Сборник задач по высшей математике 2 том
.pdf4.2.25. IIpoBepHTb, HBJUleTCH JIH BblpaJKeHHe
(3X2y + b) dx + (X3 - ~) dy
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HeKOTopofi <PYHKIIHH U (x, y) H |
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0603HaQHM P = 3x2y + b, |
Q = x 3 - ~. Tor,n:a |
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oP = 3x2 |
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TaKHM 06Pa30M, YCJIOBHe rpHHa (~~ = |
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HHTerpa.JI
(x,y)
f ( 3x2Y + b) dx + (x3 - ~) dy,
(XO,YO)
r,n:e (Xo, Yo) - npOH3BOJIbHM <pHKCHpOBaHHM TOQKa nJIOCKOCTH Oxy, He JIe-
:>KarrrM Ha OCH OX (TaK KaK Yo =f 0). IIOJIO:>KHM (xo, Yo) = (0,1), a B KaQeCTBe nYTH HHTerpHpOBaHHH BbI6epeM nYTb L = ABC, H306paJKeHHblfi Ha pHC. 47.
Tor,n:a COKparrreHHO MO:>KHO HanHcaTb
(x,y)
U(x,y)= f = f = f+ f·
(XO,Yo) ABC AB BC
y
C(x,y}
1I-A.....:.(O,..;..'.....:.1}___....., B(x, I}
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x x
Puc. 47
IIMeeM: 1) (AB): y = 1, T. e. dy = 0 H |
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~) dy= |
(x,l) |
f(3x 2y +b) dx+ (x3 _ |
f(3x 2 +1)dx=x3 +x. |
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(0,1) |
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2) (Be): x - <pHKcHpoBaHo, CJIe,lJ;OBaTeJIbHO, dx = 0, OTKY,IJ;a |
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1(3x2y + t) dx + (x3 - |
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x2) |
dy = 1(X3 - |
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= (x 3y + !f) I(X,Y) = x3y + !f - x 3 - |
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Y (x,l) |
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3) TaKHM o6pa30M, U(x, y) = x 3 + X + x 3y + ~ - x 3 - X = x3y + ~. |
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IIpoBepKa IIOKa3bIBaeT, 'ITO,lJ;efiCTBHTeJIbHO, |
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dU = d (x 3y + ~) = (3X2y + t) dx + (X3 - ~) |
dy. |
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(3,0) |
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4.2.26. |
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(x4 + 4xy3) dx + (6x2y2 - 5y4) dy. |
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(-2,-1) |
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(1,0) |
x dy - |
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4.2.27. |
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+ 2y) dx + y dy |
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4.2.28. |
(1,1)(X |
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..t) |
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4.2.29. |
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dx + ( |
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+ x) dy. |
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Jx 2 + y2 |
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B'bI."tuc.IIum'b 1CPU60.llUttefJ.tt'bl.e Uttmezpa.ll'bl., 63S1m'bl.e 6aO.ll'b npocmpattCm6etttt'bl.X 1CpU6'b1.X:
4.2.30. |
I(y - z) dx + (z - |
x) dy + (x - |
y) dz, |
r,lJ;e L - |
BHTOK BHHTOBOfi |
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JIHHHH x = acost, y = asint, z = bt, 0 ~ t ~ 211". |
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4.2.31. |
f y dx + z dy + x dz, r,lJ;e L - |
OKP)')KHOCTb,3a,lJ;aHHaJI <p0pMYJIaMH |
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x = Rcosexcost, y = Rcosexsint, z = Rsinex (ex = const). |
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B'bI."tuc.IIum'b 1CPU60.llUttefJ.tt'bl.e |
uttmezpa.ll'bl. |
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(npea6apume.!!'bttO ttafJ.aSl nep6oo6pa3tty'lO): |
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(6,4,8) |
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(a,b,c) |
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4.2.32. |
1 xdx + ydy - |
zdz. |
4.2.33. |
1 yzdx + xzdy + xydz. |
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(1,0,-3) |
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(1,1,1) |
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211
(3,4,5)I
4.2.34. xdx+ydy+zdz.
(0,0,0) J X2 + y2 + Z2
4.2.35.C nOMOIIJ;bIO <POPMYJIbI rpUHa npe06pa30BaTb KPUBOJIUHeiiHbIii UH-
TerpaJI
fJx 2 + y2 dx + y[xy + In(x + JX2 + y2)] dy
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B ,LI,BOiiHOii U C era nOMOIIJ;bIO BbIqUCJIUTb UHTerpaJI no KOHTYPY np1IMoyraJIbHUKa ABCD (puc. 48), r,lJ;e A(1,1), B(7,1), C(7,4),
D(1,4).
Pu.c. 48
a MMeeM P = Jx2 + y2, Q = y(xy + In(x + Jx2 + y2)), OTKY,IJ;a
aQ = y [Y + |
1 (1 + |
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J x2 |
+ y2 |
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=y [ y + |
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1 yJx2 |
+ y2 + 1 |
ap _ |
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J x |
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J x2 + y2 ' |
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aQ _ ap = y2 + |
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= y2. |
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ax ay |
J x2 + y2 |
J x2 + y2 |
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TaKUM 06pa30M, B CUJIY <POPMYJIbI rpUHa ,lJ;aHHbIii KPUBOJIUHeiiHbIii UHTerpaJI paseH ,lJ;BoiiHOMY UHTerpaJIY OT y2 no np1IMoyraJIbHUKY ABCD, T. e.
fPdx+Qdy = II (~~ - ~P) dxdy = |
II y2dxdy = jdX jy2dY = |
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ABCD |
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ABCD |
1 1 |
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= xl:·y; I:= 7· 6l = 147. • |
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4.2.36. |
IIpuMeH1I1I <P0PMYJIY rpuHa, BbIqUCJIUTb |
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12(x 2 + y2) dx + (x + y)2 dy, |
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L
212
r,D;e L - KOHTYP 6ABC, rrpo6eraeMbltl: B rrOJIO:lKHTeJIbHOM HarrpaBJIeHHH, H A(I,I), B(2,2), C(I,3). IIOJIYl.JeHHbltl: pe3YJIbTaT rrpoBepHTb HerrOCpe,D;CTBeHHbIM BbIl.JHCJIeHHeM KPHBOJIHHetl:Horo HHTerpaJIa.
4.2.37.dU = x 2 dx + y2 dy.
4.2.38.dU = 4(x2 - y2)(xdx - ydy).
4.2.39.dU = (x + 2y) dx + y dy .
+y)2(x
4.2.40.
4.2.41.
4.2.42. |
! |
Xdy+ydX |
OKPy:lKHOCTb (X - |
1)2 + (y - |
1)2 = 1, rrpo- |
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6eraeMaJI rrpOTHB XO,D;a l.JacOBOtl: CTpeJIKH. |
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!(xy + y + x) dx + (yx - Y + x) dy, L - |
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4.2.43. |
9JIJIHrrC x 2 + Y2 = l. |
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4.2.44. |
!(xy +x+y) dx+ (xy +x -y) dy, L - |
OKPy:lKHOCTb x 2 +y2 = ax. |
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4.2.45. |
!(2x + 3y) dx + (3x - |
4y) dy, r,D;e L COCTOHT H3 .D;yrH rrapa60JIbI |
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y = ax2 , COe,D;HHHIOIIIetl: TOl.JKH 0(0,0) H A(2, 4), H oTpe3Ka rrpHMotl:, |
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COe.D;HHHIOIIIetl: 9TH TOl.JKH. |
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4.2.46. |
!(x4 + 4xy3) dx + (6x 2y2 - 5y4) dy, |
r,D;e |
L - .D;yra BepxHetl: rro- |
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y2 |
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JIOBHHbI l"Hrrep6oJIbI x2 |
- 2" = 1 OT TOl.JKH A( -a, v'2b) ,D;O TOl.JKH |
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B(a, v'2b) H OTpe3Ka rrpHMotl:, COe.D;HHHIOIIIetl: 9TH TOl.JKH.
4.2.47. BbIl.JHCJIHTb rrJIOIIIMb 9JIJIHrrCa rrpH rrOMOIIIH KPHBOJIHHetl:Horo HH-
TerpaJIa.
Q 3arrHweM 9JIJIHrrC |
2 |
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= 1 B rrapaMeTpHl.JeCKotl: <popMe x = a cos t, |
x2 |
+ Y2 |
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y = bsin t, 0 :s:;; t :s:;; 211", |
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rrOCJIe l.Jero BOCrrOJIb3yeMcH <POPMYJIotl: .D;JIH rrJIOIIIa.D:H |
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213
06JIacTH D
S = ~ f xdy - ydx = |
211" |
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~ !(acost. bcost + bsint·asin t) dt = |
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aD |
0 |
211" |
= ~ !abdt = 7rab. • o
4.2.48.ACTPOH,IJ;Oit x = a cos3 t, y = a sin3 t.
4.2.49.Kap,IJ;HOH,IJ;oit x = a(2cost - cos2t), y = a(2sint - sin2t).
4.2.50.IIeTJIeit ,lJ;eKapTOBa JIHCTa x 3 + y3 - 3axy = 0 (a> 0).
4.2.51.KPHBOit (x + y)3 = axy.
4.2.52.IIeTJIeit (x + y)4 = x2y.
4.2.53.JIeMHHcKaTOit BepHYJIJIH (x 2 + y2)2 = 2a2(x 2 _ y2).
4.2.54.IIeTJIeit JIHHHH (.;x + ..fij)12 = xy.
4.2.55.BbIqHCJIHTb pa60TY CHJIOBOrO IIOJIH F = yi - xj IIpH IIepeMem:eHHI!
MaTepHaJIbHoit T09KH B,IJ;OJIb BepxHeit IIOJIOBHHbI 9JIJIHIICa
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~+!L=1 |
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a2 |
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H3 TOqKH C(a, 0) B TOqKY B( -a, 0). |
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Pa60Ta A CHJIOBOrO IIOJIH F = |
Pi + Qj IIpH IIepeMem:eHHH MaTepHaJIbHOA |
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TOqKH M B,IJ;OJIb JIHHHH CB paBHa
! Pdx+Qdy.
CB
3aIIHweM ,IJ;yry 9JIJIHIICa C B B IIapaMeTpHqeCKoit <p0pMe: x = a cos t, y =
= bsint, t E [0,7r). Tor,lJ;a dx = |
-asintdt, dy = bcostdt H |
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11" |
11" |
A= !ydx-xdy= !(-absin2 t-abcos2t)dt=-ab !dt=7rab. • |
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4.2.56.,LLaHa IIepeMeHHaH CHJIa F = -yi+ (3y - 8x)j. BbIqHCJIHTb pa60TY 9TOit CHJIbI IIpH IIepeMem:eHHH MaTepHaJIbHoit TOqKH B,IJ;OJIb KOHTYpa IIpHMoyroJIbHHKa C BepWHHaMH A(9,4), B( -9,4), C( -9, -4),
D(9, -4).
4.2.57.HaitTH pa60TY CHJIbI F = -yi + (3y - 8x)j IIpH IIepeMem:eHHH MaTepHaJIbHoit TOqKH B,IJ;OJIb 9JIJIHIICa
x 2 y2
81+ 16 = 1.
4.2.58.HaftTH Pa60TY CHJIbI F = -4yi+(4y-3x)j IIpH IIepeMem:eHHH MaTePHaJIbHOit TOqKH B,IJ;OJIb IIpHMoyroJIbHHKa C BepWHHaMH A(2, -6), B(2, 6), C( -2,6), D( -2, -6).
4.2.59. HaitTH pa60TY CHJIbI F = -4yi + (4y - 3x)j B,IJ;OJIb 9JIJIHIICa x 2 y2
4+36=1.
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4.2.60. |
)J;aHbI TOqKH A(2,2), B(2, 0), C(O, 2). BbIqHCJIHTb |
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j(2X + 3) dx + (x + 7y + 1) dy, |
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r.n:e |
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a) |
L = OA; |
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6) |
L= OBA; |
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L = OC; |
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r) |
L = OCA; |
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L - |
napa6oJIa, CHMMeTpHqHruI OCH Oy H COe,D;HHHIOrn:ruI TOqKH |
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OHA. |
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4.2.61. |
)J;aHbI TOqKH A(9, 0, 0), |
B(9, 4, 0), C(9, 9, 9). BbIqHCJIHTb |
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dx + (3z + 8) dy + (7z + 6) dz, |
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L = AC; |
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L = OABC. |
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4.2.62. |
)J;aHbI TOqKH A(-2, -2), B( -2,0), C(O, -2). BbIqHCJIHTb |
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j(y + 4) dx + (3x + 3y) dy, |
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L=OA; |
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L=OCA; |
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L = OBA; |
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r) |
L - ,rr,yra napa60JIbI OA. |
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4.2.63. |
)J;aHbI TOqKH A(2, 0, 0), |
B(2, 2, 0), C(2, 2, 2). BbIqHCJIHTb |
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j |
4 dx + (4z + 3) dy + (3x + 4) dz, |
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L = AC; |
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L = OBC. |
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6) |
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4.2.64. |
BbIqHCJIHTb |
j (3x2y + y) dx + (x - |
2y2) dy, r.n:e ABC - KOHTYP |
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ABC |
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TpeyrOJIbHHKa ABC C BepWHHaMH A(O, 0), B(1, 0), C(0,1). |
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4.2.65. |
BbIqHCJIHTb |
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ydx - |
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r.n:e L - |
3JIJIHnC x |
2 + Y2 = |
1, npo6eraeMbIfi B nOJIO>KHTeJIbHOM |
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HanpaBJIeHHH. |
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4.2.66. |
BbIqHCJIHTb |
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ydy |
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no OKPy>KHOCTH C u;eHTpOM B HaqaJIe Koop.n:HHaT.
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4.2.67. |
Bbl'lHCJIHTb |
ydx + xdy |
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x2 +y2 |
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4.2.68. |
no OTpe3KY npHMoii y = X OT TO'lKHX = 1 ,Il;0 TO'lKHX = 2. |
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xdy - |
ydx |
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no neTJIe ,Il;eKapTOBa JIHCTa x = ~ y = 3at2 • |
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4.2.69. |
BbI'IHCJIHTb |
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1 + t3 ' |
1 + t3 |
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xdy - |
ydx, |
y = a(1 - cos t), °~ |
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= a(t - sin t), |
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r,Il;e |
L - apKa IJ;HKJIOH,Il;bI x |
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~ t |
~ 27r. |
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Hafi.mu tjjYH'lCqUU no ux nO.l&H'btM iJutjjtjjepeHqua.l!a.M:
4.2.70.dU = ydx + xdy.
4.2.71.dU = (cosx + 3x2y) dx + (x3 - y2) dy.
4.2.72. dU = |
x |
dx + |
Y dy. |
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Jx 2 + y2 |
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Jx 2 + y2 |
4.2.73.dU = (2x + y + z) dx + (x + 2y + z) dy + (x + y + 2z) dz.
4.2.74.dU = (3x2 + 2y2 + 3z) dx + (4xy + 2y - z) dy + (3x - y - 2) dz.
4.2.75.dU = (2xyz - 3y2 z + 8xy2 + 2) dx + (x2Z - 6xyz + 8x2y + 1) dy+ +(x2y - 3xy2 + 3) dz.
4.2.76. |
dU = (1 -~) dx + (1 -~) dy + (1 -lL) dz. |
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4.2.77. |
BbI'IHCJIHTbpa60TY CHJIbI F = |
xyi + (x + y)j npH nepeMemeHHH |
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TO'lKHH3 Ha'lanaKOOp,Il;HHaT B TO'lKYA(I, 1): |
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a)no npHMoii;
6) no napa60JIe y = x 2 .
4.2.78. B KroK,Il;oii TO'lKe9JIJIHnCa x = a cos t, y = bsin t npHJIO>KeHa CHJIa
F= -xi - yj.
a)BbI'IHCJIHTbpa60TY CHJIbI F npH nepeMemeHHH TO'lKHB,Il;OJIb )J;yrH 9JIJIHnCa, JIe>Kameii B nepBoii 'IeTBepTH.
6) BbI'IHCJIHTbpa60TY, ecJIH TO'lKa06XO,Il;HT BeCb 9JIJIHnc.
Hafi.mu pa60my iJaHHoeo CU.l!060eO nOM F npu nepeMe~eHuu MamepUa.l!'bHofi. mo",,'lCu6iJO.l!'b Ompe3'ICa AB, eC.l!u U36eCmH'bt 'lCOopiJuHam'bt 'lCOHq06 Ompe3'ICa:
4.2.79.F = (x4 + 4xy3)i + (6x2y2 - 5y2)j, A(-2, -1), B(3,O).
4.2.80. |
F= |
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2j ,A(O,-I),B(I,O). |
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4.2.81. |
F = |
(x + 2y). |
+ |
y . |
A( |
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2 1 |
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2 J , |
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1,1, |
3,1. |
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4.2.82. F = ( x |
+ y) i + ( Y + X) j, A(O, 0), B(I, 1). |
J x 2 + y2 |
J x 2 + y2 |
4.2.83.F = (3x 2y - y3)i + (x3 - 3xy2)j, A(O, 0), B(I, 1).
4.2.84.F = (.jY - 3 W) i+ (~ + Vx)j, A(I,4), B(9, 1).
4.2.85. |
F = |
(x 2 +y2)Y(arctg~) i - |
(x 2 +y2)X(arctg~)j, A(I, 1), B(2,2). |
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4.2.86. |
F = 22x 2 i + |
22y 2 j , A(2, 1), B(I, 7). |
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4.2.87.F = (y3 - 6xy2)i + (3xy2 - 6x2y + 8y3)j, A(O,O), B(I, 1).
KOHTponbHble Bonpocbl III 60nee CnO>KHble 3aAaHIIIR
4.2.88. |
1 dx - arctg ~ dy, r~e OmA - |
,rr,yra IIapa60JIbI y = x2, a |
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OmAnO |
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OnA - OTpe30K IIPjlMOii y = X. |
(3,-4) |
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4.2.89. |
.1xdy +ydx. |
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4.2.90. |
1xdx + ydy. |
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(1,2)y dx ~X dy , |
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4.2.91. |
(x -I 0). |
4.2.92. |
1xdx+ydy. |
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4.2.93. |
(1'''')(1 _y2 cos '#..) dx + (sin'#.. + '#.. cos '#..) |
dy |
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x2 |
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4.2.94. |
,il;oKa3aTb, '-ITOeCJIH f(u) - |
HeilpepbIBHM <PYHKIIHjI, TO |
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1f(x 2 + y2)(xdx + ydy) = 0
L
MjI JIlOooro rJI~KOrO KOHTypa L.
HatJ.mu nepaoo6pa3HY'lO rjjYH'It;'qU'lO no iJaHHO.MY iJurjjrjjepeH'qua.!!y:
4.2.95.dz = 2xdx + 2ydy.
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x2 + y2 |
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4.2.96. dz = - |
2ydx |
+ |
2ydy |
2 |
2' |
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x 2 sin2 J.. |
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x sin J.. |
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4.2.97. dz= (x -5y)2[-ydx+xdy].
217
4.2.98. |
BbPUlCJUlTb |
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j(y2 _ Z2) dx + 2yz dy - x 2dz, |
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= t3 , °~ t ~ 1, rrp06eraeMruI B |
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r,Il;e L - |
KpHBrui X = t, Y = t2 , Z |
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4.2.99. |
HarrpaBJIeHHH B03pacTaHHH rrapaMeTpa t. |
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j(y2 _ Z2) dx + (z2 _ x 2) dy + (x2 _ y2) dz, |
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r,Il;e L - KOHTYP, KOTOPblit OrpaHH'IHBaeT'IacTbc<l>epbI |
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x2 + y2 + Z2 = 1, x ~ 0, |
y ~ 0, z ~ 0, |
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rrpo6eraeMblit TaK, 'ITOBHeWHHH CTOpOHa c<l>epbI OCTaeTCH CJIeBa. |
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4.2.100. .II:oKa3aTb, 'ITO)];JIH KPHBOJIHHeitHoro HHTerpaJIa II pO,Il;a CrrpaBe,Il;- |
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JIHBa Ou;eHKa |
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IjP(x,y)dx + Q(X,y)dyl ~LM, |
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r,Il;e L - |
)];JIHHa KPHBOit L, a |
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M = max |
y'p2(x,y) + Q2(X,y). |
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(x,v)EL |
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4.2.101. |
Ou;eHHTb HHTerpaJI |
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ydx - xdy |
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(x |
2 + xy + y2)2 ' |
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r,Il;e L - |
OKP)')KHOCTbx 2 + y2 = R2 . .II:oKa3aTb, 'ITOrrpH R -t +00 |
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,Il;aHHblit HHTerpaJI CTpeMHTCH K HYJIIO. |
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§ 3. nOBEPXHOCTHbll4l11HTErPAl1
OnpE!AeneHllle III Bbl'"lIllCneHlllenOBepXHOCTHoro IIIHTerpana nepBoro POAa
~ IIycTb f(x, y, z) - <PYHKIIHH, 3a,n;aHHaH B TO'IKaxHeKoTopoil: rJIa,!l;Koil: rrOBepx-
HOCTH 8. PacCMOTPHM pa36HeHHe rrOBepXHOCTH 8 Ha 'IacTH81, ... , 8 n C IIJIOIIIa,n;HMH
~0"1, ••• , ~O"n H ,!l;HaMeTpaMH d 1 , •• • , dn COOTBeTCTBeHHO. IIpOH3BOJIbHO BbI6paB Ha KWK,!l;Oil: H3 'IacTeil:8i TO'IKYMi(Xi, Yi, Zi), COCTaBHM CYMMY
n
L f(Xi, Yi, Zi) . ~O"i,
i=l
KOTOPaH Ha3bIBaeTCH tJ.Hme2paJIbHoit CYMMOit nepao2o poiJa)l;JIH <PYHKIIHH f(x, y, z).
ECJIH rrpH d -t 0 (r,!l;e d = m~di), cYIIIecTByeT rrpe,!l;eJI HHTerpaJIbHbIX CYMM, KOTOPbIiI: He 3aBHCHT OT crroco6a pa36HeHHH rrOBepXHOCTH 8 Ha 'IacTHH BbI60pa
218
ToqeK Mi, TO 9TOT npe,Il;e.rr Ha3bIBaeTCH n06epXXOCmX'btM UXmeepaJIOM nep60eO pOaa
II 0603HaqaeTCH
111(x, y, z) da. s
ECJIH <PYHKU;HH I(x, y, z) HenpepblBHa, TO HHTerpa.rr
111(x, y, z) da s
cyIIlecTByeT.
Onpe,IJ;e.rreHHe nOBepXHOCTHoro HHTerpa.rra nepBoro pO,Il;a aHa.rrorHqHO onpe,Il;e- JIeHHlO KpHBOJIHHeil:Horo HHTerpa.rra nepBoro pO,Il;a. CBoil:cTBa nOBepXHOCTHoro HHTerpa.rra nepBoro pO,Il;a (JIHHeil:HocTb, a,IJ;,Il;HTHBHOCTb H T.,Il;.) TaK:lKe aHa.rrorHqHbl co-
OTBeTCTBYlOIIlHM CBoil:cTBaM KPHBOJIHHeil:Horo HHTerpa.rra nepBoro pO,Il;a.
ECJIH nOBepXHOCTb S 3a,Il;aHa Ha 06JIacTH D nJIOCKOCTH Oxy <PYHKu;Heil: z = npHqeM z(x, y) HenpepblBHa BMeCTe CO CBOHMH '1acTHblMHnpOH3BO,Il;HblMH z~ = z~(x, y) H z~ = z~(x, y), TO nOBepXHocTHblil: HHTerpa.rr CBO,Il;HTCH K ,Il;BOil:HOMY C
IIOMOru;blO <P0PMYJIbl:
III(x,y, z) da = 111(x, y, z(x, y»Jl + (z~)2 + (z~)2dxdy.
S |
D |
ECJIH nOBepXHOCTb S 3a,IJ;aHa napaMeTpHqeCKH B BH,Il;e X = x(u,v), y = y(u,v), |
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z = z(u,v), r,Il;e X, y, z - |
HenpepblBHO ,Il;H<p<pepeHU;HPyeMble <PYHKU;HH B HeKoTopoil: |
06JIacTH G ITJIOCKOCTH OUV, TO |
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111(x, y, z) da = II f[x(u, V), y(u, V), z(u, v)h/EH - F2 dudv, |
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S |
G |
F = ax ax + ayay + az ~. au av au av au av
nplllnO>KeHIIIH nOBepXHOCTHoro IIIHTerpana nepBoro pOAa
IIycTb S - rJIa,IJ;KaH MaTepHa.rrbHaH nOBepXHOCTb C ITJIOTHOCTblO P = p(x,y,z).
Tor,Il;a C nOMOIIlblO nOBepXHOCTHblX HHTerpa.rrOB nepBoro pO,Il;a MO:lKHO BblqHCJIHTb:
1) CTaTHqeCKHe MOMeHTbl 9TOil: nOBepxHocTH OTHOCHTe.rrbHO KOOp,Il;HHaTHblx
IIJIOCKocTeil:
Mxy = Ilzpda, |
Myz = Ilxpda, |
Mxz = II ypdaj |
S |
S |
S |
2) KOOp,Il;HHaTbl u;eHTpa TH:lKeCTH nOBepxHocTH
Myz |
Mxy |
= Ilpda j |
Xc = --;;n:-, |
Zc = --;;n:-, r,Il;e m |
S
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