1. Интегралы ФНП Диф_ур (853736), страница 9
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dx dy dz ., M (x0 + dx y0 + dy z0 + dz ) M0M = dxi + dyj + dzk0 dx + Fy0 dy + Fz0 dzFnM(0M )xcos ' = jnj jM M j =jnj 0p = dx2 + dy2 + dz 2:,# ## # M0 { "# # %, (# $ Fx0 Fy0 Fz0 12.2/#0 F (x y z ) //0. . F ( /#0 F = dF + o() ! 0 dF = Fx0 dx + Fy0 dy + Fz0 dz:& ( ,F = F (x0 + dx y0 + dy z0 + dz ) ; F (x0 y0 z0) = 0 ; 0 = 0# ## # M (x0 + dx y0 + dy z0 + dz ) M0(x0 y0 z0) % % % # %F (x y z ) = 0: , dF = ;o() 1 o() :cos ' = jndF=;j jnj o() = 0 ,# ## ! 0 M ! M0 lim!0 limcos'=0lim'=M !MM !M2 "' #$'.,# "$, # n (15.1) (' ' # % F (x y z ) = 0 # M0(x0 y0 z0):#$ 15.3. # # % #( #$ , .( # #( ' # % ( # # % $ #.14100B# 3 ' x ; x0 = y ; y0 = z ; z0 :(15:2)Fx0Fy0Fz0#$ 15.4.
$ # % #( # $ #', # % $ # # # % ( #.=$ 15.4 , '( # # % # M0 # #'( # # % ( #, 3' Fx0 (x ; x0) + Fy0 (y ; y0) + Fz0 (z ; z0) = 0:(15:3). 8 % (15.2) (15.3) $ # M0(x0 y0 z0):? # % "(, ( # '( # % # 3 .', , ',%' 3 ' #'( #.@, %x2 + y2 ; z 2 = 0F (x y z ) = x2 + y2 ; z 2 Fx0 = 2x Fy0 = 2y Fz0 = ;2z:8 # O(0 0 0) Fx02 + Fy02 + Fz02 = 0 # ". ;# ## ( # (.
. 15.2), , ' # % # ( # , # 3 ## #'( #.;. 15.2142 ' /#0 z = F (x y) . $ # M00 (x0 y0): ? /# %'F (x y z ) = f (x y) ; z = 0:? 3' f (x0 y0) = z0 # M0(x0 y0 z0) " #( %, M00 (x0 y0) { #0 #' Oxy (. .15.3).;. 15.3H $Fx0 = fx0 Fy0 = fy0 Fz0 = ;1 # M0(x0 y0 z0) :1) 62) Fx02 + Fy02 + Fz02 = fx02 + fy02 ; 1 6= 0:&', M0 { "# # % F (x y z ) (.
15.2) . #' #' # % ( #. ? fx0 (x0 y0)(x ; x0) + fy0 (x0 y0)(y ; y0 ) ; (z ; z0) = 0:143 # M 0(x1 y1) # Oxy "$ #M00 (x0 y0) . ( # M1(x1 y1 z1) #'(# M (x1 y1 f (x1 y1)) 0 /#0 (.. % F (x y z ) = 0).B # M1(x1 y1 z1) #'( #:fx0 (x0 y0)(x ; x0) + fy0 (x0 y0)(y ; y0) ; (z ; z0) = 0:(15:4)I ', # M 0 $' .($ # M00 $ . , dx = x1 ; x0 dy = y1 ; y0, $ (15.4) fx0 (x0 y0)dx + fy0 (x0 y0)dy = z1 ; z0, ', ( //0 /#0 z = f (x y) #(x0 y0)dz = z1 ; z0 $ # # M1 M0 # 3 ' ## . # # M1 #'( # .
#0 M 0 # Oxy $ # M00 :4 , dz /#0 z = f (x y) # M00 M1 # /# ( /#0 # M0 , , dx dy.J, $' # # M M0 3 . /#0 z = f (x1 y1);f (x0 y0) ( '# //0 dz (. . 15.3).H" ' #( , $( , ' l $ # M00 M 0 (. . 15.3), #'( # M0M1: F , , " #'(( # /# /#0 z = f (x y):' ' { # ( M0M1 # l l { . $ # M00 # M 0: E # ' (. .
15.3), 144dz , # ## dz = @z l (. (14.11)), tg ' = l@l@z = tg ':@l0,# "$, @z@l # M0 ! ! l ! z=f(x,y) # M0 3.( ( # ' l:H" ' ! 0fx(x0 y0) fx0 (x0 y0) $ # M00 lx "" Ox ly "" Oy $ # M0 #' Lx Ly # /# /#0z = f (x y) 3. ( # ' lx ' ly : , $ fx0 (x0 y0) " # # lx #'( ( Lx fy0 (x0 y0) { # # ly #'( ( Ly : 1. 3 & (. 15.1).2.
& < ! (. 15.3 ! (15.2)).3. 3 < 3 (. 15.4 (15.3)).4. ! 9" 9 $ & (. 15.2).5. =# &3 3 3 9$? (. . 15.2).6. 4 $ $! $ " & "&?7. 4 $ $! $ " & "&?8. 4 $ $! $ " $"& "& &"&?145 16 &!'( )(" *+,$ 3( /#0 y = y(x(t)) $ dy 3 x $ dx $( dxdt3 t ( ,dy . ).
#, $ dxdt dx/#0( #'#% % # 3% % 3 "' ' $"$, # . $"$ / //0 3% /#0(.' /#0 z = f (x y) $ /#0( x = '(u v) y = (u v): , 3 '( z = f ('(u v) (u v)) = (u v):8 /#0 z = f (x y) $ ), /#0x = '(u v) y = (u v) { /#0. ; u v" $' , x y { ( 3( /#0. 16.1. x = '(u v) y = (u v) (u v) z = f (x y) (x y) ( z = f ('(u v) (u v)) (u v): (83 % $% " 0 #$' ).". ,# ## /#0 x = '(u v) y = (u v) //0 # (u v) % .
x y $. u v :@x v + u + vu+(16:1)x = @x11@u@v@y u + @y v + u + v(16:2)y = @u22@v 1 1 2 2 { "# /#0 u ! 0 v ! 0:146,# ## /#0 z = f (x y) //0 # (x y) .( # (u v) . z $ . , x y @z x + @z y + x + y(16:3)z = @x@y { "# /#0 x ! 0 y ! 0:=$ //0 /#0( x = '(u v) y = (u v) #(u v) % ' ( # (. 13.2), (1.4)limy = 0! x = 0 limu!u 0v 0!v0!0..
x ! 0 y ! 0 u ! 0 v ! 0 , "# /#0 u ! 0 v ! 0:8 z /( (16.3), ./#0 z = f (x y) { ( # 3( /#0z = f ('(u v) (u v)): @ "$', (16.3) ' (16.1) (16.2), " 3' . ( 3( /#0, $ . % , u v ( x y " . . 3% 3( /#0).$' # (16.1), (16.2) (16.3) 3 ' @z @x @z @y ! @z @x @z @y !z = @x @u + @y @u u + @x @v + @y @v v + u + v:(16:4) @x! @y!@z@zJ' = @x 1 + @y 2 + @u + 1 + @u + 2 { "# /#0 u ! 0 v ! 0 ## "# %6 { "# /#0 u ! 0 v ! 0 # ## 3 " 3 .,# ## 3 #"#% (16.4) $ u v $(16.4) //0' ( 3( /#0 # (u v) (. 14.2 //0 /#0).
, #$.147B , $ (16.4) 3 ' 3 % @z @z # #//0$% 3( /#0 @u@v u v (( . /#0 z (.(14.4), (14.8) (14.9)):@z = @z @x + @z @y (16:5)@u @x @u @y @u@z = @z @x + @z @y :(16:6)@v @x @v @y @v$ / (16.5):@z { $ 3( /#0 u6@u@z { $ 3 x6@x@x { $ 3 @u u6@z { $ 3( /#0 3 y6@y@y { $ 3 @u u:; / (16.6), #( v:( $ $ /' . //0.* ( ! ( !, ( ( ! !.F /#0 " #3% % .
16.1. ' z = z (x y) x = x(t) y = y(t) { //0 /#0. , 3 /#0 z = z (x(t) y(t)) //0( /#0( t $dz = @z dx + @z dy :dt @x dt @y dt148dx dy { $ /#0( . J' dzdt dt dt@z @z { % "$% '$ "" d: ; @x@y$ ( /#0 z = z (x y) # /#0(% %. "$% % $% '$ "# " @: 16.2. ' z = z (x y) y = y(x) { //0 /#0. , 3 /#0 z = z (x y(x)) //0( /#0( x dz = @z + @z dy :dx @x @y dxJ' $ 3% :@z 3 dx = 1:x = x: = 3 ', @xdxdz { $" , ( / dx3( /#0, # /#0( x@z { $ ( d ""), ( @x( /#0 z = z (x y) # /#0( % %( @ "# "). 16.3. ' w = w(x y z ) z = z (x y) { //0/#0 % .
, 3 /#0 w = w(x y z (x y)) //0( /#0( x y $ @w ! @w !@w@w@z@w + @w @z :=+=@x n @x @z @y @y n @y @z @yJ' x y $. dxdy = 0!dy = 0: H $ @w @w { $ $dx@x n @x% /#0(: { $ 3( /#0 ( $ ( $(, #'# 3 /#0 $' $( x " !@w3 z @x %#$ $n149'), { $ ( /#0 w = w(x y z ) (#( z $.
x). @w ! @w; "$ $ $ @y @y :n ' /#0 x = '(u v) y = (u v) //0 #(u v) /#0 z = f (x y) //0 .(# (x y): , 3 /#0 z = f ('(u v) (u v)) //0 # (u v) (. 16.1) //0@z du + @z dv:dz = @u@v@z @z $ (16.5) (16.6) @u@v"$, ae @z @x @z @y ! @z @x @z @y !dz = @x @u + @y @u du + @x @v + @y @v dv = @x! @z @y! @z@z@x@y@z dy:= @x @u du + @v dv + @y @u du + @v dv = @x dx + @y,# "$, //0 3( /#0@z dx + @z dy(16:7)dz = @x@y / //0 ( /#0@z dx + @z dy:(16:8)dz = @x@yF ( $ //0, # ## $ , ## ' # % 3( /#0 (u v) ( /#0 (x y):& # ' , 3 /% (16.7) (16.8) $ (## $ //0 dz dz): F $ $# , x y ( /#0 150% //0 dx dy . x y % , 3( /#0 dx dy { //0% /#0( x = '(u v) y = (u v) ".
' ( .( x y %/#0(.,# , / (16.8)dx = x dy = y / (16.7)dx 6= x dy 6= y:&( / //0 /#0 " # 3% % . ' u = u(x1 x2 ::: xn) v = v(x1 x2 ::: xn) z (u) { //0 /#0 % . , 1 d(Cu) = Cdu C { ,2 d(u v) = du dv3 d(uv!) = vdu + udv4 d uv = vdu v;2 udv (v 6= 0)5 d(z (u)) = z 0(u)du:F # #$ .' ( / //0. @, #$'4 3 /#0 z = uv 3 u v { x1 x2 ::: xn: 8 ( //0 ( /#0@z du + @z dvdz = @u@v du = du(x1 x2 ::: xn) dv = dv(x1 x2 ::: xn):151; # ##@z = 1 @z = ; u @u v @vv2dz = 1v du ; vu2 dv = vdu v;2 udv :u!uA, z = v dz = d v % # 4:&' 3 #$ ' . 16.4.
@( ( //0 $/#0!x;y2 3z = sin 5x y + x + y : ' 5 2 3 1 4 2 ae!x;y2 3dz = cos 5x y + x + y 10xy3 dx + 15x2y2dy+1!(x+y)(dx;dy);(x;y)(dx+dy)x;y23A = cos 5x y ++(x + y)2x+y 101 1002x2y @@10xy3 (x + y)2 A dx + @15x2y2 ; (x + y)2 A dyA :@z z = @z #//0 H $ z = @x@ydx dy :01@z = @10xy3 + 2y A cos 5x2y3 + x ; y ! 6@x(x + y)2x+y01@z = @15x2y2 ; 2x A cos 5x2y3 + x ; y ! :@y(x + y)2x+y L#0, # $ , $ ' , 3 .% $ /#0(, $ .
/#0 .152#$ 16.1. M, F (x y) = 0(16:9) ( ) /#0 y(x) F (x y(x)) 0(16:10)(3 (16.10) $, F (x y(x)) = 0 % % $( x). 16.5. Ax2 ; xy + 5 = 0$ /#0 (16:11)y = x + x5 (16:12) x 6= 0 # ## !52x ; x x + x + 5 = 0 , x2 ; x2 ; 5xx + 5 = 0 8x 6= 0 .. 3.J, 16.1 /#0. L#0 -3 ## 3 % 3. 8 ' "( " .8 16.5 (16.12), . /#0 , 3' $ (16.11), $' ' y:,# ". % $ /#0 (16.9) $' ' y: 16.6. y 2 ; x = 0:$ ' y ppy1 = + x y2 = ; x .
/#0.153;;; M/# % /#0( " y2 ; x = 0 $"3 . 16.1.. 16.1( #$, (16.9) 3' , #'# % /#0(. 8$3 #(', x2 + y2 + 5 = 0(16:13) ( ('( /#0, # ## ##% ('% x y:A, #% (16.9) /#0 , .( . 16.2. * F (x y) :1) F (x0 y0) = 062) Fx0 Fy0 (x0 y0)63) Fy0 (x0 y0) 6= 0:/! F (x y) = 0 x0 y(x) y(x0) = y0: 0 , x0:4#$' .8 #( ( . . #, # #( F (x y) = 0 # ( (16.13), # ##% ('% x y: =$ 2- 3- ( , (x0 y0) { "# # #( F (x y) = 0 ( ( # Fy0 6= 0 ',154Fx02 + Fy02 6= 0 # nG = Fx0 (x0 y0)Gi + Fy0 (x0 y0)Gj '# #( # (x0 y0):& "' $ ' ( /#0, ( F (x y) = 0 .( y(x0 ) = y0: H" ' , # 16.6:F (x y) = y2 ; x Fx0 = ;1 Fy0 = 2y: # # (x0 y0) # (1,1). ,F (1 1) = 06 Fx0 Fy0 # # (1,1)6Fy0 (1 1) = 2 6= 0: A 16.2 , y2 ; x = 0 p"$ # x0 = 1 /#0 y(x) = x . y(1) = 1 (.. 16.1).
4 # (1 ;1) #3 (Fy0 (1 ;1) = ;1 6= 0) y2 ; x = 0 "$ #x0 = 1 '3 , /#0 py(x) = ; x . y(1) = ;1 (. . 16.1). ; # (0,0) $ Fy0 (0 0) = 0 3-e # . % % /#0(, % y2 ; x = 0p .% p y(0) = 0 : y(x) = + x y(x) = ; x: M# ( $ ' ' #nG = Fx0 (0 0)Gi + Fy0 (0 0)Gj = ;Gi Ox: 8 % 3 % %nG = Fx0 (1 1)Gi + Fy0 (1 1)Gj = ;Gi + 2Gj 6 k Ox nG = Fx0 (1 ;1)Gi + Fy0 (1 ;1)Gj == ;Gi ; 2Gj 6 k Ox , . #'% # #( #% (1,1), (1 ;1), # $3' "( $(. . 16.1)." ' # " # $( /#0,$( . ? 3 ( # $ , # % /#0 //0 " .? 3 F (x y) = 0 $ % /#0%' y ( ' x /#0x(y)), ( # $ /#0 $3. 8 # $( 3 ' .
"$.? /#0 F (x y) 16.2, F (x y) = 0 /#0 , # ". 155 3, ..F (x y(x)) 0:4//0 3, aedF (x y(x)) 0 / //0Fx0 dx + Fy0 dy(x) = 0: //0 ( /#0 /0dy(x) = ; FFx0 dxy $( {dy = ; Fx0 :(16:14)dxFy0 16.7. @( $ /#0 y(x) ( y + sin y ; x = 0:(16:15)J' F (x y) = y +sin y ; x = 0 Fx0 = ;1 Fy0 = 1+cos y /(16.14) dy = ; ;1 = 1 :(16:16)dx1 + cos y 1 + cos yJ, ( # $ /#0y(x) $3, # ## , . , 0 $ % /#0% 'y:O ., /#0 y(x) , ". $, # % /#0'% $( 3 ' # /#0(, % .# $% " #% # /#0(, $% , $ "' , # , ' # 16.7.156' /#0 y(x) ( (16.16)," ( $ #. @ $(16.16) d2y = d dy ! = d 1 ! :dx2 dx dx dx 1 + cos y8$# " //0 x /#0 1 + 1cos y # 3 $ x $ y: @ $' y { $ /#01y(x) 1 + cosy(x) 3( /#0(.
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