1. Интегралы ФНП Диф_ур (853736), страница 28
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F (x y z ) = 0: 4 , @y = 1@ ) @y @z @x = ;1:a) @x@y @x@z @x @y< J x y z (.. x = x(y z )), :@x = ; Fy0 :@yFx0429D, y z x @y = ; Fx0 :@xFy0=0101@x @y = @; Fy0 A @; Fx0 A = 1@y @xFx0Fy0(, (x y z ) ).D :@y = ; Fz0 @ @z = ; Fx0 @ @x = ; Fy0 :@zFy0 @xFz0 @yFx0 9 , :010101@y @z @x = @; Fz0 A @; Fx0 A @; Fy0 A = ;1: >@z @x @yFy0Fz0Fx0 1)2)1)2)@z =? @z =?x2 ; 2y2 + z2 ; 4x + 2z ; 5 = 0 @x@yx2 + y2 + z2 = 1 @z =? @z =?a2 b2 c2@x@y@z = 2 ; x @x z + 1@z = ; c2x @xa2z@z = 2y :@y z + 1@z = ; c2y :@yb2 z|||||430 12 .
z = f (x y) { , , , , , . =@z = f 0 (x y) @z = f 0 (x y)@x x@y y 1- , 2- :@ 2z = @ @z ! @ 2z = @ @z ! @ 2z = @ @z ! @ 2z = @ @z !@x2 @x @x @y@x @y @x @x@y @x @y @y2 @y @y fxx00 (x y) f xyyx00 (x y) f yxxy00 (x y) fyy00 (x y):D 3- .E , , . 2 . 5, @ 3z @ 3z@x2@y@x@y@x , @ 3z @ 3z@x2@y@x@y2 x y:4312 , , ( ).2z2z@@3235 12.1. z = x + xy ; 5xy + y : , @x@y = @y@x :< 5 :@z = 3x2 + y2 ; 5y3@ @z = 2xy ; 15xy2 + 5y4:@x@y4 , :@ 2z = @ (2xy ; 15xy2 + 5y4) = 2y ; 15y2@@x@y @x@ 2z = @ (3x2 + y2 ; 5y3) = 2y ; 15y2:@y@x @y@ 2z = @ 2z : >2, @x@y@y@x3z3z@@xy 12.2.
z = e : 5 @x2@y @y@x2 :@z = exy 2xy = 2xyexy @< 7 @y@ 2z = @ (2xyexy ) = 2yexy + 2xyexy y2 = 2(y + xy3 )exy @@x@y @x@ 3z = @ 2(y + xy3 )exy = 2y3exy + 2(y + xy3 )exy y2 =@x2@y @x= 2y3(2 + xy2 )exy :-, x y, 9, , :@ 3z = @ 3z = 2y3(2 + xy2 )exy : >@y@x2 @x2@y2222222222224323w@ 12.3. w5 @x@y@z :xyz xy = xyexyz @< 7 @w=e@z= exyz :@ 2w = @ (xyexyz ) = xexyz + xy exyz xz = (x + x2yz )exyz @@y@z @y@ 3w = @ (x + x2yz )exyz = (1+2xyz )exyz +(x + x2 yz )exyz yz =@x@y@z @x= (x2 y2z 2 + 3xyz + 1)exyz : >2u @ 2u1@ 12.4. u = ln px2 + y2 : , @x2 + @y2 = 0:< F u = ; 12 ln(x2 + y2):5 :@u = ; x @@xx2 + y2@u = ; y :@yx2 + y24 , :@ 2u = @ ; x ! = ; x2 + y2 ; 2x x = x2 ; y2 @@x2 @x x2 + y2(x2 + y2)2(x2 + y2)2@ 2u = @ ; y ! = ; x2 + y2 ; 2y y = y2 ; x2 :@y2 @y x2 + y2(x2 + y2)2(x2 + y2)2 :x2 ; y2 + y2 ; x2 0:(x2 + y2)2 (x2 + y2 )2 , ..
u . >4332y2y@@2 12.5. y = '(x ; at)+ (x + at): , @t2 = a @x2 ' :< 5 :@y = '0 (x ; at) @ (x ; at) + 0 (x + at) @ (x + at) =@t@t@t= '0 (;a) + 0 a = a(0 ; '0 )@@ 2y = a 00 @ (x + at) ; '00 @ (x ; at)! =@t2@t@t= a(00 a ; '00 (;a)) = a200 + a2'00 @@y = '0 @ (x ; at) + 0 @ (x ; at) = '0 + 0 @@x@x@x@ 2y = '00 1 + 00 1 = '00 + 00 :@x2 :a200 + a2'00 = a2(00 + '00 ):: , .. y , ' { .4 , y = '(x ; at) + (x + at)& .2 , , a: 5, a > 0 y = sin(x ; at) , , y = sin(x + at) { , @ y = sin(x ; at) + sin(x + at) { , .
>434 @ 2z = @ 2z z = xy : 4 $, @x@y@y@x3@z 2) z = ln(x2 + y 2): )@x@y21@ 2u + @ 2u + @ 2u = 03) z = p 2:4 $,@x2 @y2 @z2x + y2 + z2!2u2 @1@a@u24) u = 9'(ax + y ) + (ax ; y )] : 4 $, y@x2 = y2 @y y @y :1)2);4x(3y 2 x2):(x2 + y 2)3||||{ ! z = f (x y) (x0 y0): = 9 dz = fx0 (x y)dx + fy0 (x y)dy x y dx dy:. 9 , , , dz 2- d2z = d(dz ):D 3- .4 n- n- !n@@nd z = @x dx + @y dy z (n = 1 2 :::)435 z . 5,2z2z2z@@@22d z = @x2 dx + 2 @x@y dxdy + @y2 dy2:(F , 2- 122@ z dxdy = @ z dxdy:A 9 @y@x@x@y. z = f (x y) (x0 y0) (n +1) , '2n4z = dz(x1!0 y0) + d z(x2!0 y0) + ::: + d z(nx!0 y0) + Rn+1 4z = f (x0 + dx y0 + dy) ; f (x0 y0) { , dx dy Rn+1 , n+1 z (x + dx y + dy )d00Rn+1 = 0 < < 1 ( L)(n + 1)!qRn+1 = o(n) = dx2 + dy2 ! 0 ( ).
12.6. 5 = z = x2++3xy ; y3 M0(2 ;1):< 5 @z dx + @z dy = (2x + 3y)dx + (3x ; 3y2)dydz = @x@y2z2z2z@@@22d z = @x2 dx + 2 @x@y dxdy + @y2 dy2 = 2dx2 + 6dxdy ; 6ydy2 3z3z3z3z@@@@3232d z = @x3 dx + 3 @x2@y dx dy + 3 @x@y2 dxdy + @y3 dy3 = ;6dy3 dnz = 0 n = 4 5 :::4362, n 4 .5 M0(2 ;1):dz (2 ;1) = dx + 3dyd2z (2 ;1) = 2dx2 + 6dxdy + 6dy2 d3z (2 ;1) = ;6dy3 :7, = 4z = dz(2 ;1) + 21 d2z(2 ;1) + 16 d3z(2 ;1)4z = dx + 3dy + dx2 + 3dxdy + 3dy2 ; dy3: dx = x ; 2 dy = y + 1 (x ; 2) (y + 1):4z = (x;2)+3(y +1)+(x;2)2 +3(x;2)(y +1)+3(y +1)2 ;(y +1)3: >: , dx = x ; x0 dy = y ; y0 , = f (x y) (x ; x0) (y ; y0): 12.7.
? z = xy (x ; 1)(y ; 1) .7 ( !) 1 1102:< F z x0 = 1 y0 = 1 z0 = 1:1 1102 = z (1 1@ 1 02) = z (1 1) + 4z = z0 + 4z:: .2 , :@z dx + @z dy = yxy;1 dx + xy ln xdydz = @x@y2z2z2z@@@22d z = @x2 dx + 2 @x@y dxdy + @y2 dy2 == y(y ; 1)xy;2 dx2 + 2xy;1 (y ln x + 1)dxdy + xy ln2 xdy2 4373z3z3z3z@@@@322= @x3 dx + 3 @x2@y dx dy + 3 @x@y2 dxdy + @y3 dy3 == y(y;1)(y;2)xy;3 dx3 +3N(y;1)xy;2 +yxy;2 +y(y;1)xy;2 ln x]dx2dy++3Nyxy;1 ln2 x + 2xy;1 ln x]dxdy2 + xy ln3 xdy3 (1,1):d3zdz = dx d2z = 2dxdy d3z = 3dx2dy:7 =, y ; 1) + ::: + 3(x ; 1)2(y ; 1) + R 4z = x 1!; 1 + 2(x ; 1)(42!3!4z (x ; 1) + (x ; 1)(y ; 1) + 21 (x ; 1)2(y ; 1):5 4z x = 1 1 y = 1 02:4z (1 1 ; 1)+(1 1 ; 1) (1 02 ; 1)+ 21(1 1 ; 1)2(1 02 ; 1) = 0 1021:2,1 1102 = z0 + 4z 1 + 0 1021 = 1 1021: > 12.8.
, @ 2u + @ 2u = @ 2u + 1 @ 2u + 1 @u @x2 @y2 @2 2 @'2 @ x = cos ' y = sin ':< C 2 .2u@u@u@5 @ @2 @'2 :@u = @u @x + @u @y = @u cos ' + @u sin '@@ @x @ @y @ @x@y@ 2u = @ @u cos ' + @u sin '! = @ 2u cos2 '+@2 @ @x@y@x24382u2u2u@@@+ @x@y cos ' sin ' + @y@x sin ' cos ' + @y2 sin2 '@@u = @u (; sin ') + @u cos '@@' @x@y@ 2u = @ ; @u sin ' + @u cos '! =@'2 @' @x@y2u2u@@22= @x2 sin ' ; @y@x 2 sin ' cos ' ; @u@x cos ';2u2u@@2; @x@y cos ' sin ' + @y2 2 cos2 ' ; @u@y sin ': :@ 2u + 1 @ 2u + 1 @u = @ 2u cos2 '+@2 2 @'2 @ @x20 22u2u@@1+2 @x@y sin ' cos ' + @y2 sin2 ' + 2 @ @@xu2 2 sin2 ';12u2u@@@u@u; @x cos ' ; 2 @x@y 2 sin ' cos ' + @y2 2 cos2 ' ; @y sin 'A +!1@u@u+ @x cos ' + @y sin ' =@ 2u (cos2 ' + sin2 ') + @ 2u (cos2 ' + sin2 ') = @ 2u + @ 2u := @x2@y2@x2 @y2= , .
> ;7' (2,1).f (x y) = x3 ; 2y3 + 3xy #<$ * 6# =# f (x y) = 12 + 15(x ; 2) + 6(x ; 2)2 + 3(x ; 2)(y ; 1) ; 6(y ; 1)2 + (x ; 2)3 ; 2(y ; 1)3:||||{439 13 " z = f (x y) M0(x0 y0) 4z = f (x y) ; f (x0 y0){ , M0(x0 y0) M (x y):. _(M0) , 8M 2 _(M0) 4z < 0 ( 4z > 0) M0(x0 y0) ( ) f (x y) f (x0 y0) 9 ( ) f (x y):;; f (x0 y0) = max zf (x0 y0) = min z. 13.1= ! , { ! .# ". z = f (x y) 9 M0(x0 y0) a 9 , fx0 (x0 y0) = 0 fy0 (x0 y0) = 0440(13:1), , dz (x0 y0) = 0 dx dy:=, (13.1) f (x y):: , z = f (x y) 9 M0(x0 y0) M0 { , 9 (& ! ).( ! , 2- .(x0 y0) { ,00A = fxx(x0 y0) B = fxy00 (x0 y0) C = fyy00 (x0 y0)AC ; B 2>0<0=0A ( C ) > 0 < 0 88extrmin max 9Q9 9Q.
AC ; B 2 = 0 , 9, ( 4z = , , , 9).$ "4 9 , 9, , 9 . 13.1. 7 9 z = x3 + 3xy2 ; 15x ; 12y:< 5 1- @z = 3x2 + 3y2 ; 15 @z = 6xy ; 12@x@y (13.1):8 2 2< x + y ; 5 = 0: xy ; 2 = 0:441: x = 2y , :2 !2 + y2 ; 5 = 0 () (y2 )2 ; 5y2 +4 = 0 () (y2 ; 1)(y2 ; 4) = 0:y2 9 :8>< (y ; 1)(y + 1)(y ; 2)(y + 2) = 02>: x = y:J 9 , :M1(1 2) M2(;1 ;2) M3(2 1) M4(;2 ;1):E 2- :@ 2z = 6x @ 2z = 6y @ 2z = 6x:@x2@x@y@y22 D = AC ; B 2 .4 M1 :2 z 2z 2 z @@@A = @x2 = 6 B = @x@y = 12 C = @y2 = 6@MMM111D = 6 6 ; 122 = ;108 < 0:2, 9 M1 .4 M2 :2 z 2z 2 z @@@A = @x2 = ;6 B = @x@y = ;12 C = @y2 = ;6@MMMD = 36 ; 144 = ;108 < 0:: M2 9 .4 M3 :2 z 2 z 2 z @@@A = @x2 = 12 B = @x@y = 6 C = @y2 = 12@MMM223234423D = 144 ; 36 = 108 > 0 A > 0:2, M3 , zmin = z x = 23 + 3 2 12 ; 15 2 ; 12 1 = ;28:=2y=14 M4 :2 z 2 z 2 z @@@A = @x2 = ;12 B = @x@y = ;6 C = @y2 = ;12@MMMD = 144 ; 36 = 108 > 0 A < 0:9 M4 , zmax = z x ; = (;2)3 + 3 (;2) (;1)2 ; 15 (;2) ; 12 (;1) = 28: >444= 2y=;1 13.2.
5 9 z = 4(x ; y) ; x2 ; y2 :< 5 :@z = 4 ; 2x @z = ;4 ; 2y:@x@y: 9, :8< 4 ; 2x = 0: ;4 ; 2y = 0 x = 2 y = ;2@ M (2 ;2):F 2- M :@ 2z = ;2 B = @ 2z = 0 C = @ 2z = ;2:A = @x2@x@y@y22 AC ; B 2 = (;2) (;2) ; 0 = 4 > 0@ A < 0:2, M (2 ;2) :zmax = 4 (2 + 2) ; 22 ; (;2)2 = 8: >443 13.3. 7 9 z = x4 + y4 ; 2x2 ; 4xy ; 2y2p p M1( 2 2) M2(0 1):< 2 9 M1 :@z = 4x3 ; 4x ; 4y = 4 (p2)3 ; 4p2 ; 4p2 = 8p2 ; 8p2 = 0@@x MM11@z = 4y3 ; 4x ; 4y = 4 (p2)3 ; 4p2 ; 4p2 = 8p2 ; 8p2 = 0@@y MM2, M1 .5 2- :11= 12x2 ; 4 = 20@M1M1@ 2z = ;4@@x@y M12@ z = 12y2 ; 4 = 20:@y2 M1M1@ 2z @x2 7, AC ; B 2 = 400 ; 16 = 384 > 0 A > 0 ) M1 .4 M2 :@z = 4 0 ; 4 0 ; 4 1 = ;4 6= 0@@x M@z = 4 13 ; 4 0 ; 4 1 = 0:@y M5 9 .
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