1. Интегралы ФНП Диф_ур (853736), страница 27
Текст из файла (страница 27)
- :1) u = x2y3 z 4:@u = x2 3y2z 4 = 3x2y2z 43 4< @u=2xyz@x@y@u = x2y3 4z 3 = 4x2y3z 3: >@z2) u = x3y + y2z + xz 2:@u = x3 + 2yz22< @u=3xy+z@x@y@u = y2 + 2xz: >@z3) u = yz=x:!z=x ln y @u zz=x@u1zyzyz=xz=x;1< @x = y ln y z ; x2 = ; x2 @y = x y= xy @u = yz=x ln y 1 = yz=x ln y : >@zxx415 # -/:1) u = xy + yz + zx3) u = xy=z :2) u = sin(x2 + y2 + z2)@u = y + z@u = x + z@u = x + y1) @x@y@z@u@u2222) @x = 2x cos(x2 + y2 + z2) @y = 2y cos(x2 + y2 + z2) @u@z = 2z cos(x + y + z )@u = y xy=z;1 @u = 1 xy=z ln x@u = ; y xy=z ln x:3) @xz@y z@zz2|||||! ! ! z = f (x y) (x y) 4z = f (x + 4x y + 4y) ; f (x y){ , x y, 4x 4y:@ 6 4z = A4x + B 4y + o() ! 0 A= A(x y) B = B (x y) { , 4x 4y p = 4x2 + 4y2 o() { ( ! 0 3 * z = f (x y) (x y ) 4x 4y A4x + B 4y z = f (x y) (x y) ( dz :dz = A4x + B 4y:A * * dx = 4x0 dy = 4y:416 $ @z dx + @z dy:dz = @x@y 10.3.
- z = x2y4 ; x3y3 + x4y2:< - @z = 2xy4 ; 3x2y3 + 4x3y2 @z = 4x2y3 ; 3x3y2 + 2x4y:@x@y@z dx + @z dy =), dz = @x@y= (2xy4 ; 3x2y3 + 4x3y2 )dx + (4x2y3 ; 3x3y2 + 2x4y)dy: > 10.4. - qz = x + y ; x2 + y2 x = 3 y = 4 4x = 0 1 4y = 0 2:< 5 , dx = 4x dy = 4y @z@zdz = @x 4x + @y 4y =x=3y =4x=3y =4! ! 2y2x= 1 ; 2px2 + y2 4x + 1 ; 2px2 + y2 4y =!!34= 1 ; p 2 2 0 1 + 1 ; p 2 2 0 2 = 0 08: >3 +43 +42( , dz x y 4x 4y:x=3y =4x=3y =4 1) ) ()+ yz = xx ;y2) - ) ()z = x2 xy; y2 x = 2 y = 1 4x = 0 01 4y = 0 03:4171); ydx) ; y)22(xdy(x2)1:36||||| jdxj jdyj 4z dz . 10.5. " R = 4 H = 10 .' ( R 0,1 H 0,25 ?< -( V = R2 H R H .
. R 4R H 4H , V 4V:@V 4H = 2RH 4R + R2 4H =4V dV = @V4R+@R@H= (2RH 4R + R24H ) = (2 4 10 0 1 + 42 0 25) = 12:2, ( 12 3: > 1) R = 30 , r = 20 , h = 40 . !" , # #$ R 0,3 , r 0,4 , h 0,2 ?2) & 10 2 0 1 , ! '( 44 6 0 1 .) !" $ *+, *.1) -# *!# #$ 2575 3.2) 4730 100 3.|||||4184 u = f (x y z ) @u dy + @u dz:dx+du = @u@x@y@z 10.6.
5 .1) u = x2y3 z 4:< 5 @u = 2xy3 z 4 @u = 3x2y2z 4 @u = 4x2y3z 3:@x@y@z@u dy + @u dz =dx+2, du = @u@x@y@z= 2xy3z 4dx + 3x2y2 z 4dy + 4x2y3 z 3dz == xy2z 3(2yzdx + 3xzdy + 4xydz ): >2) u = xy=z :< 5 @u = y xy=z ;1 @u = xy=z ln x 1 = 1 xy=z ln x@x z@yz z@u = xy=z ln x y ; 1 ! = ; y xy=z ln x:@zz2z27,@u dx + @u dy + @u dz =du = @x@y@z= yz xy=z ;1dx + z1 xy=z ln xdy ; zy2 xy=z ln xdz =y=zx= xz 2 (yzdx + xz ln xdy ; xy ln xdz ) : >419 11 . F (x y z ) = 0:. F (x y z ) M0(x0 y0 z0) M0 , 9 . :~n = Fx0 (x0 y0 z0)~i + Fy0 (x0 y0 z0)~j + Fz0 (x0 y0 z0)~k , Fx0 (x0 y0 z0)(x ; x0) + Fy0 (x0 y0 z0)(y ; y0) + Fz0 (x0 y0 z0)(z ; z0) = 0 {x ; x0 = y ; y0 = z ; z0 :Fx0 (x0 y0 z0) Fy0 (x0 y0 z0) Fz0 (x0 y0 z0) 11.1.
5 . M (1 2 ;1) , x3 + y3 + z 3 + xyz ; 6 = 0:< = M (1 2 ;1) , :13 + 23 + (;1)3 + 1 2 (;1) ; 6 = 0:? F (x y z ) = x3 + y3 + z 3 + xyz ; 6 R3 :5 M22x M = (3x + yz )M = 3 1 + 2 (;1) = 1@F 0 42022y M = (3y + xz )M = 3 2 + 1 (;1) = 11@02Fz M = (3z + xy)M = 3 (;1)2 + 1 2 = 5:F 0 2, ~n=(1 11 5) M , (x ; 1)+11(y ; 2)+5(z +1) = 0 ( )x + 11y + 5z = 18@ x ; 1 = y ; 2 = z + 1: >1115 11.2. 5 , z = 2x2 ; 4y2 .
M0(2 1 4):< A : 2x2 ; 4y2 ; z = 0:4 11.1:Fx0 M = 4xM = 4 2 = 8@y M = ;8y M = ;8@0Fz M = ;1 )F 0 x ; 2 = y ; 1 = z ; 4:8;8 ;1A 8(x ; 2)++(;8)(y ; 1) + (;1)(z ; 4) = 0 8x ; 8y ; z = 4:7, x ; 2 = y ; 1 = z ; 4@8;8 ;1 8x ; 8y ; z = 4: > 11.3. ' 9 x2 + y2 + 2z 2 = 1 , x ; 2y + z = 0:421< M0(x0 y0 z0) 9. = x ; 2y + z = 0 .4 (2x0 2y0 4z0) . 2, (1 ;2 1) ( x;2y+z = 0) (2x0 2y0 4z0)..2x0 = 2y0 = 4z0 2x = ;y = 4z :0001 ;2 1A , M0(x0 y0 z0) 9 (, , x20 + y02 + 2z02 = 1), (x0 y0 z0) :8< 2x0 = ;y0 = 4z0: x20 + y02 + 2z02 = 18< x0 = 2z0@ y0 = ;4z01 :p)z=0: (2z0)2 + (;4z0)2 + 2z02 = 122!!41241(1) p2(2)7 M0 ; p p M0 ; p p ; p2222 2222 2222, , 2 B1 B2 :!!!284214B1 : p x ; p ; p y + p + p z ; p = 0@222222222222!!!428421B2 : ; p x + p + p y ; p ; p z + p = 0 :222222222222A, :B1 : p2 x ; p4 y + p1 z = 21222222 222x ; 4y + z = p21 @22B2 : ; p2 x + p4 y ; p1 z = 21222222 22422;2x + 4y ; z = p2122 @= , B1 B2 2x ; 4y + z = p21 :225 .
x ; 2y + z = 0 9, , , x ; 2y + z = 0: > 11.4. x2 ; y2 ; 3z = 0: , x ; 1 = y + 3 = z ; 5:423< M0(x0 y0 z0): = (2x0 ;2y0 ;3):C (4,2,3) { . 92x0 = ;2y0 = ;3 ) x = ;2@ y = 1:00423= z0 x20 ; y02 ; 3z0 = 0( M0(x0 y0 z0) , , x2 ; 2y2 ; 3z 2 = 0). 7: z0 = 1~n = (2x0 ;2y0 ;3) = (;4 ;2 ;3) :;4(x + 2) ; 2(y ; 1) ; 3(z ; 1) = 0 4x + 2y + 3z = ;3: > 1) ) #$ *# # *0 3x4 ; 4y 3z +2+4z xy ; 4z 3 x + 1 = 0 (1 1 1):2) 2# *0 z = xy *$ #$ *#, **# * x + 2 = y + 2 = z ; 1:21;1423; 2y ; 2z + 1 = 0 x ;3 1 = y;;21 = z;;21 :2x + y ; z = 2:1) 3x2)||||| . z = f (x y) { , x = '(u v) y = (u v) z = f ('(u v) (u v)) , :@z = @z @x + @z @y @z = @z @x + @z @y :(11:1)@u @x @u @y @u @v @x @v @y @v: x y , u v { @ f (x y) , '(u v) (u v) { .@z 7 (11.1) , @u u @z @z x y @x@y@y @x@u @u u: D @z : @vA ( ).
: : , ! .424 11.5. w = w(x y z ) x = x(u v) y = y(u v) z = z (u v):= w u v { x y z:< E @w = @w @x + @w @y + @w @z @u @x @u @y @u @z @u@w = @w @x + @w @y + @w @z :@v @x @v @y @v @z @vF , , , . > 11.6. z = z (x y)@ x = x(t)@ y = y(t): G(t) = z (x(t) y(t)): 5 ddtG :< = G t . 9 "d":dz = dG = @z dx + @z dy : >dt dt @x dt @y dtd (z (x y(x))): 11.7. z = z (x y)@ y = y(x): 5 dx< z (x y(x)) { , 9dz = d (z (x y(x))) = @z dx + @z dy =dx dx@x dx @y dx@z 1 + @z dy = @z + @z dy : >= @x@y dx @x @y dx 11.8.
w = w(x y z ) z = z (x y): 5@ (w(x y z (x y)))@ (w(x y z (x y))) :@x@y425< 2 :@w ! = @ (w(x y z (x y))) = @w @x + @w @y + @w @z =@x n @x@x @x @y @x @z @x@w 0 + @w @z = @w + @w @z :1+= @w@x@y@z @x @x @z @xD @w ! = @ (w(x y z (x y))) = @w + @w @z : >@y n @y@y @z @y!!@w@wE @x @y nn . - w x y , z x y ( @w @w w@x x @y y: 11.9. z = f (u v)@ u = x2y2@ v = exy :@z @z :5 @x@y< F , 9 :@z = @f @ (x2 ; y2) + @f @ (exy ) = @f 2x + @f exy y@@x @u@x@v @x@u@v@z = @f @ (x2 ; y2 ) + @f @ (exy ) = ;2y @f + exy x @f :@y @u@y@v @y@u@v@f -, @f@u @vu = x2 ; y2 v = exy : > 11.10. , '(t) { , z = '(x2 + y2) @z ; x @z = 0:y @x@y426< 7:@z = '0 (x2 + y2) 2x@ @z = '0 (x2 + y2 ) 2y:@x@y(F '0(x2 + y2) { '(t) t = x2 + y2:)=@z ; x @z = y '0(x2 + y2) 2x ; x '0(x2 + y2) 2y = 0: >y @x@y u = sin x + F (sin y ; sin x) + F (t) { -#! 667 67@u cos x + @u cos y = cos x cos y ! 67 F:4 $, @y@x@z @z : z = x2y ; y2x + x = u cos v y = u sin v:2) ) * @u @v1) 4$.2)@z = 3u3 sin v cos v(cos v ; sin v) @z = u3(sin v + cos v)(1 ; 3 sin v cos v):@u@v||||| # , .
F (x y) = 0 y(x): (C , y(x) y F (x y(x)) 0:) =, , 9 dy = ; Fx0 (x y) :(11:2)dxFy0 (x y) (11.2) , y = y(x):2ydyd 11.11. 5 dx dx2 xy ; ln y = e:427< F F (x y) = xy ; ln y ; eFx0 (x y) = y Fy0 (x y) = x ; 1y = xy y; 1 : (11.2) dy = ; y2 = y2 @dxxy ; 1 1 ; xy01d2y = d @ y2 A = (1 ; xy)d(y2 )=dx ; y2 d(1 ; xy)=dxdx2 dx 1 ; xy(1 ; xy)2A, y = y(x) d (y2 ) = 2y dy = 2y3 dxdx 1 ; xyd (1 ; xy) = ;y ; x dy = ;y ; xy2 = ; ydxdx1 ; xy1 ; xyd2y = 2y3 + (y3)=(1 ; xy) = 3y3 ; 2xy4 : >dx2(1 ; xy)2(1 ; xy)3dy y(x) 11.12.
5 dxsin(xy) ; exy ; x2y = 0:< (11.2) :dy = ; @ (sin(xy) ; exy ; x2y)=@x =dx@ (sin(xy) ; exy ; x2y)=@yxy ; 2xyxy ; y cos xyycosxy;ye2xy+ye= ; x cos xy ; xexy ; x2 = x cos xy ; xexy ; x2 : > y (x) # y ; sin(x + y) = 0:dy # x3y ; y3x = 1:2) )dx1) )004283) )1)dy # xey + yex ; exy = 0:dx; (1 ; cos(yx + y))2 2)3x2 y3xy 2; y3 ; x3|||||3)yexy ; yex ; ey :xey + ex ; xexy F (x y z ) = 0 z = z (x y): = 9 , , @z = ; Fx0 (x y z ) @z = ; Fy0 (x y z ) :(11:3)@xFz0 (x y z ) @yFz0 (x y z ) 11.13.
z (x y) @z @z : ez ; xyz = 0: 5 @x@y< (11.3): F (x y z ) = ez ; xyz: =Fx0 = ;yz Fy0 = ;xz Fz0 = ez ; xy: =@z = ; ;yz = yz @ @z = ; ;xz = xz :@xez ; xy ez ; xy @yez ; xy ez ; xy@z @z -, ,? @x@y (x y z ) ez ; xyz = 0: = ez = xyz@z = yz = z @ @z = xz = z : >@x xyz ; xy xz ; x @y xyz ; xy yz ; y 11.14.
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