1. Интегралы ФНП Диф_ур (853736), страница 8
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1!222 2 ! 0 = lim= 2: 0 2!xy :limx! x2 + y 20y!0( 2cos sin = cos sin = 1 sin 2:lim 022!xy5 < , limx! x2 + y 2y! *. ! (0 0), ! !, $ ) $ = const , ; 21 + 12 .9 , ! ! {! .00/ ( $ , .125 13.15.3 u(M ), M0 2 Rn ( M0), M0, 9 MlimM u(M ) = u(M0):!0(13:1)( Tu = u(M ) ; u(M0) { * $ u(M ), * M0 M .
> lim u(M ) = u(M0 ) ,M M0!.. lim Tu = 0M M0!lim Tu = 0M M0!(13:2) ! $ u(M ) M0.1 $ $ , , -+: $ !, < ! ! # # .( R2 : M (x y) M0(x0 y0) u = f (x y) x ; x0 = Tx, y ; y0 = Ty { * , Tu = f (x y);;f (x0 y0) { * * $ .> (13.1) ! 9 xlim!x f (x y ) = f (x0 y0 )y!y00(13:3) (13.2) lim! Tu = 0x 0y 0!(13:4))* ! $ f (x y) (x0 y0).126 13.1. ' (0 0) $8 2 2>< x y x2 + y2 6= 0f (x y) = > x2 + y2: 0 x = y = 0.4 cos2 sin2 = lim 2 cos2 sin2 = 0.5 limf(xy)=limx! 0 02y!4 f (0 0) = 0, limf (x y) = f (0 0).x!y!9 , $ f (x y) 0 0. 13.2. ' (0 0) $8 xy>22<f (x y) = >: x2 + y2 x + y 6= 00 x = y = 0.> limf (x y) * (. 13.6), $ x!y!f (x y) (0 0).00!!0000 !" #$"1.
!! Rn (!. 13.1).2. n ") (!. 13.2).3. & ! (!. 13.5).4. "& & ! (!. 13.6, 13.7).5. ! " ""?6. !& ! (!. 13.8).7. & & ! (!. 13.10).8. (!. 13.12).9. & ! (!. 13.13).10. ! (!. 13.14).11. ." ! sin(x3 + y3) :limx!0x2 + y2y!012. "! (!. 13.15).12713. ) ! ! "! (!. (13.2)).14. K!! "! (0 0) +8 sin(x3 + y3)<22f (x y) = : x2 + y2 ! x + y 6= 010! x = y = 0.5. , ! ! & ") !! ) ) , ! !! Rn .sin(x3 + y3) = lim sin(3(cos3 + sin3 )) =11. limx!0 0x2 + y22y !0!333= j sin ! 0j = lim0 (cos 2+ sin ) =!= lim0 (cos3 + sin3 ) = 0!( ! & +).14.
limf (x y) = 0 (!. ! 11) f (0 0) = 0. , f (x y)x!0y !0" (0 0).128 14&& = : '/ =/ )*(/ %/( $$ . 9 $ < $ , , , , *, * * $ . . E $ . > !, )$ $ ) # , $ , * , , * ! ) . 14.1. ( $ u = u(M ) n M0 2 R - l { , * M0- M { ) - Tl u = u(M ) ; u(M0 ) { *$ , * M0 M 8< +jM0M j M0M "" lTl = : ;jM M j M M "# l00{ * (. . 14:1).;. 14.1$ $ u(M ) M0 & l # Tl u Tl Tl ! 0 !@u = lim Tl u :(14:1)@l l 0 Tl129!( R3 Oxyzu = f (x y z ) M0(x0 y0 z0) l Ox (..
14:2).;. 14.2> M (x0 + Tx y0 z0), * Tl = Tx *Tl u = f (x0 + Tx y0 z0) ; f (x0 y0 z0)) * $ f (x y0 z0) x $ y = y0 z = z0:Tl u = Tf (x y0 z0):( @u@l < $ f (x y z ) x (x0 y0 z0), ! @u@x @u = lim Tl u = lim Tf (x y0 z0) = d f (x y z ) :(14:2)0 0 x=x@x l 0 Tl x 0Txdx0 l "" Oy, @u@l $ f (x y z ) y, , l "" Oz , { @u z (x0 y0 z0). ! < @u@y @z @u = d f (x y z ) d f (x y z ) :(14:3)@y dy 0 0 y=y dz 0 0 z=z130!!000@u , @u ,> !, @u@x @y @z $ u = f (x y z ) , ( * ).' ! , :@u f (x y z ) = @u f (x y z ) = @ufx(x0 y0 z0) = @xy 0 0 0@y z 0 0 0 @z( "<$ # x", "<$ # y", "<$ # z ").' $ (14:2), (14:3) , $ .
14.1. 1 $ u = x2y3z 4 (x y z ).' $ (14.2) , @u@x 3 y4 z ) . (< ) y z , d (x2) = 2x, dx@u = 2xy3 z 4:@x( @u@y x z , d (y3) = 3y2, dy@u = 3x2y2z 4:@y1 , @u@z $ x y @u = 4x2y3z 3:@z4 $ ! #, 3, .000131 $$ $ ), $ , ) ( *) * $ .2 $$ $ . ( $ ! # . 14.2.
( $ z = f (x y) (x0 y0)Tx = x ; x0, Ty = y ; y0 { * Tz = f (x0 + Tx y0 + Ty) ; f (x0 y0) { * *$ .0 ) Tz = ATx + B Ty + Tx + Ty(14:4) A = A(x0 y0), B = B (x0 y0) { , * Tx, Ty(), = (x0 y0 Tx Ty) = (x0 y0 Tx Ty) { ! Tx ! 0, Ty ! 0, $ z = f (x y) (x0 y0), ( Tx, Ty) * ATx + B Ty $ z = f (x y) (x0 y0) ! dz :dz = ATx + B Ty:(14:5)( * ! $ * $$ $ .p( = Tx2 + Ty2 { ) (x0 y0) (x0 + Tx y0 + Ty) (. . 14:3).132;. 14.3 x + y = x + y :, jxj 1 jyj 1. x ! 0, y ! 0 x , y , x , y { "# ## $ "# % . ,, x ! 0, y ! 0 ! 0, aex + y = 0:lim!0&', x + y "# ( ! 0 # :x + y = o() ! 0:,' $ (14:4) .
//0( /#0 z = Ax + B y + o() ! 0:(14:6)23 #$', , ", $ (14:6) (14:4), (14:6) "% //0 /#0 z = f (x y) # (x0 y0) ( ' //0 ( /#0 # (x0 y0)).4 /#0 $ //0 #( # # . $( ( #. 4/#0( #'#% % 3 "". . "$.133 14.1. - , , , .4#$' /#0 % %.' /#0 z = f (x y) //0 # (x0 y0)6l { $' /# ', %. $ # (x0 y0)., . /#0 . $ # (x0 y0) #(x0 + x y0 + y) l (. (14:6)):l z = Ax + B y + o() ! 0 # . ll z = A x + B y + o() = A cos + B cos + o() 00lll ll cos 0, cos 0 { . # l (.
. 14:3). l ! 0 cos 0, cos 0 , A B , #o() = 0 , ', . lim l z ,## l = , liml!0 ll!0 l..(14:7)9 @z@l = A cos 0 + B cos 0:,# ## ' l " " $', /#0 z = f (x y). $ " # (x0 y0).8 , l "" Ox, $, . @z . = 0, = $ (14.7)$ @x002@z = A:(14:8)@x; , "' ' l "" Oy ( 0 = 2 , 0 = 0), @z # . ( $( @y@z = B:(14:9)@y4#$' $#.134 . 3% , # .% $ #$'.1) =$ (14:7) { (14:9) /@z = @z cos + @z cos (14:10)00@l @x@y# '$ $( .2) =$ (14:5) 3 //0 $ $@z x + @z y:(14:11)dz = @x@y3) =$ (14:10) (14:11) (14:12)dz = @z@l l# ## l cos 0 = x, l cos 0 = y:4) ? @z@l 6= 0, ! 0 (l = ! 0) ( //0dz "# ( # (. (14.12)), ###z = dz + o() ! 0(14:13)(. (14:6) (14:5)), % ( # ./#0 z dz .
@ dz $ ! . z .,# "$, ! .@ //0 /#0 //0 $% %, # % .. 4 /#0 z = f (x y) % "$ dx = x,dy = y.A , $ (14:11) ae /@z dx + @z dydz = @x(14:14)@y3 . $' 3 //0 /#0, $ //0 $% %.1354 /#0 u = f (x y z ) / "". . "$:@u dy + @u dz:du = @udx+(14:15)@x@y@z; 3 ( //0 /#0 "' %, $ $ //0 $% %." # " % //0 /#0#'#% %.B# $, /#0 "% //0 #( # .
(#() $( /#0 ( #.=$ 14:1 , #'#% % # ( #. # . &. /#0, . $ #( #, . //0 ( #.4 //0 /#0 #'#% % . . 14.2. , , .". 4#$' /#0 % %.' /#0 z = f (x y) $ fx0 (x y) fy0 (x y), # (x0 y0),4z = f (x0 + 4x y0 + 4y) ; f (x0 y0){ . /#0, . . 4x 4y. 4z 4z = Cf (x0 + 4x y0 + 4y) ; f (x0 y0 + 4y)] + Cf (x0 y0 + 4y) ; f (x0 y0)]:(14:16)136 E 3 /#0 f (x y0 + 4y) $# Cx0 x0 + 4x], 4x < 0 $# Cx0 + 4x x0], f (x0 +4x y0+4y);f (x0 y0+4y) = fx0 (x0+1 4x y0+4y)4x 0 < 1 < 1(14:17)0,# ## $ fx(x y) # (x0 y0) lim fx0 (x0 + 14x y0 + 4y) = fx0 (x0 y0)',4x!04y!0fx0 (x0 + 4x y0 + 4y) = fx0 (x0 y0) + { "# /#0 4x ! 0, 4y ! 0, $ (14:17)f (x0 + 4x y0 + 4y) ; f (x0 y0 + 4y) = fx0 (x0 y0)4x + 4x: (2:18); , / E 3 # /#0 f (x0 y) $# Cy0 y0 + 4y] , 4y < 0, $# Cy0 + 4y y0] f (x0 y0 + 4y) ; f (x0 y0) = fy0 (x0 y0 + 24y) 4y 0 < 2 < 1 ' ( $( fy0 (x y) # f (x0 y0)f (x0 y0 + 4y) ; f (x0 y0) = fy0 (x0 y0)4y + 4y(2:19) { "# /#0 4x ! 0, 4y ! 0.
' (14:18) (14:19) (14:16). , ./#0 #3 4z = fx0 (x0 y0)4x + fy0 (x0 y0)4y + 4x + 4y:& (14:4), $, fx0 (x0 y0) = A fy0 (x0 y0) = B {, $. 4x 4y6 { "# /#0 4x ! 0, 4y ! 0, , ', 14:2, /#0z = f (x y) //0 # (x0 y0): , #$. . $ 3 //0' ' /#0. F $' ( /#0 .137 14.3. - , .". 4#$' /#0 % %.' /#0 z = f (x y) //0 # (x0 y0). , . (14:4):z = Ax + B y + x + ylim! z = A 0 + B 0 + 0 0 + 0 0 = 0:x 0y 0!8 (1:4) /#0 z = f (x y) # (x0 y0)." .
&. /#0 #'#% % (## /#0 ), #( #, //0 ( # (c. #'( 7). 1. (. 14.2).2. (. 14.1).3. !" " #" ! # $?4. ! " $" " "& "&? (c. (14.14)).5. ( $" " 8 xy<22f (x y) = : x2 + y2 x + y 6= 0,(14:20)0 x = y = 0 $ (0 0).6. .!! ! (14.20) $ (0 0)? (c. #13.2 14.3).7. 0#, $ !f (x y) = jxj + y" $ (0 0), !!! 1 $.1383. 2 $! 3 .
4 , 3 l l1 { #" cos 0, cos 0 { ! " l, a cos 1, cos 1 {! " l1. 6 cos 1 = ; cos 0, cos 1 = ; cos 0 "@z = ; @z .(14.10) , $ @l@l15. fx(0 0) = 0 fy (0 0) = 0. 7 ! z = f (x y),$"& (14.2), (14.3):d (0) = 0dfx(0 0) = dx f (x 0) = dxx=0dd (0) = 0:fy (0 0) = dy f (0 y) = dyy=06. ( !!!. 4 # (13.2) , $ ! (14.20) !!!" $ (0 0), $ " 14.3 # 9"3 1 $.:! (14.20) #, $ $"& "& $ !!! $" ! 1 $.
:! (14.20) $" " $ (0 0) (. 5), !!! 1 $.7.lim!0 f (x y ) = lim!0 (jxj + y ) = 0 = f (0 0):0000xy!0xy!03, ! f (x y) " $ (0 0) (. " (13.3) (13.1)). ( $! !ddfx(0 0) = dx f (x 0) = dx (jxj) :x=0x=00;$, ! f (x y) !!! $ (0 0), < 9& (. 14.1).6 9, !, "! $ # 9"3 1 $. 7 !!! # , $ , 9! 14.3 .139 15 ! " ' %'.M0 { /# # %,M { 3 # (,nG { /#( #,' = '(M ) { 3 # M0M nG (. . 15.1).;. 15.1#$ 15.1. @( # nG $ #% # M0 :lim'=M !M2#$ 15.2. ,# M0(x0 y0 z0) % F (x y z ) = 0$ , ( #:1) $ Fx0 Fy0 Fz0 62) Fx02 + Fy02 + Fz02 6= 0: % " $ % ( # M0(x0 y0 z0)$ #( %.#3, M0(x0 y0 z0) { "# # %F (x y z ) = 0, #n = Fx0 (x0 y0 z0)i + Fy0 (x0 y0 z0)j + Fz0 (x0 y0 z0)k(15:1)0 '% # M (x0 y0 z0).q 02 # (,# ## jnj = Fx + Fy02 + Fz02 6= 0 n { ( #.:,' 3 #$', Mlim'=!M20140' # M % , # # x0 y0 z0 .
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