1. Интегралы ФНП Диф_ур (853736), страница 11
Текст из файла (страница 11)
F $, ##( # /#0 #3 "'. #( /#0 /#0z = x2 ; y2:? $ zx0 = 2x zy0 = ;2y x = 0 y = 0. &', # O(0 0) ##( (0() #( /#0. # # ( # /#0173;. 8 , /# /#0 "#( " (""), . . 18.3.. 18.3J /#0 # O z (0 0) = 0: @ "( # _ (O), ## " ", ' # M1(x 0), #%z (x 0) = x2 > 0 x 6= 0 # M2(0 y), #% z (0 y) = ;y2 < 0 y 6= 0: J, . $ # O # M1 ./#0 4z > 0 .
# M2 { 4z < 0:&', . # #' _ (O) " . $ # O " # . /#0 4z "" '# "' 0, " '# ' 0. F $, # O# /#0 . #' /#0 z = f (x y) $ # # ' #( {# 0( # M0(x0 y0)M (x0 + dx y0 + dy) { # $ ( #64z = f (x0 + dx y0 + dy) ; f (x0 y0) { . /#0, # p 2 2. $ # M0 # M 6 = dx + dy { 3 # M0 M . / ,(4z = dz (x0 y0) + 2!1 d2z (x0 y0) + o(2) ! 0:,# ## (x0 y0) { 0 # /#0 z = f (x y) dz (x0 y0) = 0:4, d2z (x0 y0) 6= 0 " 6= 0. , % 6= 0 $# 4z " ' $# d2z (x0 y0).174? d2z (x0 y0) > 0 " 6= 0, 4z > 0 % #M $ ( # _(M0).
8 /#0z = f (x y) # M0(x0 y0):? 3 d2z (x0 y0) < 0 " 6= 0, 4z < 0 { #M0(x0 y0) /#0 #.( $' /#0 " .,# "$, /#0 #'#% % 0( # $#' (3' 0' ') //0 # ( #.4 # 3 ' ( ," "( ## '$.'A = fxx00 (x0 y0) B = fxy00 (x0 y0) C = fyy00 (x0 y0):, (. (17.5)):d2z (x0 y0) = Adx2 + 2Bdxdy + Cdy2 :(18:4)A3 A:Ad2z (x0 y0) = A2dx2 + 2ABdxdy + ACdy2:J ( #:Ad2z (x0 y0) = (Adx + Bdy)2 + (AC ; B 2)dy2:(18:5),# ## (Adx + Bdy)2 0, $# $ Ad2z (x0 y0) "$' (AC ; B 2). .1: AC ; B 2 > 0:J $, AC > 0 A C # $#.4 $ (18.5) , Ad2z (x0 y0) > 0 6= 0:(? dy = 0, dx 6= 0, # ## 6= 0, Ad2z (x0 y0) = (Adx)2 > 0).175? A > 0, d2z (x0 y0) > 0 6= 0, /#0 z = f (x y) # M0(x0 y0):? A < 0, d2z (x0 y0) < 0 6= 0, /#0 z = f (x y) # M0(x0 y0):,# "$, ! # M0(x0 y0): 8 # $#A ( $# C , # ## $# C A ).2: AC ; B 2 < 0:' A 6= 0.
, $ (18.5) , 6= 0 (Adx + Bdy = 0 Ad2z (x0 y0) < 0, ( dy = 0 "' (. . 18.4).;. 18.4J# d2z (x0 y0) % % 6= 0 3, "( #( # # M0, ## " ",( #, . # $ # M0 ./#0 4z " 3 3% $#. &', # M0(x0 y0) # /#0 .? A = 0 ( B 6= 0, # ## AC ; B 2 < 0), $ (18.4):d2z (x0 y0) = 2Bdxdy + Cdy2 :@ "', 6= 0, //0 d2z (x0 y0)" $% $# C = 0 % dy = dx dy = ;dx, C dy:C 6= 0 % dx = 0 dx = ; B&', A = 0 # # M0(x0 y0) .,# "$, AC ; B 2 < 0, /#0 z = f (x y) # M0(x0 y0).1763: AC ; B 2 = 0:8 , " ' # # M0(x0 y0) 3 ' ' , # ## AC ; B 2 = 0 /#0 # #M0, (. 3 18.2, 18.3).& $' "0.(x0 y0) { ! $ f (x y).A = fxx(x0 y0) B = fxy (x0 y0) C = fyy (x0 y0)AC ; B 2>0<0=0A<0 >0 88extr f (x y) max min 9! . (x0 y0)3000000 18.1.
=' # /#0 z = y3 + 2x2 ; 12xy + 4x ; 12y + 2:1) @( 0 # /#0:zx0 = 4x ; 12y + 4 zy0 = 3y2 ; 12x ; 12:4 # # 0% # :4x ; 12y + 4 = 0 ) () x ; 3y + 1 = 0 ) :3y2 ; 12x ; 12 = 0y2 ; 4x ; 4 = 083 x $ :x = 3y ; 1 # $ :y2 ; 12y = 0:B : y1 = 0 y2 = 12:& . $ x : x1 = ;1 x2 = 35:&0 #: M1(;1 0) M2(35 12):2) # 0% #%:00 = 4 zxy00 = ;12 zyy00 = 6y:zxx1778 # M1(;1 0) : A = 4 B = ;12 C = 06 AC ; B 2 = 0 ; 144 < 0:& .' "0 , # M1 # .8 # M2(35 12) : A = 4 B = ;12 C = 726 AC ;B 2 = 472;144 > 0:&' # ' (. "0). ,# ## A > 0, (. "0) M2(35 12) { # .
& z (35 12) = 123 + 2 352 ; 12 35 12 + 4 35 ; 12 12 + 2 = ;864: 18.2. =' # /#0 z = x4 + y4:1) zx0 = 4x3 zy0 = 4y3 ) O(0 0) { 0 #.00 = 12x2 zxy00 = 0 zyy00 = 12y2 :2) zxx8 # O(0 0): A = 0 B = 0 C = 06 AC ; B 2 = 0 (. "0).," ' . , '$ 18.1 # .8 # O(0 0) z = 0, #( # _(O) ( #z = x4 + y4 > 0, # ## x y ' ". .&', "( # M (x y) 2 _ (O) 4z > 0 , 18.1, O(0 0) { # /#0, min z = 0. 18.3. =' # /#0 z = x4 ; y4:1) zx0 = 4x3 zy0 = ;4y3 ) O(0 0) { 0 #.00 = 12x2 zxy00 = 0 zyy00 = ;12y2 :2) zxx8 # O(0 0): A = 0 B = 0 C = 06 AC ; B 2 = 0 (.
"0).3) 4' ( .' 18.1).8 # O(0 0) z = 0, "( #( # _ (O), ##" ", ' # M1(x 0), #% z = x4 > 0 (x 6= 0), # M2(0 y), #% z = ;y4 < 0 (y 6= 0). ./#0 4z > 0 8M1 2 _ (O) 4z < 0 8M2 2 _(O). &',4z $#. F $, # O(0 0)# .178 1. ! , 1(. ! 18.1 18.2).2. 9&"& !& 1 (. 18.1).
# "3 9& 1 ? (. (18.3)).3. $& "& $ (. 18.3).4. 4!! $! $ !!! $ 1?5. # 9"3 ! $, $9" # 9" 3 1 1$?6. # 3 $ 1 & "& $" " !? (. 9).4. ( !! (. . 18.3 18.3).5. 0 ! # 9"3 1" (#3 3 ").179 19+!(- : +8 ( #0 # #. # /#0 3 '# # /#0 #'#% %. 8 ' , /#0 % %.' /#0 z = f (x y) #( " D ! F (x y) = 0(19:1)# $ .4, /#0 y(x): , 3 ' 3 /#0 f (x y(x)) = u(x): ? /#0 # # x0 y(x0 ) = y0 , # (x0 y0) #( ! /#0 f (x y) #( $ (19.1), ##( # /#0 f (x y) F (x y) = 0:? (19.1) 3 $' ' y ( $ /#0 y(x) # , # %# # # "%, "$% # /#0 u(x): 19.1.
=' ( # /#0 z = x2 + y2 x + y ; 2 = 0:$ $ ' y :y = 2;x # /#0 u(x) = x2 + (2 ; x)2 = 2x2 ; 4x + 4:? $u0 (x) = 4x ; 4:180(); $ # , ( 0 # x0 = 1:= $# $( (. . 19.1).. 19.1,# ## $ % $ # x0 = 1 $# ";" "+", ( # /#0 u(x) , u(1) = 2:=$ (*) y0 = 2 ; x0 = 1 $# , (1,1) { # /#0 z = x2 + y2 x + y ; 2 = 0 z (1 1) = u(1) = 2:8 #( # /#0.8 19.1 /# /#0 " .z = x2 + y2: A $ # 0xy x + y ; 2 = 0 (.
. 19.2), 0xyz #', %. $ ' Oz: ( # ". ;# # ( 3 $ /#0 z = x2 + y2 x + y ; 2 = 0: 1 ( ( 3( # ( , ( # #' 0xy ! (1,1).;. 19.2" , "( # # (1,1) ' #, #% $ /#0 ', $ # (1,1). ' ' $( /#0 #%,3% ( x + y ; 2 = 0: 8 $# . , # /#0 181 " , ' #% , ( $.? 3 /#0 z = x2 + y2 ' ( " , , #( ("$ ) "(0,0) z (0 0) = 0:,# 3 # 0 3 ' ".: # /#0 z = f (x y) F (x y) = 0 # , ( F (x y) = 0: 19.2. =' ( # /#0 z = x2 ; y2 x2 + y2 = 4:8 , 19.1, 3 , p $p /#0 y(x) : y = + 4 ; x2 y = ; 4 ; x2: B , #% x = 2 $ % /#0( "#.
"$# (2,0),(-2,0) $ $' ' x ' /#0 x(y):=$"3' #$% 3( $ ( # % #, #( $' /0$# # T.E.E 3, ( $ # 83 ' //0' . " # , $ # 19.2.$ %& #' //0 /#0 z = f (x y) ( # F (x y) = 0: L#0 F (x y) ' . ( /#0 y(x) Fy0 6= 0 # x0 y0 #. , ( # /#0 f (x y) # (x0 y0) F (x y) = 0 "$ # /#0u(x) = f (x y(x)) # x0:182,# ## /#0 u(x) //0 (. 16.2), x0 0( #( /#0 u(x) ( # u0(x0 ) = 0: , fx0 (x0 y0) + fy0 (x0 y0)y0 (x0 ) = 0:(19:2)J' y0 = y(x0 ):=$ 3 F (x y(x)) 0 (. 16.1 ( /#0) # Fx0 (x y) + Fy0 (x y)y0 (x) = 0 % x "$ # x0 .% y = y(x): 8,Fx0 (x0 y0) + Fy0 (x0 y0)y0 (x0) = 0:(19:3)A3 (19.3) 3' ( $ 3 E3) 3 $' 3 (19.2):hifx0 (x0 y0) + Fx0 (x0 y0) + fy0 (x0 y0) + Fy0 (x0 y0) = 0:8" #, "# #"# "' ', ..fy0 (x0 y0)3 = 0 = ; F 0 (x y ) : , y 0 0fx0 (x0 y0) + Fx0 (x0 y0) = 0:? # "' $, # # # (x0 y0) :F (x0 y0) = 0 % (9fx0 (x y) + Fx0 (x y) = 0 >>=fy0 (x y) + Fy0 (x y) = 0 > (19:4)>F (x y) = 0 #( (# (x0 y0 0) .,# "$, " # (x0 y0) " #( # /#0 f (x y) F (x y) = 0 , " #1830 (x0 y0)fyx0 y0 .
$ 0 = ; F 0 (x y ) " y 0 0 (19.4). &' E 3 # # (( .? (x0 y0 z0) { (19.4), # (x0 y0) "$' /#0 f (x y) F (x y) = 0: 8#, #% /#0 # " %' % 0% #. @, $, % "' #, #% % # . 8 % % # , $, ##% $ % % # 0( # . $. . 3 "', %#$ . E 3.? Fy0 (x0 y0) = 0 Fx0 (x0 y0) =6 0 ( /#0 y(x), #( $# 3, 3 ' /#0 x(y) , , /#0u(x) = f (x y(x)) { /#0 u(y) = f (x(y) y): (19.2) (19.3) . fx0 (x0 y0)x0 (y0) + fy0 (x0 y0) = 0 Fx0 (x0 y0)x0 (y0 ) + Fy0 (x0 y0) = 00 = 0 = ; Ffx0 ((xx0 yy0)) % # ( 3 ( x 0 0(19.4) # 0% #.,# "$, E 3 '. ' , # " $ Fx0 Fy0 # (x0 y0): @ # (x0 y0) "(#( F (x y) = 0 # "$ (# " ' %., #0, ., E 3 "", /#0 , $ 2!(L(x y ) = f (x y) + F (x y):(19:5)? ' /#0 "$( #, 184# 0% # # 9L0x = fx0 (x y) + Fx0 (x y) = 0 >>=L0y = fy0 (x y) + Fy0 (x y) = 0 > >L0 = F (x y) = 0# ( (19.4).,# /#0 f (x y) F (x y) = 0 2!( (19.5) , 0 # "%% #.8 ' # 19.2, # "' ' ( # /#0 z = x2 ; y2 x2 + y2 = 4: /#0 E 3 (19.5).J' f (x y) = x2 ; y2 F (x y) = x2 + y2 ; 4: L(x y ) = x2 ; y2 + (x2 + y2 ; 4):4 # 0% # 99L0x = 2x + 2x = 0 >=> x(1 + ) = 0 >>=L0y = ;2y + 2y = 0 > , y( ; 1) = 0 > L0 = x2 + y2 ; 4 > x2 + y2 = 4 > # , % = ;1 ) y = 0 x = 2 = 1 ) x = 0 y = 2 6= 1 ) x = 0 y = 0 x2 + y2 = 4:,# "$, 0 # /#0f (x y) = x2 ; y2 F (x y) = x2 + y2 ; 4 = 06M10 2(2 0) M30 4(0 2):1858 #% M10 2 $ /#0 z12 = 4 #% M30 4 z34 = ;4:@ "$', z = 4 #, z = ;4 { .
&. % ## $ #% "3( (. . 19.3).;. 19.3M/# /#0 z = x2 ; y2 "#( " (""), $ /#0 #3 x2 + y2 = 4 $"3 # # 0 x2 +y2 = 4 "# ". @ ( # M1, M2 ' #, # { # M3, M4 "' #. #0% % # #' Oxy # % #. ; # ## 0% # , "' # % #: M10 2 { #% (#0 # M12), M30 4 { # %# (#0 # M34). E 3. # /#0 "', " ".
$ # % #. 4 , /#0 "' % , ( $3 3 "' "' . 8 ". $ . "$.@( # /#0f (x1 x2 : : : xn) #( $1869F1(x1 x2 : : : xn) = 0 >>>F2(x1 x2 : : : xn) = 0 >>=:::::::::::::::::::::::::::::::::: > :Fm (x1 x2 : : : xn) = 0 >>>>(m < n)(19:6)8 $ /#0 E 3 m 3 ( ( $):L(x1 x2 : : : xn6 1 2 : : : m) = f (x1 x2 : : : xn) + 1F1 (x1 x2 : : : xn)++2 F2(x1 x2 : : : xn) + mFm (x1 x2 : : : xm ):(19:7)4 # 0% # ( /#0 (n + m) (L0x = 0 L0x = 0 : : : L0x = 06 L0 = 0 L0 = 0 : : : L0 = 0:12n12m? (x10 x20 : : : xn06 10 20 : : : m0) { ( , 0 # (x10 x20 : : : xn0) /#0f (x1 x2 : : : xn) $% (19.6), #( /#0 3 '( #.H" ' # (% 0% #% 3 ' '.@ # #'% # /#0 ("$% %) $, #% " (! .
,# $ "' ' $ /#0 #% ##%{" #, ( ", /#0.' " ' /#0 f (x y) //0 $#( " D, ( #( ( L, # $ F (x y) = 0.187=$ //0 /#0 f (x y) ' " D (. 14.3) 8( (. #0 P13) /#0 f (x y) "' ' $ " D. "$ % $ M m.@ ', "' $ M /#0 3 ' #% # ( " D) 0 L #% # /#0 f (x y) F (x y) = 0, ' $ m { #% ( "D) 0 L #% . 8 # 0 #, ' $ /#0f (x y) % % #%, M m "$' #3 % . 8 $ 3 3' .(! M m.1. @( 0 # /#0 f (x y) "D.2. @( 0 # /#0 f (x y) F (x y) = 0.3. 8' $ /#0 f (x y) % (% #% ' % "' ', " M n. 19.3. @( "' ' $ /#0z = xy; " D, ( ( L: x2 + y2 = 4 (. . 19.4).. 19.41. =. 0 # /#0 z = xy.zx0 = y = 0 y = 0 zy0 = x = 0 x = 0:&', # O(0 0) { 0 , , % " D.1882. =.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
