) = dgij ggij. 17 ?(2-46),? ln g uk = 1 2g g uk = gij 2 gij uk = gij 2 (l ikglj+ l jkgli) = l lk. 2.?u,v? u, v? ln m2= ? ? ? ? (u,v) ( u, v) ? ? ? ? 2 (ln m2) ? ? ij= kl uk ui ul uj , ? det(ij) = det(kl)det ?uk ui ? det ? ul uj ? , ? ln m2= ? ? ? ? (u,v) ( u, v) ? ? ? ? 2 (ln m2).? 3.?ds2= d2+ 2d2?i jk. ? r = (cos,sin), r1= (cos,sin),r2= (sin,cos). ?gauss?rij= k ijrk+ ijn ? r11= 0 1 11= 211= 0, r12= (sin,cos) = r2/ 1 12= 0, 212= 1/, r22= (cos,sin) = r1 1 22= , 222= 0. ? 4.?e,f,g?i jk ?(2-50)?i j. ? (1).? g11g12 g21g22 ! = ef fg ! , g11g12 g21g22 ! = 1 g gf fe ! , 18 ?g = det(gij) = eg f2.?(2-49)? 1 11= (ge1 2ff1+ fe2)/2g, 2 11= (2ef1 ee2 fe1)/2g, 1 12= (ge2 fg1)/2g, 2 12= (eg1 fe2)/2g, 1 22= (2gf2 gg1 fg2)/2g, 2 22= (eg2 2ff2+ fg1)/2g. (2).? 1112 2122 ! = lm mn ! , ? 1 1 = g1111+ g1221= (gl fm)/g, 1 2= g1112+ g1222= (gm fn)/g, 2 1 = g2111+ g2221= (em fl)/g, 2 2= g2112+ g2222= (en fm)/g. ? 5.?z = f(x,y)? ? r = (x,y,f(x,y), r1= (1,0,f1), r2= (0,1,f2), n = 1 p1 + |f|2(f1,f2,1), rij= (0,0,fij). ?gauss?rij= k ijrk+ ijn ? 0 = k ij ijfk p1 + |f|2 ?(2.5)?2? ij= fij p1 + |f|2, ? k ij= fijfk 1 + |f|2 .? 3.4?p98. 1.?r = a(u) + b(v)? ?r12= 0?m = 0.? 19 5.?r = (acosh t a cos, acosh t a sin, t)? ?88? d2= 1 a2 dt2, t = t0 a, ? r= ? acosh t0 a a cos, acosh t0 a a sin, t0 a ? .? 6.? (1)? (2)? (3)? ? ? (1).?s? ?r(s) = r(u(s),v(s)? 0 = r= r11u2+ 2r12uv+ r22v2, ?n? 0 = lu2+ 2muv+ nv2, ? (2).?r = a(u) + vl(u)?m = n = 0,? ?ldu2= 0?l 6= 0?du = 0?v? ? (3).?(1)?ldu2+ 2mdudv + ndv2= 0? l = m = n = 0.?weingarten?dn = 0? ? 10.?f(x,y,z) = 0? ? ?ij= rij n = ri (n)j? ldu2+2mdudv +ndv2= 0? ?f(x,y,z) = 0?n = (fx,fy,fz).? ?r = (x,y,z(x,y).? rx= (1,0,zx),ry= (0,1,zy); 20 nx= (fxx+ fzxzx, fxy+ fyzzx, fzx+ fzzzx), ny= (fxy+ fzxzy, fyy+ fyzzy, fzy+ fzzzy); l=fxx+ 2fzxzx+ fzzz2 x, 2m=2fxy+ 2fyzzx+ 2fzxzy+ 2fzzzxzy, n=fyy+ 2fyzzy+ fzzz2 y. ?ldx2+ 2mdxdy + ndy2= 0?dz = zxdx + zydy? fxxdx2+ fyydy2+ fzzdz2+ 2fxydxdy + 2fyzdydz + 2fzxdzdx = 0, ?hess(f)(r,r) = 0? 4.1?p101. 1.?c? kn= k cos, ?r? ? ? kn= kn n = k cos. ?r? ?r2= r2?c?r2(s) = r2? r(s) t(s) = 0,1 + r(s) k(s)n(s) = 0,k = 1 r n . ?n = r/r,? k = 1 rn n = 1 rcos . 21 ?1/r?n? ? kn= 1 r(n) n = 1 r = kcos.? 2.?c?c?c? ? ? ?0 = kn= kn n?k 6= 0?nn? ?tn?c? 3.?s1,s2?c?k?si?ki(i = 1,2)? s1,s2? k2sin2 = k2 1+ k 2 2 2k1k2cos. ? ?si?ni?n,n1,n2t?n,t?c? ?n,n1,n2?n n1 n2. ?k1= kn n1?k2= kn n2? k1n2 k2n1= k(n n1)n2 (n n2)n1 = k(n1 n2) n, ? k2 1 2k1k2cos + k 2 2 = k2sin2.? 4.? ?euler?(2-56),?kn1+ kn2= k1+ k2. ? 4.2?p102. 1.? ? ?dn = 0,?ij= 0,? ?r2= r2,?n = r/r ij= ri nj= ri rj/r = 1 rgij. ? 2.? 22 ? ? x2 a2 + y2 b2 + z2 c2 = 1, ? r = (acoscos, bcossin, csin) ( 2 2 ,0 c? ( sin2sin2 = 0, a2sin2cos2 + b2sin2sin2 + c2cos2 = a2sin2 + b2cos2. ? ( sin2 = 0, a2sin2 + c2cos2 = b2. ? a r a2 b2 a2 c2 ,0,c r b2 c2 a2 c2 ! . ?(x0,y0,z0)?(x0/a2,y0/b2,z0/c2),? ? ( x2 a2 + y2 b2 + z2 c2 = 1, xx0 a2 + yy0 b2 + zz0 c2 = 0. ? x2 b2 + y2 b2 + z2 b2 = 1.? 3.?xyz = a3? ?r = (x,y,z(x,y),?89?2 i = (1 + z2 x)dx 2 + 2zxzydxdy + (1 + z2 y)dy 2, 23 ii = 1 q 1 + z2 x+ z2y (zxxdx2+ 2zxydxdy + zyydy2). ? 1 + z2 x zxx = zxzy zxy = 1 + z2 y zyy , ?z = a3 xy ?x4y2= a6= x2y4? (a,a,a),(a,a,a).? 4.4?p106. 1.? ? ? ? (1).? ?3.4?6? (2).? ? ? (3).? ? (4).? ? ? 3.?s?s? s? ? ?p?s? s? s? r(u,v) = r(u,v) + dn(u,v), ?r(u,v)?s?d? s?s?n(u,v)?r(u,v)? ? ?p?s1,s2?r1,r2? ?t1,t2?p = r(u0,v0)?p?s?p= r(u0,v0)? ?p?si?n ti?t1 t2? p?s1?n1(p) = t2(p)?s2?n2(p) = t1(p)? 24 ? ?p?n1(p) = t2(p)?n2(p) = t1(p)? ? s?p? n(p) = n(p)? ru= ru+ dnu, rv= rv+ dnv, ?weingarten?nu,nv?ru,rv? ru rvk ru rv, ? r(u,v)? s?r(u,v)?s? 4.? ? ?1? ?7? (1).?x2/a2+ y2/b2 z2/c2= 1. ? ? ? ? ? ? ? ? ? dxdydz x/a2y/b2z/c2 dx/a2dy/b2dz/c2 ? ? ? ? ? ? ? ? = 0. (2).?x2/a2 y2/b2= 2z. ? ? ? ? ? ? ? ? ? dxdydz x/a2y/b21 dx/a2dy/b20 ? ? ? ? ? ? ? ? = 0. ?r = r(u,v)? ? ? ? ? ? ? ? ? dv2dudvdu2 efg lmn ? ? ? ? ? ? ? ? = 0. ? 6.?c : r = (s)?kn6= 0? (1) c?s1? (2)?s1?c1?c1: r1= (s) + n(s)/kn? (3)?s1?r1=? ? (1). 25 ?s1?r = (s)+tn(s)?n(s) = 0?rodriques ?(s)t(s) = 0? kn6= 0? (2). s1?r = (s) + tn(s)?c1?s1?c1? r1= (s) + t(s)n(s). ?s1?c1? r = r1+ ur 1. ?n= kn? r = (s) + u(1 tkn)+ (t + ut)n(s) ?c1?r = (s) + tn(s)? 1 t(s)kn(s) = 0,t(s) = 1 kn(s), r1= (s) + n(s) kn(s). (3). ?1.6?v(s)? b(s) = (s) + v(s)n(s) ? 0 = + vn vkn,v(s) = 1 kn . ?r1= b =? 7.?f(x,y,z) = 0? ? ?r = (s)+tn(s)?(,n,n) = 0. ?(,n,n) = 0?(,n,(n) = 0? f(x,y,z) = 0?f?(,f,(f) = 0? ? ? ? ? ? ? ? ? ? xyz fxfyfz (fx)(fy)(fz) ? ? ? ? ? ? ? ? = 0? ? ? ? ? ? ? ? ? dxdydz fxfyfz dfxdfydfz ? ? ? ? ? ? ? ? = 0.? 26 4.5?p111. 2.? ?88?ln m2= m2 0,? 4.?r = (ucosv,usinv,u + v)? ? k = 1 (2u2+ 1)2 ,h = 1 + u2 (2u2+ 1)3/2 .? 5.?n1 n2= kg n. ?weingarten?ni= k irk ? n1 n2= (1 1r1+ 2 1r2) ( 1 2r1+ 2 2r2) = det( j i) r1 r2 = kg n. ? 6.? h = 1 2 z 2 0 knd. ?euler?kn() = k1cos2 + k2sin2? z 2 0 knd= z 2 0 k1cos2d + z 2 0 k2sin2d =k1 + k2 =2h. ? 8.?r = (a(u + v),b(u v),2uv) (a,b 0)? ? ? e = a2+ b2+ 4v2, f = a2 b2+ 4uv, g = a2+ b2+ 4u2; l = n = 0,m = 2ab pa2(u v)2 + b2(u + v)2+ a2b2 . ? g = eg f2= 4(a2(u v)2+ b2(u + v)2+ a2b2), k = m 2 g = 16a 2b2 g2 ,2h = 2fm g . ? 2+ 2fm g m2 g = 0. 27 ? k1,k2= 1 2 2fm g s 4f2m2 g2 + 4m2 g ! = 1 g ? fm p (f2+ g)m2 ? = 1 g ? fm egm2 ? = m g (f eg). ? 4.6?p112. 1.?x(u,v),y(u,v),z(u,v)? 2 uv = a(u,v) u + b(u,v) v ?x2+y2+z2?r = (x(u,v),y(u,v),z(u,v) ? ? ?x(u,v),y(u,v),z(u,v)?ruv= aru+brv?m = ruvn = 0?r2= x2+ y2+ z2?(r2)uv= a(r2)u+ b(r2)v? ?f = rurv= 0? 2.? x2 a2 + y2 b2 + z2 c2 = 1? ? ?a b c 0?62?3? x2 a2 + y2 b2 + z2 c2 = 1, c? c b? b 0,k(?) = 0,k(?) 0,?gauss? ? 9.?s : r