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Év, oldalszám:2002, 133 oldal
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!"#$ # $ #$ % & " ( ) ) % # # (! #) % #(# * # +! ,- % %. " $$# / 0/ 1 2 ,3( / 4 %.) #( ( ! ! 2 & " ! 5 / ,( !. ## 0/ # 6 ) 78 # !& $$ # # ## # ### # 9 # !"#$ % &) % #(# # ++ ! #!## :2 ( $$ 9% + " % / 2 #!###
$$ ;# # # <#% $ #$ % &) " ( 2 % & # # % $* / %!( = >> > >7 >5 >? >A >H >B >D (#) =!+& >>> @+ >> 0 # > > 7 + ) +) "C $$# @ > > #( % & @ > E & %# @ > 7 "C @ @+ > 7 > 9# @ >7 G%# @ 9 C @ @+ @+ 2 + # + # y = f (x, y) I < & y = f (x, y)
> 1" >> > >7 %I G ( & GG J 2" # J 2/# & $% 2 ! > 5 E# %I %I 9"(# 7 ? A B B D D >F >> > >5 >D 7 5 A D 7F 7> 7D 5F 57 5A 5D ? 5 > 7 9"(# (# %II I K2 / 9"(# (# %I I ! "# 7> 7 77 ;$# # 7 > > ;$# !(% ## K * 7 > (# 7 =!+& 7 7 K * / ! K * $# 7 7 > K * $# $ 77 K"$* ) /" % & " 5> 5 57 55 E E 5 > E /* # I& 5 2 ! 5 7 2 # 5 5 0+ /*(# % K * 5 7 > ; 2/# (+ # 57 ; # ( (+ # 5 7 7 # 5 7 5 + F
= # 5 7 ? ;2# <#% / ( !" ? ?H ?H $ A> A? H> H> HA HB HD B B? D? DB >F> >F? >FA >FH >FD >>5 >>? >>H > > 5 > D @+ ) /*2 (# # /*(# ( "/ 0# % y = y (x) /*(#) y = y ,> >. y = −y ,> . y = y ,> 7. y +x ,> 5. x /*(# $ " % @+ ) 2 ) y = y (x) /*(# 2 ( @+ # /) <L M # $(# @+ ! M y = y @+ ) $ /*(#) " " ( /(#2 #( $C
) % c ∈ R2 y = cex I /*(# ) % " ( ) % $ y = cex I /*(# ,> >. @+ ,) % y= ? A y = cex I) #" . ,> . ") $# ( ( % ) % y (x) = cos (x) , y (x) = sin (x) /*(#2 ) % " ( !(( ) ( ) % cos (x) # sin (x) /*(# ,> . # $C ) % " 2 +& # :% " ( a, b ∈ R 2 y (x) = a · cos (x) + b · sin (x) ,> . @ # #" / ) % ,> . 2 I 9 & ) % ,> 7.
y = a · ch (x) + b · sh (x) . (( &) % / % @+2 C I 2( @+ #" /) % !(* / ) / = / " " /# #( ! (( &) % % # ) % @+ ( 2 # !% & /*(# !##( % 3 ) x f (x) = ex /*(# + # % & !( /(# x # ) % y = ex @+ + # % & ( " +# ) % $C* #% @+ 2! ) % %% "$ @+ <#% # @+ ) ( # +% E2 @+
+!C / C#" (* / 2 C# ( +%2( $ * > /*(# y = y (x) I , . yy2 = 1 ,. y + 2xy = 0 +% > 7 ,+. y − y x = x2 + 3x − 2 +% > D ,. x2 y + xy + (x2 − p2 ) = 0) % p " ( & p ∈ R, 2/# . H ,. y − 3y + 2y = 0 +% 7B ,/. y − 3y + 2y = ex + 2e3x +% 7B ,. (4x3 y 3 − 2xy) dx + (3x4 y 2 − x2 ) dy = 0 +% >F /*(# z = z (x, y) , . ∂2z ∂x2 + ∂2z ∂y 2 ,. ∂2z ∂x2 − ∂z ∂y ,+. ∂2z ∂x2 − ∂2z ∂y 2 ,. ∂z ∂x + ∂z ∂y I =0 =0 =0 = 0. / " + & @+ ) /*(# ( & /2 (# ,% + $$# ( / % ". +2 & @+ ) %
/*(# $( &) ! + 2 ( / " "/ & C ( , + ( . :% > , .) > ,) > ,+) > ,) , > ,.) > ,) > ,/) , ) ,) ,+ @+ /*2 9 (# # ( #( ,:%) % /*2 (# y ) % @+ sin (y) , y y, (y )2 , y3 . > ,.) > ,+) > ,) > ,) > ,/) , ) ,) ,+) , > , .) > , /*(# # ( & &2*% & ,% % y sin (x) , % y /*(#. > , .) > ,) > ,/) , ) ,) ,+) , ! > ,.) > ,+) > ,) > , / $$#) $(" / + /2 /+
9 B ! ()*+(, N 2 C @+ ) N 0# y = −y C @ ) / !* y = a cos x + b sin x. ! ()*+(, @+ ) ) %!( 0# y = −y @ y = −5 cos x + 6 sin x. " ! #$ % & ( N − C @+ 2 * / # ) % /*(# # " N − 1 ( " ( !( y (0) = 1 -.!/ y = −y # y (0) = 0 , N = 2 !* y = a cos x + b sin x. :%) y = −a sin x + b cos x a cos 0 + b sin 0 = 1 , " a = 1 # b = 0 & x = 02 !( −a sin 0 +
b cos 0 = 0 :% +% / y = cos x. @+ ( $(" # + # * , . ) % + * ) % ( ,. * ) % ( +% / 2 @+ 2 +% / ,+. % ( & ( ) % % ## + ( ) % %I " I ( #(%" , =!+& #". D ! f (x) / /*(# ( I1 # g (y) / /2 (# ( I2 ( ) ( g (y) = 0 % y ∈ I2 y = f (x) g (y) ,> ?. I "C $$# @+ ) 1 . :% ,> ? @+ %!(
h (y) = g(y) h (y) y = f (x) . % !# # h (y) dy = + y 2& /* #( % f (x) dx + C. + x2" /* & ( &I @ dy dx = y $(" # 2 -.!/ > y y 2 = 1. @ #( % & ( &I) ( / f (x) = 1 # g (y) = y12 ( y = f (x) g (y) . dy y = dx 2 !( # dx2 / ( ) % y 2 dy = dx. 3 y 2 dy = dx. " y3 = x + C &2 # ( √ G y = 3 3x + C ,( C " ) 32 ) 3C I " /* ) C. ln y y = 2xy . @ #( % & ( &I) x2 −1 % f (x) = x22x−1 2 # g (y) = y ln y 2 ! @ y = f (x) g (y) I @ I $#) dy % y = dx 2 !( 2xy ln y dy = 2 . dx x
−1 >F y 2 # x2 2 ( 2x dy = 2 dx. y ln y x −1 ,9 y ln y = 0. # ( 1 dy y ln y = 2x dx x2 − 1 2" &) % ln |ln y| = ln |x2 − 1| + ln C. ,G ( ln C !) %*) ln x /(# ###2 2 R. % y = eC (x −1) ln y = 02 y ≡ 1 / 7 ( k ∈ R {∅} # a ∈ R y = ky # y (0) = a. ,> A. / ) "/ # /" < (( & #( % & ( 2 dy dy &I @ y = dx 2 !( dx = ky. y 2 # x2 * $ ( k1 dy = dx. # y ( ln |y| = kx + C, " y = ky y = ec ekx . 9 ( ) % a = y (0) = Ce0 , " ,> A. +% / y = aekx .
" ! I @ ( & %#) % ∀t = 02 y = ϕ (x, y) ϕ (tx, ty) = ϕ (x, y) . (( f (u) = ϕ (1, u) $ # % &) % ϕ (x, y) = ϕ 1, xy = f xy , " $(" ( & %2 # @ u = xy % !# #( % & ( &I @ $(" # * % -.!/ x sin xy − y cos xy + xy cos xy = 0. "$ x2 ( sin y y y y − cos + y cos = 0 x x x x ,> H. >> % ( & %# @ ) % x % # tx2 # y % # ty 2 !( ,( y 2$ !(* + $ x y ) + xy ). (* u = u (x) = y(x) = xy I /*(# ( y (x) = x · u (x) , x % y = u + xu $/*# ,> H.2 % !() sin u − u
cos u + (u + xu ) cos u = 0 y " C!# du · x · cos u = − sin u dx #( % & ( &I @ "" /2 & sin u = Cx , " ,> H. @ + c y sin = . x x #$ y + p (x) y = q (x) ,> B. I @ * $(" # # * y − y = x2 + 3x − 2. x ,> D. ,G p (x) = − x1 ; q (x) = x2 + 3x − 2.) # # #" "$ Y Y− =0 ,> >F. x %# @ "/ & C % # C (x) /*(# !) % ,> D. G & ,> D. % ,> D. I ) % yi,ált = Yh,ált + yi,p ,> >>.
> ( ,> D. %# yi,ált " ,> >F Yh,ált ,> D. %# yi,p 0 Y − Yx = 0 @ #( % & ( &I Yh,ált = C · x. 0 # % C (x) /*(#) y = x · C (x) !( ,> D. ( y = C (x) + xC (x) , %2 ,> D. ) % C (x) + xC (x) − C (x) = x2 + 3x − 2. 2 G C (x) = x2 + 3x − 2 ln |x| , / * 2 & && ) + *. 3 :% yi,p = x · C (x) = x2 + 3x2 − 2x ln |x| . N ,> D x3 + 3x2 − 2x ln |x|. yi,ált = Yh,ált + yi,p = c·x+ 2 Yh,ált yi,p &# " # (*) yi,p 2 I % ) %
,> >F. %# "/ & C % !2 * C (x) /(# / n2 C %# y (n) + an−1 (x) y (n−1) + · · · + a1 (x) y + a0 (x) y = 0 ,> > . # n2 C %# y (n) + an−1 (x) y (n−1) + · · · + a1 (x) y + a0 (x) y = r (x) ,> >7. >7 @ / E* #) % / # 2 ( I ( / !( ) % C *% & 1. , # !) $$ C *% &( ( & ( % . <#* # * % -.!/ 9 2 2 y − y + 2 y = 2x x x C %# @ ,> >5. "$ (* #) % ! I ) ,> >7.2
n = 2, a1 (x) = − x2 # a0 (x) = x22 # r (x) = 2x. $(" # $$ ) C / I " # # $(" #(# ) ! C) # * 1-1 . ,> >7 %# yi,ált 2 ,> > . Yh,ált # ,> >7 %2 # yi,p $# yi,ált = Yh,ált + yi,p . ,> >?. 3 ) ,> >>. ) "C # I) % + "C @ 2) % ) " n2 C @ ) % %# " %# %# ( + # @
# C$ / # ) % "$ #*) % 2 % n2 C %# @ ,> > . , /*(#% & ) % % " ! ) &2*% & ( # ) %# @ . # ## #*) % % ,> >7. # & >5 % / % ,> > . I @ 2 % ( / "$ 2 *) % & /(# 2 2 3 I ( ( # O $ * C n (I)2( I 2 # n2 / @+ % & /*(# % ) (
/(# % ) 2 # # / n2 ( # ( *#) ,> > . # ,> >7 @ C n (I) % * 9 * /(# $ # ( & ) !) ( $ # ( & ) # C n (I)2 /*(# $ # 2 ( & #( (* ) % V C n (I)2) % ∀f, g ∈ V 2 # ∀α, β ∈ R2 αf + βg ∈ V * ) ( V #) % +& #( V # V " & /* % 2 E ) +& # V " f1 , . , fn ∈ V V ) % 4 5 f1 , . , fn /* ,( ) % α1 f1 + · · · + αn fn ≡ 0, α1 = · · · = αn = 0 # 465 ∀g ∈ V 2 # α1 , . , αn ∈ R, g = α1 f1 + · · · + αn fn
# / ) n2C %# ,> > . 2 C n (I) # ) # # , %!(. ! $(" # 1-1 . ,> > . n /*(#" n2&) ( >? ,> >5. # ,> >5 # " %# @ %# # 2 2 y − y + 2 y = 0. x x ,> >A. $C ") % y1 = x # y2 = x2 /*(# ,> >A. , ") % ) ( ) $ ) % # ) % # C +% + (## ) & *% & . :% {x, x2 } ,> >A2 % &) % y1 = x # y2 = x2 /*(# / ) % % ax + bx2
≡ 0 ) % a = 0 # b = 0. 9 ( # ) C # /*(#" :% ,> >A.2 {x, x2 } , ( ,> >A. yált = ax + bx2 . # # ) n2C %# 2 # * n # ) % /* n ##) + & *% & ) / ## 2 "## ) / ! ()*+(, , ! y1 , . yn ⎡ y2 (x) y1 (x) ⎢ y1 (x) y2 (x) ⎢ y2 (x) W (x) = det ⎢ ⎢ y1 (x) ⎣ . . (n−1) (n−1) y1 (x) y2 (x) , . yn (x) . yn (x) . yn (x) . . (n−1) . yn (x) ,> > . %# ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ,> >H. %!( 4 /*(#) # # I ( ) ### (
& % % ! ) % W (x) /*(# $(" 2 /*" y1 (x) , . , yn (x) /*(#" ) + $(" # %# >A • W (x)2 + # % ( • W (x) ≡ 0 I ( E ( + W (x)2 # % ( 4 # / *) W (x) 1-1 . y1 (x) , . , yn (x) /*(# # + ,> > . %# #) % # x ∈ I ) % W (x) = 0. , / # x2 * . -.!/ :* & y + y = 0 @ %# ) % / # % & !*) $# 2 / " ) % y1 = cos (x) # y2 = sin (x) /*(# / : # " ) % C %#
# /*(#" :% + % ) % "*) % / cos x, sin x /(# /2 2) 7 # ) I $* ) % ( ) % W (x) /*(# 2 ( ) cos (x) sin (x) = cos2 (x) + sin2 (x) ≡ 1 = 0. % W (x) = det − sin (x) cos (x) :% {cos (x) , sin (x)} y + y = 0 @ ) ( @ y = a cos x + b sin x $ -.!/ 9 & ) %* y − y = 0 C %2 # @ % &) % y1 = ch (x) # y2 = sh (x) /*(# %%) % ch (x) sh (x) ) ) % W (x) = det = sh (x) ch (x) ch2 (x) − sh2 (x) ≡ 1 = 0. :% y = ach (x) + bsh (x) . 7 -.!/ :* 1 + x2 y + xy − y = 0 ,> >B. C %# @ / ) % %
P # $) % y1 = x ,> >B.2 G % ) % >H y2 = c (x) · y1 = c (x) · x * E # c (x) = 1 /*(#) y2 = c (x) · x # ,> >B. y2 = c x + c # y2 = c x + c + c = c x + 2c . ,> >B @ % !( 1 + x2 (c x + 2c ) + x (c x + c) − xc = 0, ,> >D. " c :% @ ) + c # c 9 (* d = c I ) "2 C @ ) /*(# d. # % ) " ) % ,> >D. √ √ 1+x2 @ c (x) = , ( y = 1 + x2 . #2 2 x√ 2) # % + ) %
x, 1 + x2 ( ( C) # ! 2 " ) # ") % 4 2 √ 1 + x2 x 1 1 = −√ W (x) = det <0 2 −2 1 x (1 + x ) 1 + x2 √ x2) ! 4 0) ! x, 1 + x2 2 √ 2 # ,> >B. y = ax + b 1 + x 8 2 3 9: 3"#" & *% & / /(#% & +2 ) $ *% & /(# ) ( an−1 (x) ≡ an−1 , . , a1 (x) ≡ a1 , a0 (x) ≡ a0 :% ? # A # 2 & *% & ) ! > >B # > >A /(#% &2 E y (n) + an−1 y (n−1) + · · · a1 y + a0 y = 0 ,> F. ) % a0 , a1 , . , an−1 ( & 2 % %
+ ( % & I ; 0" "#< p (λ) = λn + an−1 λn−1 + · · · + a1 λ + a0 , ,> >. # r1 , . , rn , / # * $$". $2 0# ? # y + y = 0 λ2 + 1 $ r1 = i # r2 = −i. A # y − y = 0 λ2 − 1, $ r1 = 1, r2 = −1. >B = -.!/ :* y − 7y + 16y − 12 = 0 %# *% & @ & λ3 − 7λ2 + 16λ − 12 = (λ − 2)2 (λ − 3) . $ % r1 = 2, r2 = 2, r3 = 3. n2 C n2 /I , + ( . $ n $ ( ) ,2 $$ Q $$ %. $# /
/*(# @ # & > 9 r ( & $) % y (x) = erx /*(#R 9 r s2 +I ( & $) % $(" s /*(# erx , xerx , x2 erx , . , xs−1 erx ,> . 7 9 r = u + iv Q $) r Q ) r̄ = u − iv $ ) # " % $(" # /*(# eux cos vx # eux sin vx 5 r = u + iv s2 Q $) r Q r̄ = u − iv s2 $ 2 # " % $(" 2s /*(# eux cos vx, xeux cos vx, . , xs−1 eux cos vx eux sin vx, xeux sin vx, . , xs−1 eux sin vx -.!/ 9 %# &
*% & @ λ8 − 22λ7 + 232λ6 − 1546λ5 + 7066λ4 − 22 738λ3 + 50 752λ2 −72 670λ + 54 925 3 = (λ − (2 + 3i)) (λ − (2 − 3i))3 (λ − 5)2 @ >D e2x cos 3x, xe2x cos 3x, x2 e2x cos 3x, e2x sin 3x, xe2x sin 3x, x2 e2x sin 3x e5x , xe5x . / " % (λ − (2 + 3i))3 (λ − (2 − 3i))3 & (λ − 5)2 > -.!/ 9 y + 3y + 2y = 0 & *% &) C %# @ P "$ 2 λ2 + 3λ + 2 r1 = −2, r2 = −1 $ :% {e−2x , e−x }) ( y = ae−2x +be−x . / & # C$ & *% & %# 2 & > # # # %# ,> >7. I @
I ) % ,> > . %# # % % %# 2 "" / %# # ## & / ,> >7. I @ 2 # & (* "$ % ) + ) # (# * " I %! % ) & /*(# & # # & # "$ ,> > . %# # # " " 9 / % #(" ## y1, . , yn /*(# ,> > . %# # # / " # ( +& & ) % * c1 (x) , . , cn (x) /*(#) # y (x)
= c1 (x) y1 (x) + · · · + cn (x) yn (x) ,> >7. %# ,> 7. F % 1-1 . y (n) + an−1 (x) y (n−1) + · · · a1 (x) y + a0 (x) y = r (x) %2 # @ % y (x) = c1 (x) y1 (x) + · · · + cn (x) yn (x) ) % y1 , . , yn %# # # # c1 (x) , . , cn (x) /*(# ( $(" c1 (x) y1 (x) c1 (x) y1 (x) ··· (n−2) c1 (x) y1 (x) (n−1) (x) c1 (x) y1 + ··· + ··· ··· + ··· + ··· cn (x) yn (x) cn (x) yn (x) ··· (n−2) + cn (x) yn (x) (n−1) + cn (x) yn (x) + + = = 0 0 ,> 5. = 0 = r (x) -.!/ 9 y − x2 y + x22 y = 2x C %2 # @ P %# # ,> >A. ) ( # !) % %# #
{x, x2 } . K !( ,> 5 , % y1 (x) = x2 , y2 (x) = x. c1 (x) x2 + c2 (x) x = 0 c1 (x) 2x + c2 (x) = 2x c1 (x) = 2 # c2 (x) = −2x. % c1 (x) = 2x, c2 (x) = −x2 ,G % # " ) ;S c1 (x) , c2 (x)2 ( *#) % * . :% %# 2 2 3 yi,p = 2x·x −x ·x = x . E %# yi,ált = Yh,ált + yi,p = ax + bx2 + x3 . 6 ?2 # & + &2*% & 2 %# @ ) y (n) + an−1 y (n−1) + · · · a1 y + a0 y = r (x) , ,> ?. I @ % ) ,a0 , . , an−1 ∈ R % % & r (x) I ( & /*(# $(2 " I r (x) = eux [Pn (x) cos(vx) + Qm (x) sin (vx)] , ,> A.
> % Pn (x) # Qm (x) " n2 ( m2 /I :* / ) % u + iv Q s2 $ %# # p (λ) = λn + an−1 λn−1 + · · · + a1 λ + a0 ,> ?. %# @ % yi,p = xs eux [Pk∗ cos (vx) + Q∗k sin (vx)] ,> H. ) % k = max {n, m} # Pk∗ , Q∗k k 2 /I @ AB ACD-E 9 u + iv Q $ / 2 ) s = 0) ( xs ≡ 1, % yi,p = eux [Pk∗ cos (vx) + Q∗k sin (vx)] %* @ AB ACD-E ) % Pk∗ , Q∗k k 2 /I ) I #) % Pk∗ (x) = pk xk + pk−1 xk−1 + · · · + p1 x + p0 # Q∗k (x) = qk xk + qk−1 xk−1 + · · · + q1 x + q0 , % p0 , p1 , . , pk , q0 , q1 , qk ∈ R #* #% # "$ +
2 ) % ) @ & -.!/ :* / ) % % C %# &2*% & @ ) # %# # 2 $ λ1 = 2, λ2 = λ3 = 5. :* / ) % 5 ( & r (x) = cos 2x. ,> A2 u = 0, v = 2, Pn ≡ 1, Qm ≡ 0 # n = m = 0. ( u + iv = 2i $ ) # s = 0. :% yi,p = p cos 2x + q sin 2x *) % /I p, q 02 65 r (x) = x2 e5x . ,> A2 u = 5, v = 0, Pn (x) = x2 , Qm (x) ≡ 0 # n = 2, m = 0 ( k = max {n, m} = 2. ( u + iv = 5 # $ # s = 2. :% yi,p = x2 e5x (p2 x2 + p1 x + p0 ) * -.!/ 9
y − 12y + 45y − 50y = cos 2x %# # p (λ) = λ3 − 12λ2 + 45λ − 50 $ λ1 = 2, λ2 = 5, λ3 = 5. E % / . # ) :% yi,p = p cos 2x + q sin 2x + % ) % % p # q %% yi,p = −2p sin 2x + 2q cos 2x, yi,p = −4p cos 2x − 4q sin 2x, yi,p = 8p sin 2x − 8q cos 2x. ( !( & y − 12y + 45y − 50y = cos 2x # cos2 # sin2 () # *% & ##" && cos 2x(−8q + 48p + 90q − 50p) + sin 2x(8p + 48q − 90p − 50q) = cos 2x 1 ( 0 −2 p + 82 q = 1 −82 p + −2 q = 0 1 41 1 % p = − 3360 # q = 3360 E yi,p = − 3360 cos 2x + 41 sin 2x. :( %# yi,ált 3360 41 1 cos 2x +
sin 2x. yi,ált = c1 e2x + c2 e5x + c3 xe5x + − 3360 3360 Yh,ált yi,p % -.!/ 9 y − 6y + 13y = x + sin 3x @ P %# # λ2 − 6λ + 13. $ λ1 = 3 + 2i, λ2 = 3 − 2i E* #) % ( & r (x) = x + sin (x) % ,> A. # $( *) #) ,> A. # (" /*(#) ( r1 (x) = x # r2 (x) = sin (x) /*(# $ # % / ! & yi,p,1 y − 6y + 13y = x ! " @ ) yi,p,2 7 y − 6y + 13y = sin 3x @ # y − 6y + 13y = x + sin 3x yi,p ) yi,p = yi,p,1 + yi,p,2 2$ (## 6 9 1 1 cos 3x + sin 3x yi,ált = c1 e3x cos 2x + c2 e3x sin 2x + x + 2 + 13 13 85 170
! " # " 9 (* C @ ) % x, y, y , $2 * % % C @ ) % x, y, y $* % # ) % % > - y y y = f (x) . / f (x) # ( % -.!/ y = 5 + sin 3x. y = (5 + sin 3x) dx = 5x − cos 3x + C1 . :% y = 5x − cos33x + C1 dx = 52 x2 − sin93x + C1 x + c2 . 3 F (x, y , y ) = 0 - y % ) p (x) = y (x) % !# $ -.!/ + ) y (x) = p (x) T 1 1 + x2 y − xy = 0, 2 y (0) = 0 y (0) = 3 +% / P . ( y = p (x) # y = p (x) 1 1 + x2 p − xp = 0 2 % !# 5
#( % & ( &I @ p = c (1 + x2 ) . :% y = c (1 + x2 ) G 2 x3 + K. yált = c x + 3 +% / I ) % % !* x = 02 yált ) yált # # y (0) = K = 0 . y (0) = c = 3 :% +% / y = 3x + x3 . 7 - x * F (y, y , y ) y = p (y) % !# I y = dp dy dp dp (y (x)) dp = = y = p. dx dy dx dy dy 7 -.!/ T @" $/*# ! yy = 2 (y )2 − 2y % !* y = p # y y = dp p dy dp p = 2p2 − 2p dy #( % & ( &I @ , y ( & # p = p (y) y /*(#. p ≡ 0 # p = c2 y 2 + 1 K % ( ) % y = p ) % y ≡ C # y = 1c tg (cx + c2 ) . $ % M (x, y) dx + N
(x, y) dy = 0 # ? @+ ) % ∂N ∂M ≡ ∂y ∂x / # * ) # M (x, y)) N (x, y) /(# + 2 ( # # / ) + $/*" %!( > > , . # , $/*" 1-1 . ≡ ∂N # M (x, y) N (x, y) /*(# 2 9 ∂M ∂y ∂x + ( # # / $/*" 2 ) # F : R2 R /*(# M= ∂F ∂F # N = ∂x ∂y ( grad (F ) = (M, N ) . M (x, y) dx + N (x, y) dy = 0 @ F (x, y) = C + % 2 = -.!/ T y 3x4 y 2 − x2 + 4x3 y 3 − 2xy = 0 @ @" dy # / 9 / ) % y = dx # dx2 )
% 4x3 y 3 − 2xy dx + 3x4 y 2 − x2 dy = 0. M (x,y) N (x,y) @ ) ( M # N /*(# # +2 ( * # # ∂M = 12x3 y 2 − 2x = ∂N . :% ∂y ∂x A / # # F ) M = ∂F # N = ∂F . / F ?2 ∂x ∂y ∂F #3 " ( ∂x = M (x, y) , # F (x, y) = M (x, y) dx = 4x3 y 3 − 2xy dx = x4 y 3 − x2 y + f (y) . + f (y) /*(# % & ) % N = ∂F . / !( (3x4 y 2 − x2 ) = 3x4 y 2 − x2 + f (y) G f (y) ≡ ∂y 0 ( f (y) ≡ C. :% F (x, y) = x4 y 3 − x2 y + C / # # # % @ x4 y 3 − x2 y = C G + & ( / /* ( & t2( $ ) /(#2 x (t) # y (t)2( 9
/ D @+ 2 E x (t) = Dx (t) # y (t) = Dy (t) . 9 ! # ) % (D − 5) x ##C) % x (t) − 5x (t) . @+ $* + C ) # $(" # -.!/ T $(" @ 2x + y − 4x − y = et x + 3x + y = 0 @" D 2 (D − 2) x + (D − 1) y = et (D + 3) x + y = 0 2 # C # ( ) 2 (D − 2) D − 1 = − D2 + 1 = 0. Δ = D+3 1 & $(" # " t e D−1 2 (D − 2) et # Δy = . Δx = 0 1 D+3 0 $ % H /( − D2 + 1 x = et # − D2 + 1 y = −4et . x (t) = c1 cos t + c2 sin t − 12 et # y (t) = c3 cos t + c4 sin t + 2et . # # %) # c3 ,
c4 /%" c1 , c2 2( x(t), y (t) / # # % !( ⎛ ⎞ ⎛ ⎞ ⎝c2 + 3c1 + c3 ⎠ cos t + ⎝3c2 − c1 + c4 ⎠ sin t = 0 0 0 $(" −3c1 − c2 = c3 c1 − 3c2 = c4 E t x (t) = c1 cos t + c2 sin t − e2 . y (t) = − (3c1 + c2 ) cos t + (c1 − 3c2 ) sin t + 2et > -.!/ T x y $(" +% / − y + 3x = 0 x (0) = 1, y (0) = 3 + 2x = 0 @" K ! ,> B. D % ( (D + 3) x − y = 0 2x + Dy = 0 D + 3 −1 = D2 +3D +2 = 0. :% / & ! Δ = 2 D % & 2 & 0 −1 D+3 0 = 0 # Δy = =0 Δx = 0 D 2 0 !( x + 3x + 2x = 0 # y + 3y + 2y = 0. ,> D. B ( ) % x (t)
= c1 e−t + c2 e−2t # x (t) = c3 e−t + c4 e−2t . ( + # % # % ) ( % !2 ( ) # " &) % ⎛ ⎞ ⎞ ⎛ ⎝2c1 − c3 ⎠ e−t + ⎝2c2 − 2c4 ⎠ e−2t ≡ 0. 0 0 :% c3 = 2c1 , # c4 = c2 . ( % !( ,> D ) ) % @ x (t) = c1 e−t + c2 e−2t y (t) = 2c1 e−t + c2 e−2t . ,> 7F. +%2/ I ) % t = 02 % !* 2 # # / % y (0) = 3, x (0) = 1 / # 1 = x (0) = c1 + c2 3 = y (0) = 2c1 + c2 . # ) % c1 = 2, c2 = −1. ,> 7F.2 ( !( +%2/ x (t) = 2e−t − e−2t y (t) = 4e−t − e−2t . ,> 7>. / 66 # :* / ) % ! C $(" / #
t = 0 "2 ! A = (1, 3) % & (x, y) ) # (y − 3x, −2x) ( K ! @" ( A & # # (x , y ) = (y − 3x, −2x) ) % x # y " /*## !& x (t) , y (t) /(# / ,> B. +% / ) ( F # # x (t) = 2e−t − e−2t y (t) = 4e−t − e−2t . /*(# ! ,> 7 . & ) Y = F (X, Y ) % * % y = f (x, y) / y = y (x) /*(# # x /2 ( & ( T ⊂ R2 ! % # f : T R /*(# / y = f (x, y) ,> 77. I "C @ (
## * (2 ) % y = f (x, y) y (x0 ) = y0 ,> 75. +% / ,(x0 , y0 ) ∈ T . ) f /*(# / # ( , + ##. # % ( ) % ( ,+ ##. ( / % (* +%2/ # % ! ()*+(, ) % f : T R /(# y ( & #! . 03 <? ) % #) L ) % T & (x, y1 ) # (x, y2 ) # |f (x, y1 ) − f (x, y2 )| ≤ L |y1 − y2 | ,> 7?. * L2 +% %!( +%2/ # #! / @+ % & f (x, y) /*(# ,% T . :%) % f (x, y) # + 2 ( / T 2) f (x, y)2 * +%2/ # +%2/ # y ( & /*(# #!%) 2 # # + ( 0# 2 f (x, y) = x + |y| /*(#
@+ % & y 2 2 ) y ( & #! +%2/ # ) % |f (x, y1 ) − f (x, y2 )| = |x + |y1 | − (x + |y2 |)| = ||y1 | − |y2 || ≤ |y1 − y2 | . :2 % f (x, y) = x+|y| /*(# y ( & #! +%2/ # R # L = 1 +% $ 1-1 . , # ,(x0 , y0 ) ∈ T . . 9 y = f (x, y) f (x, y) /*(# / ) 2 y (x0 ) = y0 +% / D 7F $ % ) % / $ 2 $( %%) % # C##) + 2 2 3 ) % f (x, y) = y 3 , f / !) # 2 y = y 3 ) y (0) = 0 +% / y ≡ 0 3 y (x) = x3 /*(# 7 1-1 . ,F 9 f (x, y) /*(# / # & #! . 03 <? ) # # # C + y = f (x,
y) y ( 2 2 y (x0 ) = y0 +% / ,(x0 , y0 ) ∈ T . ,*# # % y = f (x, y) %% y = f (x, y) y (x0 ) = y0 ,> 7A. +% / ) / ) # C % &) % f / # y ( & +% / # 2 "/ % ) % /*(#) # % " ! ) ( + # ! ") " + % % $( ) % # +% / = 9 " ! ) $ !" % ) 2 " " $ ! +% / $* 2 # 1 & ; # y = f (x, y) , y (x0) = y0 +% / 2 xn = x0 + nh +% # y = y (x) 2 & &
$( $ !*) (xn , yn ) $ $) % xn = x0 + nh # yn+1 = yn + f (xn , yn ) h. $ !# % / h2 C G ## ,h %II. y = y (x) /*(# #"#( % !*) % y = f (x, y) , # (x0 , x1 ) y = y (x) % #") ( y = yn +y (x0 ) (x − x0 ) = yn +f (x0 , y0 ) (x − x0 ) /*(# ( f (x0 ,y0 ) ( 7> y y2 y1 y0 y−1 x−1 x0 x1 x2 x ) G H; <; # y = f (x, y) # y (x0 ) = y0 9 & /%) + ) ## % / h5 C G & & $$ ( $ !* y = y (x) /*(#) (xn , yn ) $ $) % xn = x0 + nh, # yn+1 = yn + 16 (k1 + 2k2 + 2k3 + k4 ) , % k1 = hf (xn , yn ) k2 = hf xn + 12 h, yn + 12 k1 . k3 = hf xn + 12 h, yn + 12 k2 k4 = hf (xn+1 ,
yn + k3 ) . /%0 / /* ( & t ∈ R) # /(# x = x (t) . $ # "" / ( , y = f (x, y).) @+ $ # ẋ = f (x, t) I) % ẋ = dx . dt / ẋ = f (x, t) @ + # %") ẋ = v (x) I @+ / E + x2" /*) Q + # / t2" , t ( & " . ẋ = v (x) I %!(2 ) % v : R R # / @+2 7 % & , / * ) (* -.!/ / v 2" . /*(# # ) ẋ = (a − bx) x. I % x = 0 # x = ab v(x) # (* I % ) % ẋ = v (x) " /
"## ! ) # % t0 " # x0 % ( ) v (x0 ) = 0, ẋ (t0 ) = v (x0 ) = 0. :% ( # " ; E ( ) % x0 % " (## & &) % x (t) ≡ x0 ẋ = v (x)) x (t0 ) = x0 +% # ) # ( v (x) # ) ! "" / # + % ,+.) % ! ()*+(, x (t) ≡ x0 I % / % (*) % ∀ε > 0 ∃δ > 0, %% ϕ (t) ẋ = v (x)2 ) |ϕ (0) − x0 | < δ, |ϕ (t) − x0 | < ε ∀t2 E % / # + ( ) %% & + + # I % " " I ( -.!/ # x : R R2 ) % x (t) :% x (t) = (x1 (t) , x2 (t)) . !
ẋ (t) = (−x2 (t) , x1 (t)) # x (0) = (u, 0) , u ∈ R v(x)=(−x2 ,x1 ) E ) v(x) = ẋ (t) # x (t) " # %I % % v (x) = 0) x = 0 ,% &. N I % x (t) ≡ 0. / +% / ẋ1 = −x2 ⇒ ẍ = −x1 ⇒ x1 = a cos t + b sin t ẋ2 = x1 u = x1 (0) = a, % x1 = u cos t + b sin t ) −x2 = ẋ1 = −u sin t + b cos t. x2 (0) = u sin 0 − b cos 0 = 0, ! b = 0, ( x1 = u cos t. ! x2 = −ẋ1 = u sin t. E +% / x (t) = (u cos t, u sin t) . ( 77 y (u, 0) x ( |x (t)| = u, % x (t) & $# I u I $$ ( G & )% x (t) ≡ 0 I % (2 =!+& " δ 2 ( % δ = ε2 $ ! ()*+(, 2 x (t) ≡ x0 I % % 0 (2 2 lim ϕ (t) = x0 * t∞
ẋ = v (x) ϕ (t) ) ϕ (0) x0 2% # $ ( -.!/ x (t) & % ) % & 2 % % ( ,+ I & / # *. < () % & % *) $$ ) % I % 9 ( + !) I & I % % / ) % I % % -.!/ ẋ = x v (x) = x, 75 :% I x ≡ 02 ( @+ 2 x (t) = cet . :% ẋ = x, x (0) = u +% / x (t) = uet . E % u +) F ) t ∞ lim uet = ∞, ( x ≡ 0 I % t∞ (2 ,# ! . U ẋ = kx @+ , ! !
&* ) % % k > 0, x ≡ 0 I ẋ = kx # /) % ( / $(## , &) 2 #%) #.) +& t " I ( cekt ) # / @+ 9 ( +& ) ẋ = (a − bx) x +& t " I ( a, b > 0, ! (a − bx) & Q +$" /*(#) # / ) % % x ab , I ( # % # ) ! ẋ < 0 ) ( +$ 9 ẋ = (a − bx) x @+ " # x ##2 ## #*) ẏ = (1 − y) y @+ % G I y ≡ 1, y ≡ 0. ẋ = (1 − x) x # x (0) = u +% / et−c et−c x (t) =
et−c , % u < 1 # x (t) = et−c ,% u>1 +1 −1 y I % x(t) = et−c et−c −1 x(t) = et−c et−c +1 t ! & ) % x ≡ 1 ) # x ≡ 0 ,# # . -.!/ +& % $(" # # ( ) % # % & ) % 2 ( 7? ,. ( # # # ,. % c # % ẋ = x (1 − x) − c . "$ I ) ( v (x) = 1 2 1 x (1 − x) − c /*(# $ ( v (x) = − x − 2 + 4 − c, # % * $$ y 1 4 −c v(x) A 1 2 x B , . c < 14 , ( ) v (x)2 # !( $ ( ) % c > 0. A # B E A # B I 9 x (0) < A ( x (0) > B, x (t) = B + B−A B−A . , % K = 1 +
1−4ct x (0) − B Ke −1 √ 9 0 < x (0) < A. 9 B < x (0) . :% % x (0) + A, % +& 9 x (0) > B, lim x (t) = B. t∞ 9 A < x (0) < B, x (t) = B − E B−A √ Ke 1−4ct +1 / % ) % A ) , % K = x(0)−A . B−x(0) B 2 7A x x(t) B A ln 2B−A A t x x(t) B A t y B x(t) A x , . c = 14 , A = B = 12 % = # 2 % ) ## ) /) ( I ,+ . c > 14 , + I +& (# " * ( 7H % # x x(t) t 7B 1 2- u u(x, t) 0 x L x :* l %II $# %I 9 %I x2 [0, l]
# ! x− " 2 %I x " (x, u) ! ) % %I x +I % ( t " I ( u (x, t) /*(# " %I % % ) % u (x, t) /*(# = / & &) u (x, t) /*(# #! 2 ∂ 2u 2∂ u = c ∂t2 ∂x2 7D , >. 5F ) C + @+ ) % c > 0 / , >. @+ " %I @+ # %!( %%) % " %I % # !& u (x, t)2 ) # %I t = 0 " ,K / *) % x2 [0, l] ( ) % % f (x) /*(# . :( *# # ) % t = 02 %I 2 ∂2u E = c2 ∂∂xu2 # ! u ∂t2 @+
u(x, 0) = f (x) / # ∂u | = g (x) ∂t (x,0) * / # $ # u (0, t) = u (l, t) = 0 ∀t2 K % ∂u | ∂t (x,0) 7 ! ()*+(, U ) % F /(# # H # # 2 , *. #% ) F |H F ! H 2 E F |H /*(#) # # H # F |H /*(# H % F /(# 9 # H C % ) H = {P }) F |P = F (P ) :% 2 ∂2u 2∂ u = c ∂t2 ∂x2 @+ u (x, 0) = f (x) , ut (x, 0) = g (x) # u (0, t) = u (l, t) = 0 / # & ! ( 2 %* u (x, t) = X (x) T (t) ∂∂xu2 = X (x) T (t) 2 # ∂∂t2u = X (x) T (t) . ( !( ) % X (x) T (t) = C 2 X (x) T (t) , ( T (t) = c2 T (t) X (x) = −α2 . X (x) "* 5> 3
+ t2" ) + x2" /*) " &) % " &( ") = & 2 !( / # * N % $$# @+ (t) (x) ,>. TT (t) = −α2 ; # , . c2 XX(x) = −α2 . E 2 ,> . T (t) + α2 T (t) = 0, # , X (x) + αc X (x) = 0, T (t) = A cos(αt) + B sin(αt), # X (x) = C cos Cα x + D sin Cα x G α α x + D sin x u (x, t) = X (x) T (t) = (A cos (αt) + B sin (αt)) C cos c c T (t) X(x) :)% u (0, t) = (A cos αt + B sin αt) C ≡ 0. G C = 0, # α (l, t) ≡ 0 = (A cos αt + B sin αt) D sin αc l , ! sin ( l) = 0, " c kπ αk = kcπ , l k = 1, 2, . " % &) % u(x, t) = ∞ ! sin k=1 kcπ kcπ Ak · cos t + Bk · sin t l l kπ x l {Ak }k=1 # {Bk }k=1 *% & / # " ) 2
f (x) # g(x) /*(# K2 !##2 ∞ ( !% " ( u (x, 0) = f (x) # u (x, 0) = Σ Ak sin( kπ x) ! Ak l k=1 f (x) /*(# [0, l] ( ( K2 k 2 *% & #) ∞ ut (x, t) = Σ (−Ak αk sin αk t + Bk αk cos αk t) sin k=1 α k c x . 9 !( t = 02 # / % ( ) % sin(0) = 0; cos(0) = 1 2 ) % ) 5 ∞ ut (x, 0) = Σ αk Bk sin k=1 kπ x l = g (x) . :% % g (x) f üggvény K2 [0, l] ( ∞ g (x) = Σ bk sin k=1 ∞ Σ αk Bk sin k=1 kπ x l = g (x) # ) % αk Bk = bk . :% k = kπ x . l ∞ g (x) = Σ bk sin bk αk = k=1 lbk . kcπ kπ x l $ -.!/ " &2 utt = c2 uxx 2 " c = 1. F + %I # (## $! %I /!* I) %
) # * 9 %I !& u (x, t) /(#P y y = 0.05x y = 1 − 0.05x x "* 57 . ( %I % , #.) # %I # F :% g (x) ≡ 0. G ∀k ≥ 12 Bk = 0, % g (x) ≡ 0 /*(# K2 bk % & F {Ak } *% & % % % 0 ≤ x ≤ 10 0, 05x, f (x) = , . 1 − 0, 05x, % 10 ≤ x ≤ 20 /##P Ak = 40 (kπ)2 2 l 20 0 f (x) sin kπl xdx = 2 20 10 0 0, 05x sin kπ x 20 20 2 dx+ 20 (1 − 0, 05x) sin kπ xdx = 20 10 sin kπ 2 , %& # ! +% 7 B % . :% ∞ 40 kπ kπ kπ cos t sin x u (x, t) = Σ 2 2 sin k=1 k π 2 20 20 E π π 40 1 t sin x − ( cos π 2 12 20 20 1 3π 1 5π 3π 5π − 2 cos t sin x + 2 cos t sin x − .) 3 20 20 5 20 20 u(x, t) =
&& )* +# 2 2 = g (x) ; u (0, t) = ∂∂t2u = c2 ∂∂xu2 u (x, 0) = f (x) ; ∂u ∂t (x,0) u (l, t) = 0 / # J #(# ) % % u1, u2 : R R # @+ % & " /*(#2 ) u (x, t) = u1 (x + ct) + u2 (x − ct) 2 ∂u #! " %I ∂∂t2u = c2 ∂x + @+ # 2 1 % u (x, t) = u1 (x + ct)+u2 (x − ct) , ∂u = cu1 (x + ct)− ∂t (x,t) cu2 (x − ct) . :( ∂ 2 u 2 2 (x,t) = c u1 (x + ct) + c u2 (x − ct) 2 ∂t ) 55 # ∂u ∂x = u1 (x + ct) + u2 (x − ct) # ∂ 2u = u1 (x + ct) + u2 (x − ct) 2 ∂x / # " & 2 ∂ 2 u 2 2∂ u (x,t) = c [u1 (x + ct) + u2 (x − ct)] = c (x,t) ∂t2 ∂x2 % / # " % u1, u2 /*(# . u (x, 0) =
f (x) # ut (x, 0) = g (x) / # # u (x, t) = u1 (x + ct) + u2 (x − ct) t = 02 !( u (x, 0) = u1 (x) + u2 (x) . 9 u (x, t) = u1 (x + ct) + u2 (x − ct) /# t ( ) t = 02 % !*) ∂u (x,0) = cu1 (x) − cu2 (x) . ∂t :% ) % f (x) = u (x, 0) = u1 (x) + u2 (x) # g (x) = ut (x, 0) = c (u1 (x) − u2 (x)) ! u1 (x) + u2 (x) = f (x) u1 (x) − u2 (x) = g(x) c $$# @+ ) ,> A /. ( & &) % ⎧ ⎪ ⎪ ⎨ u1 (s) = 12 f (s) + 1 2c ⎪ 1 ⎪ ⎩ u2 (s) = 2 f (s) − 1 2c s 0 s 0 ⎫ ⎪ g (τ ) dτ + A ⎪ ⎬ ⎪ g (τ ) dτ + B ⎪ ⎭ "* " 5? ) % A + B = 0 N % u (x, t) = u1 (x + ct) + u2 (x − ct) = 1 2c x+ct 1 [f (x + ct) + f (x − ct)] + 2 g (τ ) dτ x−ct E* #) % (# %
* / # ) % u (0, t) = u (l, t) = 0, % / (# M(# %IM %I # C$ 7 -.!/ :* "" # ) # J &#( %I ( %I %I !& u (x, t) /*(# y y = 0.05x y = 1 − 0.05x x . 1 2 u (x, t) = [f (x + t) + f (x − t)] , % f (x) = # 0.05x % 1 − 0, 05x % y f (x) x 0 ≤ x ≤ 10 10 ≤ x ≤ 20 2 ) 5A :% ) % 10 "# I ( ## I % ) % x = 2 cm2 ⎡ u (2, 10) = %I ⎞⎤ ⎛ 1 1 ⎝ [f (12) − f (−8)] = ⎣1 − 0.05 05 · 8⎠⎦ · 12 − −0, 2 2 f (12) f (−8) 1 [1 + 0, 4 − 0, 6] = 0, 4 = 2 * ) +, ! -+ . ( 2 ! ! u (x, 0) = f (x) , ut
(x, 0) = g (x) , ! u (0, t) = u (l, t) ≡ 0 / % 2 3( % ! u (x, t) = 12 [f (x + ct) + f (x − ct)]+ 1 2c x+ct g (τ ) dτ / # = (#2 x−ct #( 9 * / # f # g /*(# # ## [0, l]2" R2 $(" # = % & /*(#) [0, l] ( % 2 & & *$ ! [−l, l]2 # /*(# 2l y −3l −2l −l l 2l 3l 4l x J & C$ (# %I %I #) # + u (0, t) = u (l, t) ≡ 0 * / # ) # f, g /*(# # ( 3( % ! / ) % $2( ("## # %
,x = 0 # x = l.2# E # (x0 , t0 ) ) t = − 1c (x − x0 )+t0 # t = 1c (x − x0 ) + t0 / "* 5H t (x0 , t0 ) (0, t0 − x0 c (l, t0 + x0 c (l, t0 − (0, t0 + x0 c − 2l c x0 c − cl ) t0 − 1c (x − x0 ) + t0 t + 1c (x − x0 ) + t0 (l, t0 + −2l − cl ) −l l 2l x0 c − 3l ) cx 3l x00 + ct0 x0 − ct0 x0 − ct0 + 2l 4l − x0 − ct # /) x + ct = const1 # x − ct = const2 I /*(# "$ # ) % ( " f (x0 + ct0 ) & &) % x0 − ct0 −f (−x) −f (−x + 2l)−f (−x + 4l) f (x) −2l −l f (x + 2l) l 2l 3l 4l x f (x − 2l) x0 + ct0 :% f (x0 + ct0 ) = −f (4l − x0 − ct0 ) # ! f (x0 − ct0 ) = f (x0 − ct0 + 2l) . , 4l −x0 −ct0 # x0 −ct0 +2l :% [0, l]2" ( . ) 5B u (x0 , t0
) = 1 1 f (x0 − ct0 + 2l) − f (4l − x0 − ct0 ) 2 ⎡ 2 ⎢ −l 0 l 1 ⎢ ⎢ g (y + 2l) dy + −g (−y) dy + g (y) dy + ⎢ 2c ⎢ ⎣x−ct 0 −l 0 ⎤ x+ct ⎥ 2l 3l ⎥ + −g (−y + 2l) dy + g (y − 2l) dy + −g (−y + 4l) dy ⎥ ⎥ ⎦ l 2l 3l 0 x+ct % ( ) % g (y + 2l) dy = g (s) ds # −g (−y + 4l) dy = x−ct 2l+x−ct 3l 4l−x0 −ct0 s = −y + 4l, y 3l x0 + ct0 = g (s) ds. ds = −dy, s l 4l − x0 − ct0 l :% u (x0 , t0 ) = 1 (f (x0 − ct0 + 2l) − f (4l − x0 − ct0 )) + 2 ⎡ ⎤ 4l−x l 0 −ct0 1 g (s) ds + g (s) ds⎦ + ⎣ 2c 2l+x0 −ct0 l ⎡4l−x −ct ⎤ 0 0 1 1 = (f (x0 − ct0 + 2l) − f (4l − x0 − ct0 )) + ⎣ g (s) ds⎦ . 2 2c 2l+x0 −ct0 # % ) (x0 − ct0 + 2l) , (4l − x0 − ct0 ) , (4l − x0 − ct0 ) # (2l + x0 − ct0 ) [0, l] ( & ( < /) % / (x0 , t0 )2 ( ) &
& /*") % % M( ("# M % (x, t) & [0, l] ( ) / {(x, y) : 0 ≤ x ≤ l, 0 ≤ y} ( x + ct = kl I x − ct = kl "* 5D t 4 cl 3 cl 2 cl l c x l "" / (x0 , t0 ) ! (2, 3) ( ) x − ct = kl ) x + ct = kl 2 3"# *$"# ( :% (2, 3) #(" 2 (x, t)2 u (x, t) 12 (f (x0 − ct0 + 2l) − f (4l − x0 − ct0 )) + 1 2c 4l−x0 −ct0 2l+x0 −ct0 g (s) ds Q + / ( % &) 2 Q + / * $$" /* utt = c2 uxx , u (x, 0) = f (x) , ut (x, 0) = g (x) # u (0, t) = u (l, t) ≡ 0 * / # C ( / " 0 0 u (0, t) = u (l, t) ≡ 0 * / #
% % 2 utt = c uxx , u (x, 0) = f (x) , ut (x, 0) = g (x) , x ∈ R. & ) ) ) ( I / /% ) % ) ?F J &# ) 1 1 u (x, t) = (f (x + ct) + f (x − ct)) + 2 2c = -.!/ x+ct g (s) ds x−ct 9 f (x) ≡ 0 # (x) = cos x, u (x, t) = ,9 1 1 (sin (x + ct) − sin (x − ct)) = cos x sin ct. 2c c sin (α + β) − sin (α − β) = 2 cos α sin β $/*# . -.!/ :* (# %I %I) % 2 / (−a, 0) ; (a, 0) ; (0, b) ) * ! / %I !& u (x, t) /*(# u t=0 B −A A x ha |x| ≤ a b − b|x| a # g (x) ≡ 0, % 0 ha |x| > a %I % u (x, t) /*(# . f (x) = u (x, t) = 1 [f (x + ct) + f (x − ct)] , 2 > 5 # > -.!/
+*# "* ?> u t=0 B b 2 b 2 a 2c x t= a c x 2a t= c b 2 −3a t= −2a −a a 2a x x 3a u −a a x !" %I x /) # # ( 2 ++ *# $# x = 0 +*# t = 02 |x| < a g (x) = 1 # ) $ 2 # N / %I !& u (x, t) /*(#P . f (x) ≡ 0; g (x) = 1 u (x, t) = 2c 1, 0, x+ct g (s) ds = x−ct ha |x| < a . ha |x| ≥ a 1 {% 2c (x − ct, x + ct) ∩ (−a, a) 2 } . ? ) a 1 ) = u x, % 2c 2c * a a x − ,x + ∩ (−a, a) 2( . 2 2 * $$" |x| < a2 2R a2 < x < 3a 2 # x > 2 % 3a 2 1 1 u (x, t) = (f (x + ct) + f (x − ct)) + 2 2c x+ct g (s) ds x−ct & ) % (x0 , 0) / # % c # ) ( / c
# ! %# 0 0 0 # * ) # (## 00 C 2 (! % # I % l V # *) % I x ( ) # (# I %"## # f (x) /*(# u (x, t) /*(#) I %"## # ! x t " I ( %"## t = 02 > ( %" % " % / #) # # " I ( ( x I #% 2 ) #% + %"## " "% ( 9 x t " ( # u (x, t) %"## % % ) / % K" & #(# ) % %"# ut = kuxx , (0 < x < l; 0 < t) C + @+ ! , (## 2 % =
I$# +!C $(# . ! I %"## t = 02 ) ( u (x, 0) = f (x) . ! & &) % I # (# 00 C 2 ( & ( ) I& & %" " ?7 !"! !" " # $ 0◦ C 0◦ C % f (x) l / ! ,# ) % . :% ! u (0, t) = u (l, t) ≡ 0. E ⎧ 0 < x < l, 0 < t ⎨ ut = kuxx , u (0, t) = u (l, t) ≡ 0 ⎩ u (x, 0) = f (x) # K2 % ( , %"(# ) ?5 # # K ) # & ( # +# & ( >B 2 . . u(x, t) /*(#) + I " %I # u (x, t) = X (x) T (t) * (# $( "
%I @+ # (### # # E X T = = −λ = . kT X G :( T = −λkT ⇒ T (t) = Ae−λkT . −X (x) = λX (x) , 0<x<l 2 # X (0) = X (l) = 0. # ) # / " ) X (x) = sin :% ∞ nπx . l 2 nπ u (x, t) = Σ An e−( l ) kt n=1 / #() % " %I sin nπx l ∞ nπx n=1 l f (x) /*(# K2 ⎫ ⎧ (0 < x < l, 0 < t) ⎬ ⎨ ut = kuxx , u (0, t) = u (l, t) ≡ 0 ⎭ ⎩ u (x, 0) = f (x) # % I ) % f (x) /*(# K2 /* [0, l]2 , " %I # . G * {An }∞ n=1 *% & ) !##( / ! u (x, t) # # {An }2 ## % !##( f (x) = Σ An sin ∞ 2 nπ u (x, t) = Σ An e−( l ) n=1 kt sin nπx l " ??
$/*# 2 λn = nπ Xn (x) = sin nπx /*(#2 l l ! %!( ) % % A R R2 # " # / @+ % & /*(# # 2 I ) Ag = −g , $ # AXn = λn Xn . E* #) % A2 (# ## ( π 2 4π 2 9π 2 ; ; ;. l2 l2 l2 % ) % A2 ## ⎧ (0 < x < l, 0 < t) ⎨ ut = kuxx , -.!/ T ⎩ u (t, 0) = u (l, t) ≡ 0 / 2 u (x, 0) = f (x) 20x ha 0 ≤ x ≤ 5 ) % l = 10 # f (x) = 200 − 20x ha 5 ≤ x ≤ 10 , > . k = 1. E t = 02 I $ # %"## 1000 C 2) # (# / # +$ 02% . : K2 / f (x) /*(# ∞ nπx E {An }∞ n=1 ##) f (x) = Σ An sin l (0 ≤ x ≤ l) . & ) % An = 1 An = 5
= −200 5 0 1 nπ xdx + 20x sin 10 5 −2 sin( nπ ) 2 + nπ cos n2 π 2 nπ 2 5 (200− 20x) sin 5 +200 2 l n=1 l 0 f (x) sin nπ xdx. l nπ xdx = 10 + 2 sin( nπ ) −2 sin(nπ) + nπ cos nπ 2 2 = 2 2 nπ nπ 800 = 2 2 sin( ). , 7 nπ 2 ) ?A y y = 20x y = 200 − 20x ⎧ 0, ⎪ ⎪ ⎨ 1, ( sin nπ = 2 0, ⎪ ⎪ ⎩ −1, + xπ sin 10 f (x) = 800 − π2 12 9 ( ha ha ha ha n = 4k n = 4k + 1 , ! n = 4k + 2 n = 4k + 3 sin( xπ 3) 10 32 + sin( xπ 5) 10 52 − sin( xπ 7) 10 72 + ··· . ∞ 2 nπ u (x, t) = Σ An e−( l ) kt n=1 $/*# , sin nπx l u (x, t) = π )2 t −( 10 800 e [ π2 12 2 − 5π t 10 e ( ) sin xπ 10 − sin( xπ 5) 10 2 5 2 t − 3π 10 e ( ) − ···] sin( xπ 3) 10 32 + x " ?H 1+ , $(" # % K2 / % ) #
$/ =!+&) # f : R R, / , *#) + # / ) # % .) ( ∞. f (x) 4 ! ∞ F [f ] (λ) = g (λ) = ∞ −∞ |f (x)| dx < f (t) e−iλt dt. −∞ #*) % i # ,i2 = −1. > % &) % g (λ) # / R2) # lim g (λ) = 0. |λ|∞ ∞ 9 f / ) # f (x) dx < ∞, F [f ] (λ) = iλF [f ] (λ) , .−∞ # ! ∀n2 F f (n) (λ) = (iλ)n F [f ] (λ) * ) % f (n) / ) # ∞ f (n) (x) dx < ∞. −∞ %# 0 0 C# k = 1 %"(# ut = kuxx , # N $(" # ( ut = u”xx −∞ < x < ∞ # t ≥ 0 . u (x, 0) = f (x) :* / ) % ∞ −∞ |f (x)| dx; ∞ −∞ 2 |f (x)| dx; # ∞ −∞ |f (x)| dx
# (f ∈ C (R)) . / u (x, t) /*(# $# ) u ∈ C 2 ,( # / @+ % &. # ,. ∀x ∈ R2 ∞ −∞ |u (x, t)| dx; ∞ −∞ |ux (x, t)| dx; ∞ −∞ ” uxx (x, t) dx < ∞. ) ?B ,. ut (x, t) /*(# [0, r] ( ( (t2" /* ) % & g (x) |ut (x, t)| ≤ g (x) # ∞ g (x) dx < ∞. −∞ . E* ut = uxx # x2 K2 / ∞ v (λ, t) = u (x, t) e−iλx dx = F [u] . −∞ F [ut ] = ∞ ∂ ut (x, t) e−iλx dx = ∂t −∞ ∞ u (x, t) e−iλx dx = vt (λ, t) , −∞ %% / ,. / # u (x, t)2 ( F [uxx ] = −λ2 v (λ, t) , ! ∀λ2 t ( & & ⎫ vt (λ, t) = −λ2 v (λ, t) ⎬ ∞ )
$ ! f (x) e−iλx dx ⎭ v (λ, 0) = −∞ t ( & ( +% / −λ2 t ∞ v (λ, t) = e 2 f (x) e−iλx dx = e−λ t F [f ] (λ) . −∞ , + / π − λ2 2 "" & ) % F e−ax = e 4a . a = a % !# $(" x2 e− 4t 2 (λ) = e−λ t . F √ 2 πt / % ( & 2t v (λ, t) = e−λ $(" ∞ 2 f (x) e−iλx dx = e−λ t F [f ] (λ) 2 −∞ x2 x2 e− 4t e− 4t (λ) F [f ] (λ) = F √ ∗ f (x) , F [u] (λ) = v (λ, t) = F √ 2 πt 2 πt 1 4a " , % ) [h ∗ g] (x) = :% ∞ ?D h (u) g (x − u) du. −∞ 1 u (x, t) = √ 2 πt ∞ x2 e− 4t f (x − v) dv. −∞ :% K2 / !##( + @+ $$# @+ (* ( AF )
30 % 0 5 C $!( % ! & 0 = ! ()*+(, r : [a, b] R3 @+ % & /(# < C $# (*) % $(" , . r ) ( t1 = t2 2 r (t1 ) = r (t2 )R ,. r ! R ,+. ṙ = 0 r $ " " ## # $ # ,. r (t) = (x (t) , y (t) , z (t)) = ix (t) + jy (t) + kz (t) ,. / *) % r (t) /(# x (t) , y (t) , z (t) , $$ . @+ % & ,. [a, b] ( ( -.!/ A> A , . C $ 0 % ) P0 (x0 , y0 , z0 )) # ( ) v = (v1 , v2 , v3 ) . 2 , ) .) ## ⎞ ⎛ r (t) = ⎝x0 + tv1 , y0 + tv2 , z0 +
tv3 ⎠ x(t) ,. $ :* y(t) z C # a z(t) I $%2 z a a y x % / * # % " r (t) = (a cos t, a sin t, ct) , % 0 ≤ t, 0 < a, 0 < c $# (* ,+. - ) & ) % R I $ / * # % # $("# P R I $/ * ) ( P0 P xy ! ( & (* Ψ OP /# # z $ (0 ≤ Ψ ≤ π), # ϕ OP0 /# # x $ (0 ≤ ϕ ≤ 2π) . % P =(x0 , y0 , z0 ) ) % y0 = R sin Ψ sin ϕ z0 = R cos Ψ. :( . x0 = R sin Ψ cos ϕ E $/ * A7 ## r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ) . ) #!& ) % ϕ2 $!*) # Ψ2 2 # (0 ≤ Ψ ≤ π).
I ) % Ψ2 $!*) # ϕ2 2 # ,0 ≤ ϕ ≤ 2π . E R I $/ * (ϕ0 , Ψ0 ) #% & P0 % & ) ! * r (Ψ) = (R sin Ψ cos ϕ0 , R sin Ψ sin ϕ0 , R cos Ψ) . P0 2 % & r (ϕ) = (R sin Ψ0 cos ϕ, R sin Ψ0 sin ϕ, R cos Ψ0 ) . 6 % √ √ N / R = 4 I $ P0 = 6, 6, 2 % & %I # # # $$P √ 3 π ( 0 ≤ Ψ ≤ π # 4 cos Ψ = 2, ! Ψ = . :% sin Ψ = , ! 0 0 3 2 √ √ √ √ 3 3 π 4 2 cos ϕ0 = 6 # 4 2 sin ϕ0 = 6. # ϕ0 = 4 , % P0 ϕ0 = π4 # Ψ0 = π3 #% N ) ! 1 0 √ √ 2 2 sin Ψ, 4 sin Ψ, 4 cos Ψ , (0 ≤ Ψ ≤ π) , r (Ψ) = 4 2 2 r (ϕ) = 1 √ 1 3 3 cos ϕ, 4 sin ϕ, 4 , (0 ≤ ϕ ≤ 2π) 4 2 2 2 0 √ # # $ ( # C $!() r (0) = r (2π) , % # ( [0, 2π] (
#( # ( ) C $# ! ()*+(, # $/*" % % (*) % # $) % C $!( > ! ()*+(, @+ % & $# (*) % # (# ,0 $ ( . " A5 & -.!/ $* @+ % & $#W ) # ) @+ % & $ ) # B ) + ) C # ) $ $) % C $!( / ) +I+ 2 # @+ % & + *3" ! ()*+(, γ C $!( ## r (t) ; a ≤ t ≤ b. a = t0 < t1 < < tn−1 < tn = b, A = r (t0 ) , B = r (tn ) # Pi = r (ti ) , % 0 < i < n. $ / P1 A P2 γ P3 P4 Pn−1 B r(t) t0 = a a1 a2 a3 an−1 tn = b
L , / " . $ %# AP1 P2 . Pn−1 B % 9 L (# ) γ $ 7 0 # !(% L. A? = 1-1 . 9 γ C $!( ,/ *) % / @+2 % &.) γ = % &) # !(% b |ṙ (t)| dt L= a * γ ### % -.!/ 9 r (t) = (cos t, sin t, t) !( % / ## . 0 ≤ t ≤ 2π, ṙ (t) = (− sin t, cos t, 1) , |ṙ (t)| = :% L = 2π 0 |ṙ (t)| dt = 2π √ 0 / sin2 t + cos2 t + 1 ≡ √ 2. √ 2dt = 2 2π. 2- . 3 r (t) , a ≤ t ≤ b " !( s (t) = t ## |ṙ (τ )| dτ ) s (t) γ C $2 t " / a !( % ( s (t) I ) ! # ( /*(#) # I # / @+ % &
t (s) s (t) ( /*(# % L $ γ %) t : [0, L] [a, b] , r : [a, b] γ, (r ◦ t) : [0, L] γ :% 2 r (s) = (r◦t) (s) = r (t (s)) γ $ ## 2 r (s) ## γ $ 0 ( %!( -.!/ !(% r (t) = (cos t, sin t, t) , 0 ≤ t ≤ 2π + ( ( ###P . t s (t) = 0 |ṙ (τ )| dτ = t √ 2dτ = 0 (/*(# t (s) = √12 s. E s s √s √ √ # 2 r (s) = cos 2 , sin 2 , 2 , √ 2t. :% s (t) = √ 2t /*(# + ( ( # #2 √ 0 ≤ s ≤ 2 2π $C ) % AA 2 r (s)| ≡ 1. r (s) #" ( &) |2 $# % 1-1 . γ C $!() # 2 r (s) , (0 ≤ s ≤ L) r (s)| ≡ 1. ## |2 γ !(% I " 2 r (s) = ṙ (t (s)) dr (t
(s)) ṙ (t (s)) = ṙ (t (s)) t (s) = = ds s (t (s)) |ṙ (t (s))| #( & # # / % ) % s (t) = t |ṙ (τ ) dτ | , a ! s (t) = |ṙ (t)| . E !(% ## ##) % #" ,#. ( % > > 1-1 . 2 r (s) , 0 ≤ s ≤ L γ $ # # 2 r (s) * 0 ≤ s ≤ L2 r (s) ⊥ 2 #2 I " :) % |2 r (s)| ≡ 1, % 2 r (s) ≡ 1 ( ( r (s) 2 r (s) + 2 r (s) 2 r (s) ≡ 0, ! 22 r (s) 2 r (s) ≡ 0. # 2 r (s) 2 E 2 r (s) ∀s2 r (s) ⊥ 2 E #; γ C $ !(% ## 2 r (s) , 0 ≤ s ≤ L. :* / ) % 2 r (s0 ) % (I P0 2 r (s0 ) = 0. ) % P1, P2 $# P0 2% $ ) P1 P0 P2 * KP1 P0 P2 $ % & KP1 P0 P2 γ P1 P0 P2 P1 , P2 P0 KP1 P0 P2 $$
# % % ) 1 ) P0 2 #" " |r̃ (s I $ $ γ $ 0 )| P0 2 (* &$ $# % ( r̃ (s0 ) + |r̃r̃ (s(s0)|)2 . 0 AH γ r(s0 ) + r (s0 ) |r (s0 )| r(s0 ) r (s0 ) |r (s0 )| P0 r (s0 ) ! ()*+(, &$ + γ $ P0 = r̃ (s0 )2 ( % %!( O κ (s0 ) , % κ (s0 ) = |r̃ (s0 )| . % ) % r̃ (s) ⊥ r̃ (s)2) ) % r̃ (s) #(2 ) # r̃ (s) I #() n (s) = |r̃r̃ (s) (s)| ! (* #" I #( t (s) = r̃ (s)) ! ( 2 b (s) = t (s) × n (s) , % t (s) , n (s) , b (s) #(& & I P = r̃ (s) %!( #" (t (s)) # /"
#( (n (s)) ! * (u − r (s0 )) b (s0 ) = 0. /" (n (s)) # (b (s)) ! 0 (u − r (s0 )) t (s0 ) = 0. #" # ! 7 ) (u − r (s0 )) · n (s0 ) = 0. A 6 3 3 0 ! I $ = & ! %2 ) ! # #"! ( P0 = r̃ (s0 ) $* $ ( 2 &! % 2 = &! ! ) ! 2 ! 8 ! % / ! P0 ( ) % & ,=!+& . # $* ) AB b n t γ ! ( $! n b b t n t b 1-1 . $* #"( !(% $# 2 γ $ # ## r̃ (s) , a ≤ s ≤ b. I
" Δα s0 # s0 + s ( t (s) # t (s0 + s) #" $ t(s0 ) t(s0 + δs) δα γ #% & t(s0 + δs) r(s0 + δs) r(s0 ) s0 s0 + δs #"( !(% $# r̃ (s0 )2 =!+& 2 lim α = lim sinα , ( !( ) % % ) % sin α = s s s0 s0 AD |r̃ (s0 ) × r̃ (s0 + s)| = |r̃ (s0 ) × [r̃ (s0 + s) − r̃ (s0 )]| , ) % α sin α = lim = s0 s s0 s r̃ (s + s) − r̃ (s ) 0 0 = lim r̃ (s0 ) × = lim r̃ (s0 ) × r̃ (s0 ) = s0 s0 s = r̃ (s0 ) = χ (s0 ) . ,7 > lim C $!( * $$" % 2 I 2 &! % &! #( # ( #( $) 2 # $ !$#" ###
* ! ()*+(, r̃ (s) (% % & r̃ (s) = ##C γ $ s #2 τ (s) := − ḃ (s) , n (s) E* #) % #( ( ) ḃ (s) , % /" #( (n (s) 2 ) . 3 b (s) b (s) ≡ 1, ( 2ḃ (s) b (s) ≡ 0, ḃ (s) ⊥ b (s) . # b (s) t (s) ≡ 0, ( ḃ (s) t (s) + b (s) ṫ (s) = 0, χ(s)n(s) 0 ! ḃ (s) ⊥ t (s) . ḃ (s) ⊥ b (s) # ḃ (s) ⊥ t (s) " #" $() % ḃ (s) n (s) . :% ḃ (s) = −τ (s) n (s)) ) % ḃ (s) = |τ (s)| . 1-1 . |τ (s)| . τ (s) =!+& " $(2 #( !(% $# I " HF Δβ s0 és s0 + Δs #% & b (s0 ) # b (s0 + Δs)
#( $ 2 #( !(% $# s0 #% & r̃ (s0 ) " Δβ sin Δβ |b (s0 ) × b (s0 + Δs)| = lim = lim = Δs0 Δs Δs0 Δs Δs0 Δs |b (s0 + Δs) − b (s0 )| = lim b (s0 ) × = Δs0 Δs · · = b (s0 ) × b (s0 ) = b (s0 ) = |τ (s0 )| . ,7 lim <?"#; K2/ t = n = b = −κt κn +τ b −τ n ⎤ ⎡ ⎡ ⎤⎡ ⎤ t 0 κ 0 t τ ⎦⎣ n ⎦ K2/ ⎣ n ⎦ = ⎣ −κ 0 b 0 −τ 0 b I " " # % / I / n (s) = b (s) × t (s) # ( ( n (s) = b (s) × t (s) + b (s) × t (s) = −τ (s) n (s) × t (s) + b (s) × −b(s) × (κ (s) n (s)) = τ (s) b (s) − κ (s) t (s) , b (s) × n (s) = −t (s) . $(" # $* # & # 2 1-1 . 4A 6 0
5 $# # C 2 % !(% ## κ (s) # τ (s) /*(# :() % κ (s) > 0 # τ (s) / /*(# [a, b] ( & R2 # ) # $) !(% #2 # # $* κ (s) , & τ (s) 1" 62 " 0 # % 1-1 . r (t) a ≤ t ≤ b γ $ " # $* t #% & r (t) κ (t) = |ṙ (t) × r̈ (t)| |ṙ (t)|3 #2 + H> I " :) % !(% t " s (t) = t |ṙ (τ )| dτ. a r (t) = r̃ (s (t)) , ! 2 ṙ (t) = r̃ (s (t)) s (t) # r̈ (t) = r̃ (s (t)) s (t) + r̃ (s (t)) s (t) . |ṙ (t) × r̈ (t)| = |ṙ (t)|3 |r̃ (s (t)) × r̃ (s (t))| |s (t)|3 = |r̃ (s (t))| = κ (r̃ (s (t))) = κ (r (t)) . = 3 |r̃ (s (t))| |s (t)|3 1 ,7 7. 1-1 .
r (t) , a ≤ t ≤ b $ r (t) & τ (t) = γ " ## ṙ (t) r̈ (t) r (t) |ṙ (t) × r̈ (t)|2 45 / * #% (" * 2 # % z = f (x, y) z = x2 − y 2 F (x, y, z) = 0 0# x2 + y 2 + z 2 = 1 x2 a2 + x2 a2 + x2 a2 − y2 =1 ) − z2 c2 =1 $ C % ) − z2 =1 #$ C % + y2 b2 b2 $) c2 b2 y2 z2 c2 H + r (u, v) = (x (u, v) , y (u, v) z (u, v)) , 0# r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ) R I $) r (u, v) = ((a + b cos v) cos u, (a + b cos v) sin u, b sin v) * & /2 "?2 > , . xz ! z = f (x) /*(#) % / 2 x # xy ! && z $* ) / ( ( ) ( x2+ y 2
( E z C / // * z = f x2 + y 2 , % f " /*(# (f : R R) . z = x2 + y 2 ( z = sin (x2 + y 2 ) . ,. xz ! $ (x (u) , z (u)) , a ≤ u ≤ b + H7 z b y a x $ z $* / ( /2 a≤u≤b . / * r (u, v) = (x (u) cos v, x (u) sin v, z (u)) , 0 ≤ v ≤ 2π , . K % & K $ z $* z (a + b cos u, b sin u) b u a x #* & / ) K $ u (a + b cos u, b sin u) , 0 ≤ u ≤ 2π. N & / * r (u, v) = ((a + b cos u) cos v, (a + b cos u) sin v, b sin u) , 0 ≤ v ≤ 2π. ,. K % & /# $ z $* H5 z R cos ψ R P ψ R x R sin ψ R /# $ Ψ (R sin Ψ, R cos Ψ) , 0 ≤ Ψ ≤ π $ # r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ) , 0 ≤ ϕ ≤ 2π, $/ !
## E " :* / ) % / z = f (x, y) ) z = x + y . ∀c ∈ R2 # * c2% & % z = x2 + y 2 $("# {(x, y) |f (x, y) = c} . # c < 0 % & % 23 # "J c = 02% & % = {(0, 0)} . c > 02% & % " 00" : √c ; : % ) $#) ! #% " %!( " 2 2 y 0.251 √ 2 x :* / ) % z = f (x, y) / ( r(t) = (x (t) , y (t)) , a ≤ t ≤ b $ f (r (t)) ≡ c, ( c ∈ R # a ≤ t ≤ b2 f (r (t)) ≡ c # t2 ( ( ) + H? % f (r (t)) ṙ (t) ≡ 0. E ∀t ∈ [a, b]2 grad|r(t) ⊥ ṙ (t) :% ) % ( # #"# ( % ## ) % z = f (x, y) / *
%# / ) # ( $#) #(" % / $ 1-1 . z = f (x, y) / * # / * $(# #) xy ! " (* r (t) $ r (t)2 ( & f 2 / # # "## r (t) 2 % + ) % r (t) # % & ( z z = f (x, y) $!" y r(t) x $ -.!/ z = f (x, y) ( # r (t) $# 2 / # # "## A # B % 9 / * r (u, v) = (x (u, v) , y (u, v) , z (u, v)) , (u, v) ∈ D ⊂ R2 , ) % / * #! "" " < # !##( U + / * HA !! y 10 20 30 40 50 B A r(t) x # % /# # & / *
9 ( ( ) / * r (u, v) = (x (u, v) , y (u, v) , z (u, v)) 9 z = f (x, y) ( ) # ) I) % u = x, v = y # 2 !% r (u, v) r (u, v) = (u, v, f (u, v)) . :% & ) u, v r (u, v) I / * # ( & z = f (x, y) 2 / * (%" % /# & I + 2 2 2 2 2 2 x2 +y 2 +z 2 = 1 ( xa2 + yb2 − zc2 = 1 ( xa2 − yb2 − zc2 = 1 / / * & ! r (u, v) " *) % $/ * $ C %2 y2 x2 z2 a2 + b2 − c2 = 1 r (u, v) = (a cos uchv, b sin uchv, c · shv) , 0 ≤ u ≤ 2π, v ∈ R 2( #%* + / * ) r (u, v) # 2 / !) # / " # !##( 4.
! % ! ()*+(, r : T (⊂ R2) R3 # # F ### # # ?2 %!() % T ⊂ R2 $/*" ,+ . ) r : 2 1 − 1 # C) 2 (# @+ % &) + HH 2 ( / ) 2 ru # rv ( /* # ?2 0 # ! ()*+(, 2 ( " r # % ) $/*") # ( $$) % 2 / * 7 -.!/ r (u, v) = (u2 , v 2 , u3 + v 2 ) # ru = (2u, 0, 3u2 ) ; rv = (0, 2v, 2v) . :% ru (0, 0) = (0, 0, 0) # rv (0, 0) = (0, 0, 0) , ( r (u, v) # % / * T ) & P ") u) v T 2 %%) % r (u, v) / * % P ru E* #) % / # ) % ru # rv / ) × rv = 0 / #
(( $ ! ()*+(, ru (u0, v0) × rv (u0, v0)2 ?2 "# " %!( v z / * (u0 , v0 )2% & ru (u0 , v0 ) × rv (u0 , v0 ) rv (u0 , v0 ) v = v0 ru (u0 , v0 ) u = u0 y u x ( / * ( " u r (u, v0 ) $ r (u0 , v0 ) #"( ) v r (u0 , v) $ rv (u0 , v0 ) #"2 ( ) ! ru × rv / * ( " r (u0 , v0 )2 #"! ) % ( :% % r (u0 , v0 ) = (x0 , y0 , z0 ) # ru (u0 , v0 ) × rv (u0 , v0 ) = n = (n1 , n2 , n3 ) , ?2 r (u, v0 )<6 9 (n1 , n2 , n3 ) ((x, y, z) − (x0 , y0 , z0 )) = 0. HB ! n1 x + n2 y + n3 z − n1 x0 − n2 y0 − n3 z0 = 0. |ru × rv | / * % 2 2 2 2 |ru × rv | = (ru × rv ) (ru × rv ) = (ru ) (rv ) − (ru rv ) E |ru × rv | = F2 G
√ EG − F 2 / E = (ru )2 ; F = ru rv ; G = (rv )2 # G 0# %!( A ;<? < 1 5 , . "$ * # ,T 2 uv !. % ( / " 7 / * /& "" "< / * / !# !% + / ( % / !#) / ! $ / * / !# " !( z v δu · ru (u0 , v0 ) δv · rv (u0 , v0 ) r(u0 , v0 ) δv v0 δu y u0 u x ( /! * $ !2 " ) % Δuru # Δvrv % ΔuΔv |ru × rv | * # , $ !# ) # Δu # Δv. :% F / * / ! $ !" √ lim ΣΔuΔv |ru × rv | =
|ru × rv | dudv = EG − F 2 dudv, Δu,Δv0 T T + % 0 HD Σ2 $ # (* T ) % " =!+& :% ?2 ? √ EG − F 2 dudv. Δu, Δv T = -.!/ 9 R I $ / !#P . r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ)) T : (0 ≤ ϕ ≤ 2π; 0 ≤ Ψ ≤ π) rϕ = (−R sin Ψ sin ϕ, R sin Ψ cos ϕ, 0) rΨ = (R cos Ψ cos ϕ, R cos Ψ sin ϕ, −R sin Ψ) R2 E = R2 sin2 Ψ sin2 ϕ + R2 sin2 Ψ cos2 ϕ = R2 sin2 Ψ F = −R2 cos Ψ sin Ψ cos ϕ sin ϕ + R2 cos Ψ sin Ψ cos ϕ sin ϕ = 0 G = R2 cos2 Ψ cos2 ϕ+R2 cos2 Ψ sin2 ϕ+R2 sin2 Ψ = R2 cos2 Ψ + sin2 Ψ = 2 E = rϕ F = rϕ rΨ G = (rΨ )2 :%) R2 I $ / ! 2π π √ EG − F 2 dΨdϕ = ϕ=0Ψ=0 2π π / R4 sin2 ΨdΨdϕ = ϕ=0Ψ=0 = 2πR 2 π Ψ=0 sin ΨdΨ = 4R2 π ,7 5. 2 45 0 F / * ## r :T R3 .
( t (u (t) , v (t)) T 2 /& γ $ ## r (u (t) , v (t)) , a ≤ t ≤ b ) F / * /& γ $ ## -.!/ ! / γ $ !(%P BF r v z F γ t r(u(t), v(t)) T γ t (u(t), v(t)) y u x %> -.!/ 9 γ1 , γ2 # / & = / * $) P0 = r (u0 , v0 ) ) % γ1 # γ2 $# $#P . ! Φ (t) = r (u (t) , v (t)) , a ≤ t ≤ b. :) % γ !(% b L = Φ̇ (t) dt. a ( Φ̇ (t) = ru u̇ + rv v̇ ,$ /*(# ( .) # L= b / (ru u̇ + rv v̇) (ru u̇ + rv v̇)dt = a E u̇2 + 2F u̇v̇ + Gv̇ 2 dt. a 7 ! ()*+(, E u̇2 + 2F u̇v̇ + Gv̇ 2 /# 0?"# 0" K P ,9 EF FG X|ru × rv |2 > 0 # ru "3 % 0,
! E > 0. (* " b √ . / ! Q :* / ) % t (u1 (t) , v1 (t)) # t (u2 (t) , v2 (t)) 2 ##C γ1 # γ2 T 2 $# (u0 , v0 ) Φ1 (t) = r (u1 (t) , v1 (t)) # Φ2 (t) = r (u2 (t) , v2 (t)) ##C + B> a ≤ t ≤ b $# γ1 # γ2 r (u0 , v0 ) = P0 γ1 # γ2 $ P0 2 =!+& Φ̇1 # Φ̇2 $ P0 2 9!( % & $ α2 Φ̇1 Φ̇2 (r u̇1 + rv v̇1 ) (ru u̇2 + rv v̇2 ) = cos α = = u |ru u̇1 + rv v̇1 | |ru u̇2 + rv v̇2 | Φ̇1 Φ̇2 E u̇1 u̇2 + F (u̇1 v̇2 + u̇2 v̇1 ) + Gv̇1 v̇2 / =/ 2 . ,7 ? E u̇1 + 2F u̇1 v̇1 + Gv̇1 E u̇22 + 2F u̇2 v̇2 + Gv̇2 z F Φ̇2 P0 γ Φ̇1 γ α y x " &) % 7 1-1 . / * ( ( " F = 0. P0 = r (u0, v0 ) % & #2 (u0,
v0 ) " " ) % I " 0 #( # u̇1u̇2 = v̇1v̇2 = 0. :% / # & F 2 # u̇1 v̇2 # u̇2 v̇1 $* F) # / $ 0) % F = 0. " # (E, F, G) / * ,!(%) $) / !. % ( ) % / *2 % ! 9 / * # ( ) *# # L, M # N. B 9 O $ * / * #( m2 P E ru × rv . m= |ru × rv | 2 2 2 2 ∂ r ∂ r ∂ r ∂ r :( ruu := ∂u 2 ; ruv := ∂u∂v ; rvv := ∂v 2 ; rvu := ∂v∂u . 2 # mv := ∂m . & ) % ruv = rvu , ( mu := ∂m ∂u ∂v = ! ()*+(, L = ruum; M = ruv m; N = rvv m # #" G A ;<? ?# %!(
! ()*+(, (* Lu̇2 +2M u̇v̇ +N v̇ 2 /# #" 0?"#< 1 5 - - 5 = 1-1 . F / * ## r : T R3 . :( γ T 2 $) t (u (t) , v (t)) , a ≤ t ≤ b ## Φ (t) = r (u (t) , v (t)) ##C / * $ γ $* 1 Lu̇2 + 2M u̇v̇ + N v̇ 2 κ= 2 2 , mn E u̇ + 2F u̇ v̇ + G v̇ ) % γ $ /" #( ) n + / * #"! ) ( nm =0, ( / ( # ! !(% ## ## ) # $ /*(# ( & & $(# ) % 1-1 . 9 / * % & # $# &! ,# / * ( #"2 !( .) $* I " / * T γ1 : t (u1 (t) , v1 (t))
γ : t (u (t) , v (t)) $# F 2 2 2 # ) % (u0, v0 ) + B7 $$ &! P0 = r (u0 , v0 ) ! * $$ P0 2 #"!& P γ1 : Φ1 (t) = r (u1 (t) , v1 (t)) , a ≤ t ≤ b $# t1 (u0 , v0 ) = γ2 : Φ1 (t) = r (u2 (t) , v2 (t)) ±t2 (u0 , v0 ) # n1 (u0 , v0 ) = n2 (u0 , v0 ) ( #" # /" #2 ( Φ̇1 (t) = ru u̇1 + rv v̇1 |(u0 ,v0 ) % Φ̇2 (t) = ru u̇2 + rv v̇2 . :% (u0 , v0 )2 {ru , rv } /* # (u̇1 , v̇1 ) = λ (u̇2 , v̇2 ) ( 2 2 2 1 Lu̇ + 2M u̇v̇ + N v̇ λ = 02 E κ = mn 2 2 / P E u̇ + 2F u̇v̇ + Gv̇ n1 = n2 (u0 , v0 )2 :() % u̇1 = λu̇2 # v̇1 = λv̇2 % !# % ) λ2 2 && ("" ( # λ2 2 C!( )
% 1 Lu̇21 + 2M u̇1 v̇12 + N v̇12 1 Lu̇22 + 2M u̇2 v̇2 + N v̇22 κ1 = = = κ2 mn1 E u̇21 + 2F u̇1 v̇1 + Gv̇12 mn2 E u̇22 + 2F u̇2 v̇2 + Gv̇22 (u0 , v0 )2 > 1-1 . / * $ P0 2 $ $ &! / * " !$ $ #( ,K #() % &! #"!& P0 2 * $ . I " "" # " )"## J ?# & ( r : T R3 F / * P0 F ##) # > ! ()*+(, ! ) P0 ponton, # "2 P0 2 #"! ,% P0 " & / # m (. "#" J ! F / * " 2 $# "## (* ! ?# %!( 1-1 . 4@; < 4@ 55 9 $* κn , !& 0 < ϕ <
π2 $ / /# " ! / $* κn . κ= cos ϕ B5 z F m P0 y ! x I " / * $# $ # & 1 Lu̇ + 2M u̇v̇ + N v̇ 1 Q2 κ = mn 2 2 # " κ = mn Q1 , % Q2 , Q1 E u̇ + 2F u̇ v̇ + G v̇ 2 . ( " / # m n, % κn = Q Q1 ( ϕ = (m, n) ( ϕ = π − (m, n) , ! cos ϕ = |mn| . :% κ = 1 Q2 = cos1 ϕ κn G / % ) % κ # κn =!+&& && |mn| Q1 + !( %P :% / $* cos ϕ2( ( & , % ϕ 2 ! $. % % & " # + "## % ( / ) # $(# 2 (## κ 2 ! κn % $* # " ! ) % n = m (
n = −m, κ !( ( !( 9 $* % κ= κ 2 Q2 . Q1 " $* # $ ## % + B? 1# - 5 ,# 1-1 . t P0 " I #( F / * P0 2 #"! t|(u0 ,v0 ) = ξru + ηrv |(u0 ,v0 ) . t2% & $* κ = Lξ 2 + 2M ξη + N η 2 . z F S rv P0 ξ · ru t ru η · rv y x I " / * $ !(% ## ( Q1 ≡ 1 # t̄ = r = u ru + v rv , ! u = ξ, v = η. :% κ = Lξ 2 + 2M ξξ + N ξ 2 . 1-1 . 2# P0 2 S #"! % & (e) # ) + P0 2& /*" κ1 , κ2 ) % e2" ϕ $ / % & $* κ = κ1 cos2 ϕ + κ2 sin2 ϕ. I " : , "" # " .) t = ξru
+ ηrv , κ = Lξ 2 + 2M ξη + N η 2. E* / S 2 " {i, j} 2 ,i⊥j # i = j = 1.) :* / ) % t = xi + yj , t {i, j}2 x # y). 2 Q / !( L M ξ κ = [ξ, η] . M N η BA S j ru P t f rv e i φ 9 #* {ru, rv } " {i, j} ) x ξ =B ( 2 × 22 B Q ! [ξ, η] = [x, y] B T . y η L M T B 2 E A := B M N κ = [x, y] A x y = a11 x2 + 2a12 xy + a22 y 2 M B Q) a11 x2 +2a12 xy+a22 y 2 ( B T N ( /" %% & " "" /+&( ) ( % & {e, f } P E ∃C x h h =C . :% κ = [h, k] D Q) , % D = y k k ξ h κ1 0 T C AC Q) D = . ( = BC , 0 κ η k 2 h #
t ( {e, f } ( t = k 1, # % ϕ (t, e) , h = cos ϕ # k = sin ϕ. N % κ = cos ϕ κ1 0 , ( [cos ϕ, sin ϕ] 0 κ2 sin ϕ L M κ = κ1 cos2 ϕ + κ2 sin2 ϕ. ! : # ( cos2 ϕ + sin2 ϕ = 1, # κ κ1 # κ2 $# # Q # κ1 # κ2 ) % κ = κ1 + (κ2 − κ1 ) sin2 ϕ, # 0 ≤ sin2 ϕ ≤ 1. + BH f t sin φ p cos φ e ! ()*+(, κ1és κ2 $ F / P0 ? < 62 J % & #" ? " J /"2 % & ?# (* ( @ "L 62 : H := κ1 + κ2 . ! ()*+(, $ ( A ; 62 K := κ1 κ2 % 1-1 . 2 LN − M , EG − F 2 / % ) # K= " # H= EN − 2F M + GL . EG − F 2 " #3
" P0 2 #"! (ξ, η) % % &) % ξru + ηrv (ξ, η) ) $ , $(" . 1-1 . # /" & (ξ, η) 2 η −ξη ξ 2 E F G = 0 L M N /I $ 2 0" " " " : 2# $(# r (u, v) = r (u0 , v0 ) + (u − u0 ) ru (u0 , v0 ) + (v − v0 ) rv (u0 , v0 ) + . 1+ (u − u0 )2 ruu (u0 , v0 ) + 2 (u − u0 ) (v − v0 ) ruv (u0 , v0 ) (v − v0 )2 rvv (u0 , v0 ) + 2 + Ψ (u, v) , ,7 A. BB z F S rv P0 ξ · ru ξ · ru + η · rv ru η · rv y x 7 > (ξ, η) % S #"! % (u−u Ψ(u,v) 2 2 0. 0 ) +(v−v0 ) r̂ (u, v) = r (u, v) − Ψ (u, v) . r (u, v) # C r̂ (u, v) / * P0 2 $$ #2
"! 9 ( {ru , rv , m} , r̂ (u, v) I Z = 12 [Lx2 + 2M xy + ny 2 ] & , r̂ =!+& m2 ) # #(*) % {ru , rv , m}2 r (u0 , v0 ) = 0, # ru , rv ⊥m. " $() % r̂ (u, v) / * 2 )( 2 % 2 )( %) & /*") % Lx2 + 2M xy + N y 2 ( / K J K J ( # K ,6 Q (x, y) ( / 0" K ) % Q (x, y) > 0 ∀ (x, y) = 0, 0" # K J % Q (x, y) ≥ 0 ∀ (x, y)2 ) #2 (x, y) = 0, % Q (x, y) = 0. ( K J % ∃ (x1 , y1 ) # (x2 , y2 ) , % Q (x1 , y1 ) > 0; Q (x2 , y2 ) < 0. + BD #; 0 6"" r̂ (u, v) / * P0 %!( {ru , rv , m} * & z = 12 # z = − 12 !
, #"! P0 2 % ! . $# (!* " ,{ru , rv , m}2 2 . ru , rv ! , P0 2 #"! (* $# {ru , rv , m} Lx2 + 2M xy + N y 2 = 1 % 2 % & 2 ) 2 Lx2 + 2M xy + N y 2 = −1 # P0 0 ; ) % ) P0 3 06" ; 0 6" ; 9 P0 κ1 = κ2 = 0, $ 1-1 . P0 0 0" P0 ) % ) ( ,. κ , * $$" &) F2 / ( ) P0 . &) ( &) ( 2 ,. Q2 =) =) ( = ,. K > 0, K < 0, K = 0, κ12 + κ22 = 0 ,(. LN − M 2 > 0, LN − M 2 < 0, LN − M 2 = 0 DF & t O1 O2 P0 "?2
0" 9 " " K g $# t $* ! " F // g $# # 6 (* /" # $ %% & ! &$ $# (* O1 T 2 /" ! &$# $# O2 / * " # P0 0 ;J % O1 # O2 #"! " " " ( ) / " #( $ " P0 3 06" ;J % O1 # O2 #"! " ( ) ( $ / $ #( (Q P0 0 ) % $ $* # , YQ& #" " . % -.!/ % -.!/ H P E PL % r (u, v) = (u cos v, u sin v, v) , 0 ≤ u ≤ 1, 0 ≤ v ≤ 2π. / * I &) % + ( ( , x !
% . $$* z 2 . + D> H P E PL z 2π 1 y 1 x 4 ! /"$* P / * /" # . > # # / 0# #3 " ru (u, v) = (cos v, sin v, 0) ; rv (u, v) = (−u sin v, u cos v, 1) ruu (u, v) = (0, 0, 0) ; ruv (u, v) = (− sin v, cos v, 0) ; rvv (u, v) = (−u cos v, −u sin v, 0) 2 2 E = r2 u ≡ 1; F = ru rv ≡ 0; G = rv = u + 1 v m = |rru ×r = √u12 +1 (sin v, − cos v, u) u ×rv | i j k sin v 0 = (sin v, − cos v, u) ru × rv = cos v −u sin v u cos v 1 D |ru × rv | = √ 2 √ sin v + cos2 v + u2 = u2 + 1 L = ruu m = 0m ≡ 0; M = ruv m = = (− sin v, cos v, 0) (sin v, − cos v, u) √ 1 u2 +1 = −√ 1 u2 +1 ; ,7 H. N = rvv m = (−u cos v, −u sin v, 0) (sin v, − cos v, u) √u12 +1 ≡ 0 # # κ1 , κ2 #3 "
:) % − u21+1 1 LN − M 2 =− K = κ1 κ2 = = 2 . 2 2 EG − F u +1 (u + 1)2 # ) % 2 1 · 0 − 2 · 0 · √u−1 EN − 2F M + GL 2 +1 + (u + 1) · 0 ≡ 0. H = κ1 +κ2 = = EG − F 2 1 (u2 + 1) − 0 /"$* ) κ1 , κ2 % % /I # $ $/*#" ) κ1 , κ2 #! κ 2 − Hκ + K = 0 ) ( H = κ1 + κ2 # K = κ1 κ2 . :% κ1 , κ2 ### ) % κ 2 − Hκ + K = 0 1 1 1 H = 0 # K = − (u2 +1) # κ 2 − (u2 +1) 2 2 2 = 0 ⇒ κ1 = u2 +1 ; κ2 = − u21+1 . 7 # # η 2 " #3 " :) % /" t = ξru + ηrv () # 2 η E L −ξη F M ξ2 G N ξ, = 0. ξ ( % t #( ) η ! % ) ( " +
η 2 2 ( D7 ) % 1 E L % !( 1 1 0 E ξ1 = − ηξ 0 − √u12 +1 − ηξ F M G = 0. N ξ2 η2 u +1 =0⇒ 0 ξ2 η2 2 ξ η ξ η √ u2 + 1 ( √ . 2 =− u +1 = √ √ u2 + 1, η1 = 1 # ξ2 = − u2 + 1, η2 = 1. / ? √ u2 + 1 (cos v, sin v, 0) + (−u sin v, u cos v, 1) = √ √ u2 + 1 cos v − u sin v, u2 + 1 sin v + u cos v, 1 . ,7 B = ξ1 ru + η1 rv = / #" ? √ ξ2 ru + η2 rv = − u2 + 1 (cos v, sin v, 0) + (−u sin v, u cos v, 1) = √ √ = − u2 + 1 cos v − u sin v, − u2 + 1 sin v + u cos v, 1 . ,7 D D5 ! 6" % -.!/ K z $* ω $# P z F (P ) P y x 5 > z $* ω $#C / ω , 2π $* / +# . # P xy 2! % $ 9
F (P ) P #( ) = F (x, y, z) = (−ωy, ωx, 0)) % P (x, y, z) I( = E # (x, y, z) % % * ( (x, y, z) (−ωx, ωy, 0) D? ! DA z y x 5 E# # % ( * %% -.!/ f (x, y, z) = xyz ,( f R3 R. < (2 #) % f # ) (xyz)2 / /*(# %!( grad(f ) R3 R3 <( grad(f ) (x, y, z) = (#) % grad(f ) I ( (yz, xz, xy) . F = grad(f ) , % f : R3 R %!( ( % r = (x, y, z) ,5 >. $ # E r (x, y, z) % ( # (* ) % ( / ! C r % r2 $ * :% / r = |r| = x2 + y 2 + z 2 P
# $C ) (x, y, z) # & K$ $# ) ( 9 F (P ) ") M ( P 2 −− <L $(# F (P ) P M # F (P ) % /!( r2 2 " F (r) = −k rr3 , % k > 0 E 1 0 −ky −kz −kx . F (x, y, z) = 3 , 3 , 3 (x2 + y 2 + z 2 ) 2 (x2 + y 2 + z 2 ) 2 (x2 + y 2 + z 2 ) 2 ! DH z r (x, y, z) y x 5 7 r =!+& z y x 5 5 F (# ( " %!(% : R3 R3 (#) / /% ) 4 9 F # & / # ##) ( ( /# "2 ! # ) ( ! / # = ! I # ## / ( # # "#) F 9 % γ 2 /& "# # (# W ,E "# (#
W. ;*" γ ! $ ! DB F z F F F γ F B A y x 5 ? γ ! $ # # F (# 2## & / # ," <*" F ## F / * "# W &. & / # #/2 CC F x 5 A #/ K Q / * "# z FF F y % & / # " #/ =!+& F 2 F 2 ( & => %!( / ;* ( ) ! < / !##( ( 6" %# γ ! $) ## r (t) = (x (t) , y (t) , z (t)) , a ≤ t ≤ b # F : R3 R3 . , /*(# + ! DD ( #) # # / ( % & γ $# . / # ME M
/ % ! ()*+(, F "# γ 2 A2& B 2 & (# ( (* , F ( 2 F (r) dr γ 2 O γ F F z A F F B γ y x 5 H E :% = / / #) M" #"9; #; N X , " ".·, 6 :* / "$) % γ $ AB ) F "# F0 " − − AB F0 2 AB " " (* ( F0 − ) AB | | − " ") ( AB ) ) − AB − − W = F0 − AB = F0 AB AB . (# γ $) # F # # " # # ( ((
γ $ [a, b] # ( n #) a = t0 < t1 < · · · < tn−1 < tn = b I) % max {ti − ti−1 } − 0. 1≤i≤n n∞ ! >FF F0 F0 F0 B A 5 B (# ( 1 ≤ i ≤ n 2 Pi ) % ( r (ti ) . −−−− γ $# (Pi−1 , Pi ) !(# ri = Pi−1 Pi ( % !* ri ( " ## ri · F (r (ti−1 )) (# $ !" ) % F 2 * ri 2 :% n2 ti−1, ti #% & r (ti−1 ) , r (ti ) !( (# $2 !" F (r (ti−1 )) [r (ti ) − r (ti−1 )] ≈ F (r (ti−1 )) ṙ (ti−1 ) (ti − ti−1 ) . $( b n 3 W = lim F (r (ti−1 )) ṙ (ti−1 ) (ti − ti−1 ) = F (r (t)) ṙ (t) dt. n∞ i=1 a F (r) dr ( :% ) % # !%
$("2 γ 7 1-1 . b F (r) dr = γ % -.!/ F (r (t)) ṙ (t) dt a F (x, y, z) = (x, yz, y) # γ ## ! >F> # z r(t2 ) Δri r(ti−1 ) r(ti ) r(t1 ) A B y x t0 = a t1 t2 ti−1 ti 5 D tn = b $# r (t) = (t, t2 , t3 ) 0 ≤ t ≤ 1 , 5 >F. 1 F (r) dr = 0 γ = 5 2 t, t , t F (r(t)) 1, 2t, 3t 2 ṙ(t) t2 2 7 3 5 + t + t 2 7 5 1 = 0 1 dt = t + 2t6 + 3t4 dt = 0 ,5 . 97 1 2 3 + + = . 2 7 5 70 / "$ F 2 γ 2 ) F (r (t))2 2 I $#) % F (x, y, z) = (x, yz, y) # # ∀x % # r (t) " /*(##) t2 !) ∀y % # r (t) /*(##) t2 2 !) ∀z % # r (t) % /*(##) t3 2 ! F (r (t)) = (t, t2 · t3 , t2 )) ṙ (t)2 ! ṙ (t) = (1, 2t, 3t2 )
% F (t) · ṙ (t) ) t2" /*" ( & ##C /*(#) # & [a, b]2 / " ,5 0! ## ) % / # # ) % $! A # B # A # B 2 $$" # $# F (# ! >F z B = (1, 1, 1) γ A y x 5 >F γ $ ( ) ( /* I& ) + "#" # A # B (# & /*W γ1 B A γ2 γ3 5 >> E I& ( & /* # % ! ()*+(, ) % F (# ) % F (# E F = grad f ( f : H ⊂ R3 R /*(#) f /(# F (# ! ! ()*+(, ) % F ! ) ) % A # B #
$$" # γ1, γ2 ! C $!() " A, # (# B ,5 7. F (r) dr = F (r) dr γ1 γ2 ! >F7 * ) / #() % γ1 # γ2 (# F # # % @ AB ACD-E E* #) % F ( /2 I& ) ∀ γ $# F (r) dr = 0, % F (r) dr = γ F (r) dr − γ1 γ F (r) dr = 0 γ2 γ2 A A γ1 B γ γ γ γ B 5 > E I& ( & /* # # $# ( " ( / +# ) % / # ) ! $##) % (# ( /* 2 I& %% # * $% $(" / (# # $ ! ()*+(, curl F ( rot F =!+& 4 5 2 ! (#) % F = (F1 , F2 )) F 9 F : R2 R 2 +& rot F = ∂F − ∂F 1 <#%
!) % F ∂x ∂y rot F % curl F ( rot F , . ## % ! (# # ( & 465 3 3 9 F : R # (#) % F = (F1, F2, F3 ) R curl F = rot F ( j k i ∂ ∂ ∂ curl F = rot F = ∂x ∂y ∂z F1 F2 F3 = ∂F3 − ∂y ∂F2 ∂F1 ; ∂z ∂z − ∂F3 ∂F2 ; ∂x ∂x − ∂F1 ∂y ! >F5 %$ -.!/ z $* ω $# ( & / # +& F (x, y, z) = (−ωy, ωx, 0) . j k i ∂ ∂ ∂ rot F = ∂x ∂y ∂z = (0, 0, 2ω) −ωy ωx 0 :% z % # / / # & ω %I ( z ω 2ω rot(F) y x 5 >7 z $* ω $#C / +& %7 -.!/F (r) ## curl F = −kr r3 (+& (#) % k > 0 ( & ##
−ky −kx −kz F (x, y, z) = 3 , 3 , 3 2 2 2 2 2 2 2 2 2 (x +y +z ) 2 (x +y +z ) 2 (x +y +z ) 2 i j k ∂ ∂ ∂ rot F = ∂x ∂y ∂z −ky −kz −kx (x2 +y2 +z2 ) 32 (x2 +y2 +z2 ) 32 (x2 +y2 +z2 ) 32 5 2 2 2 −2 3 (x + y + z ) k (yz − yz, − xz + zx, xy − xy) ≡ (0, 0, 0) = 0 ( / # # % &) % +& $(# # + ) % (+& (## # ( & / # G + $(# #) # rot F = 0. ! /& !& " # (# +& C % ! >F? %= -.!/ F (x, y) = (−ωy, ωx)) ω $#C & $* /2 # xy ! curl F = ω + ω = 2ω. % -.!/ f : R3 R " # / @2 + % & /*(# # F = grad (f ) j k i ∂ ∂ ∂ rot F = ∂x ∂y ∂z f f
f x y z ⎛ ⎞ ⎠ = ⎝fzy = 0, − fyz , fxz − fzx , fyx − fxy 0 0 0 ( ( + ( ( / +# %" :2 % / ! *) % ( !( , . (# +& G< G; Z 13 E :T1P = 1-1 . F (# ( /* I& ) % F ( !( # + :% /" # ) % () ( /* I& &) % ( !( ( % $(" ## /" F " M" 92 J 3" " <N ( + 2 ( M # + M ! I) % ##) *# / # + 2# #2 " !) ( / ) ( C $/*" % / / 6
+ . F : R2 R2 ! (#) F = (F1 , F2 ) . ( (( curl F + #& * $$" ## ( / ) F < ( !( :* / ) % curl F ≡ 0 9 ! >FA / 1 F /*(## + ( ∂F , ∂x ∂F2 ∂F1 ∂F2 , ∂y , ∂y ! G< < = ( ( F ∂x * I E ) + ( = ) % + 2 ++ ( ,% ) . %) % F 2 ( !() %) % / 6 + F : R3 R3 (# ( (( # F = (F1, F2 , F3 ) curl F + #& * $$" ## ( / ) F < ( !( :* / ) % curl F ≡ 0 9 / 1 1 1 2 3 9 + + ( ∂F , ∂F , ∂F , ∂F , .
, ∂F ∂x ∂y ∂z ∂x ∂z # ( ( ) 0 F * I 9 / + + ( $* + # 2 ( (# ) + 2 ++ () % %) % F ( !() # ) % + 2 $ y x > -.!/ F (x, y) = x2+y2 , x2+y2 F ! (2 # curl F = 0, F + # ( & ) ! GG[ /) ( + 2 ++ ( -.!/ F : R3 R3 , F (r) = −k rr3 . ) % curl F ≡ 0 ) % F + # ( & $ $ F ) + + ( # ( ( ) ! F ( !( ! // ,5 >FH 9 F ( !( (#) ( + /*(#) ( # f : R3 R) F = grad (f ) + /*(# % # #
-.!/ F (x, y, z) = (1 + 4y + 5z, 2 + 4x, 3 + 5x) i j k ∂ ∂ ∂ ≡ (0, 0, 0) ≡ 0. curl F = ∂x ∂y ∂z 1 + 4y + 5z 2 + 4x 3 + 5z # $ + ( * # ) ! F 2 ( !( + 2 :) % fx = 1 + 4y + 5z ⇒ f (x, y, z) = (1 + 4y + 5z) dx + g (y, z) = x + 4xy + 5xz + g (y, z) # fy = 2 + 4x ⇒ ∂ (x + 4xy + 5xz + g (y, z)) = 4x + gy (x, y). ∂y E 4x + gy (y, z) = 2 + 4x ⇒ gy (y, z) = 2 ⇒ g (y, z) = 2y + h (z) . :% f (x, y, z) = x + 4xy + 5xz + 2y + h (z) . N) ( fz = 3 + 5x ⇒ 5x + h (z) = 3 + 5x ⇒ h (z) = 3 ⇒ h (z) = 3z + c. # f (x, y, z) = x + 4xy + 5xz + 2y + 3z + c. , "" / % ( . x -.!/ F (x, y, z) = x2−y+y2 ; x2+y ; −4 ## ) % ( !(2 2 F W ≡ 0, ( P F 2 . curl F = z
) % (# ) ! GG[ /) + 2 % !P γ x2 + y 2 = 1 # z = 12 ! >FB $) F (r) dr ###P 9 !( ! ! γ F (r) dr =0, " * ) % &) % F ( !( γ : , 7 #.) % ( !( ( $# ( ( # γ ## x (t) = cos t, y (t) = sin t, z (t) ≡ 12 # 0 ≤ t ≤ 2π; ( r (t) = (cos t, sin t, 12). :% t cos t , 1 , −4 , ṙ (t) = (− sin t, cos t, 0) . F (r (t)) = − sin 1 F (r(t))· ṙ (t) = sin2 t+cos2 t ≡ 1. F (r) dr = γ 2π 0 F (r (t))· ṙ (t) dt = 2π 0 1dt = F (r) dr = 0, ! F ( !( 2π = 0. :% γ $(" # ) % ( !( (# # # $$ (# + /*(# ( ## * $# 1-1 . F : R3 R3
# 2 ( # <(* F + /(## f 2 :% F = grad(f ) γ $) " A # (# B 2 F (r)dr = f (B) − f (A). γ I " γ ## r(t)) a ≤ t ≤ b :* g(t) = f (r(t)) /*(# ( R R / @+ % & /(#) % <L2 # 9 ( ) % g (t) = grad(f )(r(t)) · ṙ(t) = F (r(t)) · ṙ(t). :% b f (B) − f (A) = g(b) − g(a) = F (r(t)) · r(t) = g (t)dt = a b a F (r)dr. γ 2 ! + >FD % -.!/ γ $ ( r (t) = (t, 2t3 ) 3t2 ) , % 0 ≤ t ≤ 1, ## ! F (r) dr ###) % γ F (x, y, z) = grad(ex+y+z+1 )! & $ $ A (1, 2, 3) ) # F + /*(# f (x, y, z) = ex+y+z+1 . :% F (r) dr =f (1, 2, 3) − f (0, 0, 0) = e7 − e. . γ $ γ 45#
F / * ) # r :T R3 ## F 2 ( F : R3 R3 (#) # / @+ % & # * ) % / # I ) % # P F (P ) / # # " F / * !% ( ru × rv , ( −ru × rv ( / * ! / # ( CC z m x 5 >5 F y ( m = ru × rv ( m = −ru × rv / * ! I ) % ) % / * ! M/ / # & M , # . % ! F / * ) # % F 2 , ! . "# / & / # " #/ ! >>F 7 ! ()*+(, F 2 "# / &) F ## 2 & / # #/ F (# F 2 ( ! 2( $
* ) ( => %!() # F dA F :% 2% YQ !() / * ! $ / # / R 2% YQ !() ! # / $ / #R 2% YQ #) / # / ! " # # "# " 9 % YQW 0 :* / ) % F (#) F ≡ F0 # F ) a # b ( /! ) # a × b2 ( ! a × b z F0 · a×b |a×b| r0 F0 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx b xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx F xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx a F0 y x 5 >? K Q ! F / * a × b "# / & / # % & $ :% F 2 F 2 ! + >>> ( & YQ #/ ) alapterület × magasság. * F = a × b ) F0 2 b a × b2 ( " (* ) F0 · a× " a×b| | b #/ a × b · F0 · a× = F0 · (a × b) |a×b| 2 E ) % F = F0 ) # F a, b /! a × b2( = F0 · (a × b). ! ) F dA F 0 F 4> F v Δu z F xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx v0 + Δv v0 Δv F r(u0 , v0 ) u0 u0 + Δu 5 >A u x y K Q F 2 F / * ## r :T F
2( !2 ##( + ) # + / * # $ !# ) % C) % / * / % I =! ( $ !# ) % /#( ! / * % #" ! / " * ( $ !# ) % F 2 ) F ≡ F (r (u0 , v0 ))2 (*) ! % ; ## 9 " / * !) / ;* ## ( ) % $ !" F (r (u0 , v0 )) · (ru × rv ) uv ,5 5. m̄(P ) >> ! r̄u Δu F̄ (P ) r̄v Δv v0 + Δv u0 0 u0 + Δu 5 >H v0 K Q YQ # / * I ) % / ( YQ $*) # u, v 0 E = lim ΣF (r (u0 , v0 )) · (r × r ) uv F dA u v u, v0 F =
F (r (u, v)) · (ru × rv ) dudv, ,5 ?. T / #() % F ru × rv 2( ( !( 9 F 2 − ru × rv 2( !) F dA = − F (r (u, v)) · (ru × rv ) dudv. F T F (# F / * / * " " ( ) 4 9 F (p) · m (p) ≡ q, ! + >>7 = q · f elszı́n(F). F dA ,5 A. F % & # ) % F !) # F (# 2 -.!/ F / * ) T : 0 ≤ u ≤ 3, 0 ≤ v ≤ 1 ### r (u, v) = (u + 2v, −v, u2 + 3v) # F 2 / / # & ! F (x, y, z) = (xy, 2x + y, z) . =? F dA " F 0 F 2 F / * F (x, y, z) = (xy, 2x + y, z) # % & x % # u + 2v 2) y % # −v 2 # z % # u2 + 3v 2 ! N F (r (u,
v)) = ((u + 2v) (−v) , 2(u + 2v) − v, u2 + 3v) . G F (r (u, v)) = (−uv − 2v 2 , 2u + 3v, u2 + 3v) . 0 % ( %% ! ru ×rv = i j k 1 0 2u = 2ui + (4u − 3) j − k. ( k *% & !( 2 −1 3 # ru × rv / # ) % n = −ru × rv = (−2u, 3 − 4u, 1) . 0 ! F (r (u, v))·n F (r (u, v))·n =2u2 v+ 4uv + 6u + 3v − 8u2 − 4uv + u2 + 3v 2 %0 −21 F = F dA 3 1 4uv 2 + 2u2 v − 7u2 + 6v + 6u − 4uvdvdu = · · · = u=0 v=0 ) % "## 21 # $ / # F 2 * M / #M) M/ / #M $ -.!/ A (1, 0, 1) ; B (1, 1, 1) ; C (2, 0, 3) F ABC %$) # ! F 2 M / #M & 2 ( F ≡ (5, 4, 3) . ! >>5 " ( " F dA? F )
# F !) ! . ( F YQ F 2 = (0, 1, 0) ; b = AB " (* ( * #( c = AC = (1, 0, 2) i j k b × c = 0 1 0 = (2, 0, −1) , ( & 1 0 2 b × c / " ( c F (* b×c2 F · b× . %$ * |b×c| 5 4 3 b× c | | F ·( b× c) c 1 F bc = 1 5 0 1 0 = = = F · b× · 2 2 2 2 2 1 |b×c| 1 0 2 !() # |b×c| 2 . :% YQ 3 = 2 7 2 = F dA. F / 2+, = ! ()*+(, F : R3 R3 . F div F (x, y, z) = (x, y, z) ∂F1 ∂F2 ∂F3 + + , ∂x ∂y ∂z % F = (F1, F2, F3 ) . 7 -.!/ div F = F (x, y, z) = (xy, y 2 sin z, z 2 + y) . ∂ ∂ ∂ (xy) + y 2 sin z + z 2 + y = y + 2y sin z + 2z. ∂x ∂y ∂z (# (+ % ( & ##C /*(#) ( (2 +
) , # +&( ) (. (+ /"# ### % 9 #) $(" # ! ()*+(, 9 K #) ∂K 2( $ * K % G / *) % ∂K ! / ! ! + / 2 >>? K # ∂K 2) K % ,/ * #. ! = F dA div F dxdydz, / # & K ∂K % F K # ( ) # div F ( & ##C /*(# ( ρ + , ρ 0 .) # # P $* ρ I Bρ $$ z m̄ Bρ P y x 5 >B ρ (+ # ! K % / # G! Bρ 2 ∂Bρ / * 2 # / # & ) % ρ +) / %") % div F |Bρ ≡ div F (P ) . , + % (#*) ρ
02 P. ; # div F dxdydz ≈ div F (P ) · térf ogat(Bρ ). Bρ ! >>A :% ; # ≈ div F (P ) · térf ogat(Bρ ) F dA ∂Bρ F dA ( div F (P ) ≈ ∂Bρ térf ogat(Bρ ) . 0 F dA div F (P ) = lim ρ0 ∂Bρ térf ogat(Bρ ) . :% (+ P 2 # + $ / * # "2 # & / # #/ ) ( + $ #/ ( / div F (P ) < 0, $ / # / ) / P $# I + I $ # P I div F (P ) > 0, P $* + $" $ / # / ) / ) % P ! + div F (P ) = 0, / # / ) / P $* + $" = -.!/ T 5 >D % & $$ &) ∂T / * # / #
& ! F (x, y, z) = (x, y, z) = (2x + y 2 , y + sin z, z − x3 ) 9 (yz, xz, xy) # G 5 65 # F dA ∂T A ###P Gd ∂T ≡ 0, div G = 2 + 1 + 1 = 4. :% ( . div F # /*(#) # ; # " = div F dxdydz = 0, ,div F . F dA T ∂T ≡ 4. div Gdxdydz = 4 · térf ogat (T ) , ,div G A = Gd ∂T " T T $$ & #/ W (+ ! + >>H z m̄ ∂T T 1 3 y x 5 >D :& ?B 0# % :) % T 2 I ) % D = (x, z) : (x − 3)2 + z 2 ≤ 1 $ / z $* D2 % & (x − 3)2 + √ 2 z 2 = 1 $ z 2% $ √ /# $!( x = f1 (z) = 3− 1 − z , # ( x = f2 (z) = 3 + 1 − z 2 . & #/ I ) % #/ & ) x = f2 (z)2 z $* / & π x =
f1 (z) z $* / ( 1 T érf ogat(T ) = π −1 f22 (z) − f12 (z) dz = π √ 9 − 6 1 − z 2 + 1 − z 2 dz = π 1 −1 1 −1 f22 (z) dz ) ( #/ π 1 −1 1 −1 f12 (z) dz √ 9 + 6 1 − z2 + 1 − z2 − √ 12 1 − z 2 dz = 12π 1 −1 √ 1 − z 2 dz = 6π 2 / > ! ()*+(, ) % F / * ) # ∂F % % ( !( ) % (## ∂F 2 ∂F ! >>B z xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx 1 xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx D x = f (z) 5 F x = f2 (z) 4 x 3 :$$ & #/ I) % / % # F ! / " ( ) # / * # -.!/ F 5 % & % ) ∂F % # /" $ 9 F / # & !) % ! % % ) ∂F 2 & & # / " $$W . & $ ! & &( / " $ ! & &(
> 1-1 . 4E " 5 # :* / ) % F / # ∂F % % ! ) % F + ( F = ) F dr = curl F dA ∂F F / * % F ∂F ) $$ $ ! + >>D n̄ 5 > K * # % % ! %) % F % ) ∂F # $ $ & ) % # "/ )% F % *% 0# % F $ / * ) ( F / ∂F = ∅ ( F dr = 0, ! % F / * ) ∂F curl F dA = 0, $( # " F $> -.!/ γ 5 5 % & F (x, y, z) = (x, x, x) . ( " F dr? K; ! $) # γ . F / * ) γ % ) γ = ∂F. G!2 F 2 γ 2( % ( j k i ∂ ∂ ∂
= curl F = ∂x ∂y ∂z = (0, 0, 0) = 0 ) % curl F dA F x y z = 0. # curl F dA F dr = 0. 9 ( ∂F F $ -.!/ F = {(x, y, z) | x2 + y 2 ≤ 1, z = 0} , ( F xy ! # I $ ) / / # & !( 2 F (x, y, z) = √ 21 2 (y, −x, 3z) . ( " F dr? x +y ∂F ! > F ∂F F ∂F 5 9 # % % ! F dr2 . 5 ( ∂F / ( ∂F #) r (t) = (cos t, sin t, 0) , % 0 ≤ t ≤ 2π. F (r (t)) = (sin t, − cos t, 0) ; ṙ (t) = (− sin t, cos t, 0) :% F (r (t)) · ṙ (t) = − sin2 t − cos2 t = −1. E 2π F dr = ∂F 9 ( 2π F (r (t)) · ṙ (t) dt = 0 2# −1dt = −2π. 0 ( F dr2 ) ∂F ! + > > % 5 7 / * %
%) % # $ z H / * % n = (4, 1, 5) G F F E x 5 5 y K;9 % "$ i j k ∂ ∂ ∂ curl F = ∂x ∂y ∂z −x 3z √ y √ √ x2 +y2 x2 +y 2 x2 +y 2 −3/2 = −3zy x2 + y 2 , 3zx x2 + y 2 −3/2 ,0 ,5 H. ## % " ( & ) % F dr = 0) % ∂F xy ! , % z = 0.) curl F = 0 E % % ( ) % " F dr = −2π & ∂F % & &) % < % & # # ) 2# / # ) % F ! > + ( # F / * ( F z # // ) * ) * F 4 7 # ) # Sρ P $# I) + I $) P n. ∂Sρ # Sρ % ! z
xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxx xxxx
xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx n̄ Kρ Sρ P ρ y x 5 ? + F = # /% Kρ = ∂Sρ . 9 ρ +) curl F $ !"
" curl F (P ) Sρ 2 :% |n| = 1. U curl F dA ≈ curl F (P ) ·ndA = curl F (P ) ·n · terület (Sρ ) . Sρ Sρ F dr = :% 2# " Kρ Sρ ∂Sρ n · terület (Sρ ) " $() % n · curl F (P ) ≈ F dr 0 Kρ ter ület(Sρ ) , % |n| = 1. ≈ curl F (P ) · curl F dA F dr = F dr Kρ terület (Sρ ) . ! + > 7 curl F 2 n2 " " (* " ,$ !"2 ) % ρ > 0, +. F Kρ 2 ( & (# ) ( Sρ * #( E $ -.!/ F (x, y, z) = (z 2 , x + x2 + z 2 , y 2 + x) / # 2 #) " & % * / # I # + P (1, 1, 1) , ! 2 # −n & ( n = (−2, 5, −1) . #* n̄ P ρ 5 A <
+ # K2 #) # % ) & & ( " ( # W + curl F = (2y − 2z, 2z − 1, 2x + 1) , ! curl F (P ) = (0, 1, 3) , % P = (1, 1, 1) ( curl F 2 n = (−2, 5, −1)2 √ ( & (* (0, 1, 3) · |nn| = (0, 1, 3) · (−2,5,−1) = √230 > 0. :% 4+25+1 F dr Kρ n ·curl F (P ) ≈ terület(Sρ ) $/*#" F !( (# |n| Kρ 2) % / # !() & & ( # / 9 # + P (1, 1, 1) ( $!() % ( n ( %%) % # ! > 5 /R 6 / & & ( " , !(. W # /) % F "# (# ) ( ) % F dr = 0. 9 ( ) % + ?? Kρ F dr
curl F (P ) · n ≈ Kρ terület (Sρ ) $( ) % curl F (P ) ⊥ n. :% # /) % n " () " curl F (P ) = (0, 1, 3)2 0# n = (0, −3, 1) # ∀n⊥ (0, 1, 3) . (P ) · n ≈ curl F F dr Kρ / 2 / # & & ( " ) % I ##C !( (# / # 2 $( ) % curl F (P ) · n %" I ##C !( :% n −curl F (P ) + ?%? ter ület(Sρ ) & #( E n = (0,−1,−3) √ 1+9 = √1 10 (0, −1, −3) . /8 2+ 2# + # ;2# P S xy ! ) % ∂S C $) # F S % @ % F xy ! (#) ( F (x, y, z) =
(P (x, y) , Q (x, y) , 0) , curl F = 0, 0, Qx − Py . # 2# " ) % = curl F dA n · curl F dxdy = Q − P dxdy. F dr = x ∂S S S S y ! + > ? z n̄ x x x x x S x x x x ∂S x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 5 H y S xy ! ( / ! F (#) S xy ! ( ) # / % ! !#
# 1-1 . 4A < 5 % D !) 2 C C $ y C = ∂D xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D 5 B x D ,D P. :( F ! (#) + ( D # ) % F (x, y) = ! > A (P (x, y) , Q (x, y)) : F dr = C Qx − Py dxdy D $ -.!/ F (x, y) = (2y, x) # C (x − 3)2 + 9 (y − 1)2 = 9 !( ! ( " F dr? C Qx . P (x, y) = 2y, Q (x, y) = x, ! Qx = 1 # Py = 2 :% − Py = −1. E Qx − Py dxdy = −terület (D) = −π · 3 · 1 = −3π. F dr = C D D 5 D C = ∂D (3, 1) A7 / 12 # 9 P (x, y) ≡ 0 # Q (x, y) = x) ( P (x, y) = −y # Q (x, y) ≡ 0 ⇒ Qx − Py ≡ 1. :% / terület (D) = xdy − ydx.
G % C C $ #) % F = (F1 , F2 ) (# (F1 dx + F2 dy) := F dr γ γ ! + G > H &) % 1 terület (D) = 2 xdy − ydx. ,5 B. C y C = ∂D xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D 5 7F :* x D + / C $ r = r(ϕ), % a ≤ ϕ ≤ b x (ϕ) = r (ϕ) cos ϕ, y (ϕ) = r (ϕ) sin ϕ. :% 5 B xdy − ydx = (x (ϕ) · ẏ (ϕ) − y (ϕ) · ẋ (ϕ)) dϕ . = r2 (ϕ) cos2 ϕ + sin2 ϕ dϕ = r2 (ϕ) dϕ E 5B 1 T erület (D) = 2 b a r2 (ϕ) dϕ. ,5 D. ! > B $% -.!/ r = sin 3ϕ I( M% ( C &%M , 5 7> . * # , sin 3ϕ < 0, r / !( ) % ) % r $ && ( ( I
/ ) % (|r| , ϕ + π) . . # " ! " R * % 0≤ϕ≤ π . 3 / # T erület (R) = 1 2 π 3 r2 (ϕ) dϕ = ϕ=0 1 2 π 3 T erület (R) = 1 2 · 1 2 ϕ=0 $(" ϕ=0 ! cos2 3ϕ + sin2 3ϕ = 1 # cos2 3ϕ − sin2 3ϕ = cos 6ϕ 2 sin2 3ϕ = 1 − cos 6ϕ ⇒ sin2 3ϕ = π 3 (sin 3ϕ)2 dϕ, (#( G 1 2 (1 − cos 6ϕ)) & .π (1 − cos 6ϕ) dϕ = 14 ϕ − 16 sin 6ϕ 03 = π . 12 π % ( C &% * 3 12 = π4 . $(" / # * ## # #" / % $ -.!/ (# C ) " (a, b) , (c, d) . xdy − ydx ##W C ## $(" 0 ≤ t ≤ 1, r (t) = (a, b) + t · (c − a, d − b) = (a + t (c − a) , b + t (d − b)) , F (x, y) = (−y, x) , % F (r (t)) = (−b −
t (d − b) , a + t (c − a)) , ṙ (t) = (c − a, d − b) . F (r (t)) · ṙ (t) = −b (c − a) − t (d − b) (c − a) + a (d − b) + t (c − a) (d − b) = ad − bc. . C F dr = E C $$ -.!/ * P 1 0 F (r (t)) · ṙ (t) dt = 57 1 0 ad − bcdt = ad − bc. % & . " / ## # n ;2# " ) % % C = ∪ Ci i=1 $ T ;2# % !! " T erület (t) = 1 2 xdy − ydx = C n 1 Σ 2 i=1 xdy − ydx = Ci > D n 1 Σxy 2 i=1 i i+1 − xi+1 yi, % xn+1 = x1 , # yn+1 = y1 . ,K % ( "" / #) ) % xdy − ydx = xi yi+1 − xi+1 yi . Ci + T erület (t) = 1 [(x1 y2 − x2 y1 ) + (x2 y3 − x3 y2 ) + . 2 + (xn−1 yn − xn yn−1 ) + (xn y1 − x1 yn )] # + ) % (x1 , y1 ) , (x2 , y2 ) , (xn, yn )
% > P % P # + !##( $ P ,2 7% % % "% % 0 $7 -.!/ F = rr3 , % r = (x, y, z) # r = |r| . S x2 + y 2 + z 2 = b2 $/ * / # & !( ## =? F dA S @" ( ( ( F %( ( $/ * F (r) ) # F 2 I (* |r| F (r) = r3 = r12 2 " ( b I $ / * # ( ) r = b % I ( ) % , ,5 A.) 1 b2 = 1 · $/ ! = 1 4b2 π = 4π F dA b2 b2 S /* $= -.!/ $# b2" div F dxdydz ) % K & K I) b I $) % F "" / P ! >7F ⎛ @" "$ div F 2
( F (x, y, z) = ⎞ r r3 = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x y z ⎜ ⎟ , , 3 3 3 ⎟ # / / ⎜ / 2 2 2 2 2 2 2 2 2 ⎜ ⎟ x +y +z x +y +z x +y +z ⎝ ⎠ F1 F2 ∂F1 = ∂x F3 / 3 / 2 2 2 x +y +z − x 32 x2 + y 2 + z 2 2x (x2 + y 2 + z 2 )3 / x2 + y 2 + z 2 (y 2 + z 2 − 2x2 ) . (x2 + y 2 + z 2 )3 √2 2 2 2 2 2 √2 2 2 2 2 2 x +y +z (x +z −2y ) x +y +z (y +z −2z ) ∂F2 ∂F3 3 ! ∂y = # ∂z = . G 3 (x2 +y 2 +z 2 )√ (x2 +y 2 +z 2 )3 2 2 2 2 2 2 2 2 2 2 2 2 x +y +z (y +z −2x x +z −2y +y +z −2z ) 1 2 3 div F = ∂F + ∂F + ∂F = = 0. ∂x ∂y ∂z (x2 +y 2 +z 2 )3 :% div F ≡ 0 # # div F dxdydz = 0. = K % @ AB ACD-E / # / " (2 + # # & * $$"# ) % (+ # + % ) % F ( # K # ( ( , / F = rr3 # K # ( * .
!! " >7> ϕ= π 3 y xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0.4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0.2 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0.2 0.4 0.6 0.8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx R ±0.8 ±0.6 ±0.4 ±0.2 0 ±0.2 ±0.4 ±0.6 ±0.8 ±1 5 7> 9 ( C &% x ! >7 C4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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" 6>8 O&/ K C# 6 8 31O 1 % 7 5) ( + ( + G 3( / 4 % ,>DDA. 678 31O 1 Q++ ( E+ + ) +; L29 ) G+ ,>DDF. 658 (+ K+ ! #$% & FF> & + 6?8 1 0## $$# @+ : $( &) 2 >DDF 6A8 1 O&/ GG[> : $( & 6H8 4 1 ! >DHB 6B8 +% E >DBD ) C $( & 2 @+ C $( & 2 >77
$$ ;# # # <#% $ #$ % &) " ( 2 % & # # % $* / %!( = >> > >7 >5 >? >A >H >B >D (#) =!+& >>> @+ >> 0 # > > 7 + ) +) "C $$# @ > > #( % & @ > E & %# @ > 7 "C @ @+ > 7 > 9# @ >7 G%# @ 9 C @ @+ @+ 2 + # + # y = f (x, y) I < & y = f (x, y)
> 1" >> > >7 %I G ( & GG J 2" # J 2/# & $% 2 ! > 5 E# %I %I 9"(# 7 ? A B B D D >F >> > >5 >D 7 5 A D 7F 7> 7D 5F 57 5A 5D ? 5 > 7 9"(# (# %II I K2 / 9"(# (# %I I ! "# 7> 7 77 ;$# # 7 > > ;$# !(% ## K * 7 > (# 7 =!+& 7 7 K * / ! K * $# 7 7 > K * $# $ 77 K"$* ) /" % & " 5> 5 57 55 E E 5 > E /* # I& 5 2 ! 5 7 2 # 5 5 0+ /*(# % K * 5 7 > ; 2/# (+ # 57 ; # ( (+ # 5 7 7 # 5 7 5 + F
= # 5 7 ? ;2# <#% / ( !" ? ?H ?H $ A> A? H> H> HA HB HD B B? D? DB >F> >F? >FA >FH >FD >>5 >>? >>H > > 5 > D @+ ) /*2 (# # /*(# ( "/ 0# % y = y (x) /*(#) y = y ,> >. y = −y ,> . y = y ,> 7. y +x ,> 5. x /*(# $ " % @+ ) 2 ) y = y (x) /*(# 2 ( @+ # /) <L M # $(# @+ ! M y = y @+ ) $ /*(#) " " ( /(#2 #( $C
) % c ∈ R2 y = cex I /*(# ) % " ( ) % $ y = cex I /*(# ,> >. @+ ,) % y= ? A y = cex I) #" . ,> . ") $# ( ( % ) % y (x) = cos (x) , y (x) = sin (x) /*(#2 ) % " ( !(( ) ( ) % cos (x) # sin (x) /*(# ,> . # $C ) % " 2 +& # :% " ( a, b ∈ R 2 y (x) = a · cos (x) + b · sin (x) ,> . @ # #" / ) % ,> . 2 I 9 & ) % ,> 7.
y = a · ch (x) + b · sh (x) . (( &) % / % @+2 C I 2( @+ #" /) % !(* / ) / = / " " /# #( ! (( &) % % # ) % @+ ( 2 # !% & /*(# !##( % 3 ) x f (x) = ex /*(# + # % & !( /(# x # ) % y = ex @+ + # % & ( " +# ) % $C* #% @+ 2! ) % %% "$ @+ <#% # @+ ) ( # +% E2 @+
+!C / C#" (* / 2 C# ( +%2( $ * > /*(# y = y (x) I , . yy2 = 1 ,. y + 2xy = 0 +% > 7 ,+. y − y x = x2 + 3x − 2 +% > D ,. x2 y + xy + (x2 − p2 ) = 0) % p " ( & p ∈ R, 2/# . H ,. y − 3y + 2y = 0 +% 7B ,/. y − 3y + 2y = ex + 2e3x +% 7B ,. (4x3 y 3 − 2xy) dx + (3x4 y 2 − x2 ) dy = 0 +% >F /*(# z = z (x, y) , . ∂2z ∂x2 + ∂2z ∂y 2 ,. ∂2z ∂x2 − ∂z ∂y ,+. ∂2z ∂x2 − ∂2z ∂y 2 ,. ∂z ∂x + ∂z ∂y I =0 =0 =0 = 0. / " + & @+ ) /*(# ( & /2 (# ,% + $$# ( / % ". +2 & @+ ) %
/*(# $( &) ! + 2 ( / " "/ & C ( , + ( . :% > , .) > ,) > ,+) > ,) , > ,.) > ,) > ,/) , ) ,) ,+ @+ /*2 9 (# # ( #( ,:%) % /*2 (# y ) % @+ sin (y) , y y, (y )2 , y3 . > ,.) > ,+) > ,) > ,) > ,/) , ) ,) ,+) , > , .) > , /*(# # ( & &2*% & ,% % y sin (x) , % y /*(#. > , .) > ,) > ,/) , ) ,) ,+) , ! > ,.) > ,+) > ,) > , / $$#) $(" / + /2 /+
9 B ! ()*+(, N 2 C @+ ) N 0# y = −y C @ ) / !* y = a cos x + b sin x. ! ()*+(, @+ ) ) %!( 0# y = −y @ y = −5 cos x + 6 sin x. " ! #$ % & ( N − C @+ 2 * / # ) % /*(# # " N − 1 ( " ( !( y (0) = 1 -.!/ y = −y # y (0) = 0 , N = 2 !* y = a cos x + b sin x. :%) y = −a sin x + b cos x a cos 0 + b sin 0 = 1 , " a = 1 # b = 0 & x = 02 !( −a sin 0 +
b cos 0 = 0 :% +% / y = cos x. @+ ( $(" # + # * , . ) % + * ) % ( ,. * ) % ( +% / 2 @+ 2 +% / ,+. % ( & ( ) % % ## + ( ) % %I " I ( #(%" , =!+& #". D ! f (x) / /*(# ( I1 # g (y) / /2 (# ( I2 ( ) ( g (y) = 0 % y ∈ I2 y = f (x) g (y) ,> ?. I "C $$# @+ ) 1 . :% ,> ? @+ %!(
h (y) = g(y) h (y) y = f (x) . % !# # h (y) dy = + y 2& /* #( % f (x) dx + C. + x2" /* & ( &I @ dy dx = y $(" # 2 -.!/ > y y 2 = 1. @ #( % & ( &I) ( / f (x) = 1 # g (y) = y12 ( y = f (x) g (y) . dy y = dx 2 !( # dx2 / ( ) % y 2 dy = dx. 3 y 2 dy = dx. " y3 = x + C &2 # ( √ G y = 3 3x + C ,( C " ) 32 ) 3C I " /* ) C. ln y y = 2xy . @ #( % & ( &I) x2 −1 % f (x) = x22x−1 2 # g (y) = y ln y 2 ! @ y = f (x) g (y) I @ I $#) dy % y = dx 2 !( 2xy ln y dy = 2 . dx x
−1 >F y 2 # x2 2 ( 2x dy = 2 dx. y ln y x −1 ,9 y ln y = 0. # ( 1 dy y ln y = 2x dx x2 − 1 2" &) % ln |ln y| = ln |x2 − 1| + ln C. ,G ( ln C !) %*) ln x /(# ###2 2 R. % y = eC (x −1) ln y = 02 y ≡ 1 / 7 ( k ∈ R {∅} # a ∈ R y = ky # y (0) = a. ,> A. / ) "/ # /" < (( & #( % & ( 2 dy dy &I @ y = dx 2 !( dx = ky. y 2 # x2 * $ ( k1 dy = dx. # y ( ln |y| = kx + C, " y = ky y = ec ekx . 9 ( ) % a = y (0) = Ce0 , " ,> A. +% / y = aekx .
" ! I @ ( & %#) % ∀t = 02 y = ϕ (x, y) ϕ (tx, ty) = ϕ (x, y) . (( f (u) = ϕ (1, u) $ # % &) % ϕ (x, y) = ϕ 1, xy = f xy , " $(" ( & %2 # @ u = xy % !# #( % & ( &I @ $(" # * % -.!/ x sin xy − y cos xy + xy cos xy = 0. "$ x2 ( sin y y y y − cos + y cos = 0 x x x x ,> H. >> % ( & %# @ ) % x % # tx2 # y % # ty 2 !( ,( y 2$ !(* + $ x y ) + xy ). (* u = u (x) = y(x) = xy I /*(# ( y (x) = x · u (x) , x % y = u + xu $/*# ,> H.2 % !() sin u − u
cos u + (u + xu ) cos u = 0 y " C!# du · x · cos u = − sin u dx #( % & ( &I @ "" /2 & sin u = Cx , " ,> H. @ + c y sin = . x x #$ y + p (x) y = q (x) ,> B. I @ * $(" # # * y − y = x2 + 3x − 2. x ,> D. ,G p (x) = − x1 ; q (x) = x2 + 3x − 2.) # # #" "$ Y Y− =0 ,> >F. x %# @ "/ & C % # C (x) /*(# !) % ,> D. G & ,> D. % ,> D. I ) % yi,ált = Yh,ált + yi,p ,> >>.
> ( ,> D. %# yi,ált " ,> >F Yh,ált ,> D. %# yi,p 0 Y − Yx = 0 @ #( % & ( &I Yh,ált = C · x. 0 # % C (x) /*(#) y = x · C (x) !( ,> D. ( y = C (x) + xC (x) , %2 ,> D. ) % C (x) + xC (x) − C (x) = x2 + 3x − 2. 2 G C (x) = x2 + 3x − 2 ln |x| , / * 2 & && ) + *. 3 :% yi,p = x · C (x) = x2 + 3x2 − 2x ln |x| . N ,> D x3 + 3x2 − 2x ln |x|. yi,ált = Yh,ált + yi,p = c·x+ 2 Yh,ált yi,p &# " # (*) yi,p 2 I % ) %
,> >F. %# "/ & C % !2 * C (x) /(# / n2 C %# y (n) + an−1 (x) y (n−1) + · · · + a1 (x) y + a0 (x) y = 0 ,> > . # n2 C %# y (n) + an−1 (x) y (n−1) + · · · + a1 (x) y + a0 (x) y = r (x) ,> >7. >7 @ / E* #) % / # 2 ( I ( / !( ) % C *% & 1. , # !) $$ C *% &( ( & ( % . <#* # * % -.!/ 9 2 2 y − y + 2 y = 2x x x C %# @ ,> >5. "$ (* #) % ! I ) ,> >7.2
n = 2, a1 (x) = − x2 # a0 (x) = x22 # r (x) = 2x. $(" # $$ ) C / I " # # $(" #(# ) ! C) # * 1-1 . ,> >7 %# yi,ált 2 ,> > . Yh,ált # ,> >7 %2 # yi,p $# yi,ált = Yh,ált + yi,p . ,> >?. 3 ) ,> >>. ) "C # I) % + "C @ 2) % ) " n2 C @ ) % %# " %# %# ( + # @
# C$ / # ) % "$ #*) % 2 % n2 C %# @ ,> > . , /*(#% & ) % % " ! ) &2*% & ( # ) %# @ . # ## #*) % % ,> >7. # & >5 % / % ,> > . I @ 2 % ( / "$ 2 *) % & /(# 2 2 3 I ( ( # O $ * C n (I)2( I 2 # n2 / @+ % & /*(# % ) (
/(# % ) 2 # # / n2 ( # ( *#) ,> > . # ,> >7 @ C n (I) % * 9 * /(# $ # ( & ) !) ( $ # ( & ) # C n (I)2 /*(# $ # 2 ( & #( (* ) % V C n (I)2) % ∀f, g ∈ V 2 # ∀α, β ∈ R2 αf + βg ∈ V * ) ( V #) % +& #( V # V " & /* % 2 E ) +& # V " f1 , . , fn ∈ V V ) % 4 5 f1 , . , fn /* ,( ) % α1 f1 + · · · + αn fn ≡ 0, α1 = · · · = αn = 0 # 465 ∀g ∈ V 2 # α1 , . , αn ∈ R, g = α1 f1 + · · · + αn fn
# / ) n2C %# ,> > . 2 C n (I) # ) # # , %!(. ! $(" # 1-1 . ,> > . n /*(#" n2&) ( >? ,> >5. # ,> >5 # " %# @ %# # 2 2 y − y + 2 y = 0. x x ,> >A. $C ") % y1 = x # y2 = x2 /*(# ,> >A. , ") % ) ( ) $ ) % # ) % # C +% + (## ) & *% & . :% {x, x2 } ,> >A2 % &) % y1 = x # y2 = x2 /*(# / ) % % ax + bx2
≡ 0 ) % a = 0 # b = 0. 9 ( # ) C # /*(#" :% ,> >A.2 {x, x2 } , ( ,> >A. yált = ax + bx2 . # # ) n2C %# 2 # * n # ) % /* n ##) + & *% & ) / ## 2 "## ) / ! ()*+(, , ! y1 , . yn ⎡ y2 (x) y1 (x) ⎢ y1 (x) y2 (x) ⎢ y2 (x) W (x) = det ⎢ ⎢ y1 (x) ⎣ . . (n−1) (n−1) y1 (x) y2 (x) , . yn (x) . yn (x) . yn (x) . . (n−1) . yn (x) ,> > . %# ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ,> >H. %!( 4 /*(#) # # I ( ) ### (
& % % ! ) % W (x) /*(# $(" 2 /*" y1 (x) , . , yn (x) /*(#" ) + $(" # %# >A • W (x)2 + # % ( • W (x) ≡ 0 I ( E ( + W (x)2 # % ( 4 # / *) W (x) 1-1 . y1 (x) , . , yn (x) /*(# # + ,> > . %# #) % # x ∈ I ) % W (x) = 0. , / # x2 * . -.!/ :* & y + y = 0 @ %# ) % / # % & !*) $# 2 / " ) % y1 = cos (x) # y2 = sin (x) /*(# / : # " ) % C %#
# /*(#" :% + % ) % "*) % / cos x, sin x /(# /2 2) 7 # ) I $* ) % ( ) % W (x) /*(# 2 ( ) cos (x) sin (x) = cos2 (x) + sin2 (x) ≡ 1 = 0. % W (x) = det − sin (x) cos (x) :% {cos (x) , sin (x)} y + y = 0 @ ) ( @ y = a cos x + b sin x $ -.!/ 9 & ) %* y − y = 0 C %2 # @ % &) % y1 = ch (x) # y2 = sh (x) /*(# %%) % ch (x) sh (x) ) ) % W (x) = det = sh (x) ch (x) ch2 (x) − sh2 (x) ≡ 1 = 0. :% y = ach (x) + bsh (x) . 7 -.!/ :* 1 + x2 y + xy − y = 0 ,> >B. C %# @ / ) % %
P # $) % y1 = x ,> >B.2 G % ) % >H y2 = c (x) · y1 = c (x) · x * E # c (x) = 1 /*(#) y2 = c (x) · x # ,> >B. y2 = c x + c # y2 = c x + c + c = c x + 2c . ,> >B @ % !( 1 + x2 (c x + 2c ) + x (c x + c) − xc = 0, ,> >D. " c :% @ ) + c # c 9 (* d = c I ) "2 C @ ) /*(# d. # % ) " ) % ,> >D. √ √ 1+x2 @ c (x) = , ( y = 1 + x2 . #2 2 x√ 2) # % + ) %
x, 1 + x2 ( ( C) # ! 2 " ) # ") % 4 2 √ 1 + x2 x 1 1 = −√ W (x) = det <0 2 −2 1 x (1 + x ) 1 + x2 √ x2) ! 4 0) ! x, 1 + x2 2 √ 2 # ,> >B. y = ax + b 1 + x 8 2 3 9: 3"#" & *% & / /(#% & +2 ) $ *% & /(# ) ( an−1 (x) ≡ an−1 , . , a1 (x) ≡ a1 , a0 (x) ≡ a0 :% ? # A # 2 & *% & ) ! > >B # > >A /(#% &2 E y (n) + an−1 y (n−1) + · · · a1 y + a0 y = 0 ,> F. ) % a0 , a1 , . , an−1 ( & 2 % %
+ ( % & I ; 0" "#< p (λ) = λn + an−1 λn−1 + · · · + a1 λ + a0 , ,> >. # r1 , . , rn , / # * $$". $2 0# ? # y + y = 0 λ2 + 1 $ r1 = i # r2 = −i. A # y − y = 0 λ2 − 1, $ r1 = 1, r2 = −1. >B = -.!/ :* y − 7y + 16y − 12 = 0 %# *% & @ & λ3 − 7λ2 + 16λ − 12 = (λ − 2)2 (λ − 3) . $ % r1 = 2, r2 = 2, r3 = 3. n2 C n2 /I , + ( . $ n $ ( ) ,2 $$ Q $$ %. $# /
/*(# @ # & > 9 r ( & $) % y (x) = erx /*(#R 9 r s2 +I ( & $) % $(" s /*(# erx , xerx , x2 erx , . , xs−1 erx ,> . 7 9 r = u + iv Q $) r Q ) r̄ = u − iv $ ) # " % $(" # /*(# eux cos vx # eux sin vx 5 r = u + iv s2 Q $) r Q r̄ = u − iv s2 $ 2 # " % $(" 2s /*(# eux cos vx, xeux cos vx, . , xs−1 eux cos vx eux sin vx, xeux sin vx, . , xs−1 eux sin vx -.!/ 9 %# &
*% & @ λ8 − 22λ7 + 232λ6 − 1546λ5 + 7066λ4 − 22 738λ3 + 50 752λ2 −72 670λ + 54 925 3 = (λ − (2 + 3i)) (λ − (2 − 3i))3 (λ − 5)2 @ >D e2x cos 3x, xe2x cos 3x, x2 e2x cos 3x, e2x sin 3x, xe2x sin 3x, x2 e2x sin 3x e5x , xe5x . / " % (λ − (2 + 3i))3 (λ − (2 − 3i))3 & (λ − 5)2 > -.!/ 9 y + 3y + 2y = 0 & *% &) C %# @ P "$ 2 λ2 + 3λ + 2 r1 = −2, r2 = −1 $ :% {e−2x , e−x }) ( y = ae−2x +be−x . / & # C$ & *% & %# 2 & > # # # %# ,> >7. I @
I ) % ,> > . %# # % % %# 2 "" / %# # ## & / ,> >7. I @ 2 # & (* "$ % ) + ) # (# * " I %! % ) & /*(# & # # & # "$ ,> > . %# # # " " 9 / % #(" ## y1, . , yn /*(# ,> > . %# # # / " # ( +& & ) % * c1 (x) , . , cn (x) /*(#) # y (x)
= c1 (x) y1 (x) + · · · + cn (x) yn (x) ,> >7. %# ,> 7. F % 1-1 . y (n) + an−1 (x) y (n−1) + · · · a1 (x) y + a0 (x) y = r (x) %2 # @ % y (x) = c1 (x) y1 (x) + · · · + cn (x) yn (x) ) % y1 , . , yn %# # # # c1 (x) , . , cn (x) /*(# ( $(" c1 (x) y1 (x) c1 (x) y1 (x) ··· (n−2) c1 (x) y1 (x) (n−1) (x) c1 (x) y1 + ··· + ··· ··· + ··· + ··· cn (x) yn (x) cn (x) yn (x) ··· (n−2) + cn (x) yn (x) (n−1) + cn (x) yn (x) + + = = 0 0 ,> 5. = 0 = r (x) -.!/ 9 y − x2 y + x22 y = 2x C %2 # @ P %# # ,> >A. ) ( # !) % %# #
{x, x2 } . K !( ,> 5 , % y1 (x) = x2 , y2 (x) = x. c1 (x) x2 + c2 (x) x = 0 c1 (x) 2x + c2 (x) = 2x c1 (x) = 2 # c2 (x) = −2x. % c1 (x) = 2x, c2 (x) = −x2 ,G % # " ) ;S c1 (x) , c2 (x)2 ( *#) % * . :% %# 2 2 3 yi,p = 2x·x −x ·x = x . E %# yi,ált = Yh,ált + yi,p = ax + bx2 + x3 . 6 ?2 # & + &2*% & 2 %# @ ) y (n) + an−1 y (n−1) + · · · a1 y + a0 y = r (x) , ,> ?. I @ % ) ,a0 , . , an−1 ∈ R % % & r (x) I ( & /*(# $(2 " I r (x) = eux [Pn (x) cos(vx) + Qm (x) sin (vx)] , ,> A.
> % Pn (x) # Qm (x) " n2 ( m2 /I :* / ) % u + iv Q s2 $ %# # p (λ) = λn + an−1 λn−1 + · · · + a1 λ + a0 ,> ?. %# @ % yi,p = xs eux [Pk∗ cos (vx) + Q∗k sin (vx)] ,> H. ) % k = max {n, m} # Pk∗ , Q∗k k 2 /I @ AB ACD-E 9 u + iv Q $ / 2 ) s = 0) ( xs ≡ 1, % yi,p = eux [Pk∗ cos (vx) + Q∗k sin (vx)] %* @ AB ACD-E ) % Pk∗ , Q∗k k 2 /I ) I #) % Pk∗ (x) = pk xk + pk−1 xk−1 + · · · + p1 x + p0 # Q∗k (x) = qk xk + qk−1 xk−1 + · · · + q1 x + q0 , % p0 , p1 , . , pk , q0 , q1 , qk ∈ R #* #% # "$ +
2 ) % ) @ & -.!/ :* / ) % % C %# &2*% & @ ) # %# # 2 $ λ1 = 2, λ2 = λ3 = 5. :* / ) % 5 ( & r (x) = cos 2x. ,> A2 u = 0, v = 2, Pn ≡ 1, Qm ≡ 0 # n = m = 0. ( u + iv = 2i $ ) # s = 0. :% yi,p = p cos 2x + q sin 2x *) % /I p, q 02 65 r (x) = x2 e5x . ,> A2 u = 5, v = 0, Pn (x) = x2 , Qm (x) ≡ 0 # n = 2, m = 0 ( k = max {n, m} = 2. ( u + iv = 5 # $ # s = 2. :% yi,p = x2 e5x (p2 x2 + p1 x + p0 ) * -.!/ 9
y − 12y + 45y − 50y = cos 2x %# # p (λ) = λ3 − 12λ2 + 45λ − 50 $ λ1 = 2, λ2 = 5, λ3 = 5. E % / . # ) :% yi,p = p cos 2x + q sin 2x + % ) % % p # q %% yi,p = −2p sin 2x + 2q cos 2x, yi,p = −4p cos 2x − 4q sin 2x, yi,p = 8p sin 2x − 8q cos 2x. ( !( & y − 12y + 45y − 50y = cos 2x # cos2 # sin2 () # *% & ##" && cos 2x(−8q + 48p + 90q − 50p) + sin 2x(8p + 48q − 90p − 50q) = cos 2x 1 ( 0 −2 p + 82 q = 1 −82 p + −2 q = 0 1 41 1 % p = − 3360 # q = 3360 E yi,p = − 3360 cos 2x + 41 sin 2x. :( %# yi,ált 3360 41 1 cos 2x +
sin 2x. yi,ált = c1 e2x + c2 e5x + c3 xe5x + − 3360 3360 Yh,ált yi,p % -.!/ 9 y − 6y + 13y = x + sin 3x @ P %# # λ2 − 6λ + 13. $ λ1 = 3 + 2i, λ2 = 3 − 2i E* #) % ( & r (x) = x + sin (x) % ,> A. # $( *) #) ,> A. # (" /*(#) ( r1 (x) = x # r2 (x) = sin (x) /*(# $ # % / ! & yi,p,1 y − 6y + 13y = x ! " @ ) yi,p,2 7 y − 6y + 13y = sin 3x @ # y − 6y + 13y = x + sin 3x yi,p ) yi,p = yi,p,1 + yi,p,2 2$ (## 6 9 1 1 cos 3x + sin 3x yi,ált = c1 e3x cos 2x + c2 e3x sin 2x + x + 2 + 13 13 85 170
! " # " 9 (* C @ ) % x, y, y , $2 * % % C @ ) % x, y, y $* % # ) % % > - y y y = f (x) . / f (x) # ( % -.!/ y = 5 + sin 3x. y = (5 + sin 3x) dx = 5x − cos 3x + C1 . :% y = 5x − cos33x + C1 dx = 52 x2 − sin93x + C1 x + c2 . 3 F (x, y , y ) = 0 - y % ) p (x) = y (x) % !# $ -.!/ + ) y (x) = p (x) T 1 1 + x2 y − xy = 0, 2 y (0) = 0 y (0) = 3 +% / P . ( y = p (x) # y = p (x) 1 1 + x2 p − xp = 0 2 % !# 5
#( % & ( &I @ p = c (1 + x2 ) . :% y = c (1 + x2 ) G 2 x3 + K. yált = c x + 3 +% / I ) % % !* x = 02 yált ) yált # # y (0) = K = 0 . y (0) = c = 3 :% +% / y = 3x + x3 . 7 - x * F (y, y , y ) y = p (y) % !# I y = dp dy dp dp (y (x)) dp = = y = p. dx dy dx dy dy 7 -.!/ T @" $/*# ! yy = 2 (y )2 − 2y % !* y = p # y y = dp p dy dp p = 2p2 − 2p dy #( % & ( &I @ , y ( & # p = p (y) y /*(#. p ≡ 0 # p = c2 y 2 + 1 K % ( ) % y = p ) % y ≡ C # y = 1c tg (cx + c2 ) . $ % M (x, y) dx + N
(x, y) dy = 0 # ? @+ ) % ∂N ∂M ≡ ∂y ∂x / # * ) # M (x, y)) N (x, y) /(# + 2 ( # # / ) + $/*" %!( > > , . # , $/*" 1-1 . ≡ ∂N # M (x, y) N (x, y) /*(# 2 9 ∂M ∂y ∂x + ( # # / $/*" 2 ) # F : R2 R /*(# M= ∂F ∂F # N = ∂x ∂y ( grad (F ) = (M, N ) . M (x, y) dx + N (x, y) dy = 0 @ F (x, y) = C + % 2 = -.!/ T y 3x4 y 2 − x2 + 4x3 y 3 − 2xy = 0 @ @" dy # / 9 / ) % y = dx # dx2 )
% 4x3 y 3 − 2xy dx + 3x4 y 2 − x2 dy = 0. M (x,y) N (x,y) @ ) ( M # N /*(# # +2 ( * # # ∂M = 12x3 y 2 − 2x = ∂N . :% ∂y ∂x A / # # F ) M = ∂F # N = ∂F . / F ?2 ∂x ∂y ∂F #3 " ( ∂x = M (x, y) , # F (x, y) = M (x, y) dx = 4x3 y 3 − 2xy dx = x4 y 3 − x2 y + f (y) . + f (y) /*(# % & ) % N = ∂F . / !( (3x4 y 2 − x2 ) = 3x4 y 2 − x2 + f (y) G f (y) ≡ ∂y 0 ( f (y) ≡ C. :% F (x, y) = x4 y 3 − x2 y + C / # # # % @ x4 y 3 − x2 y = C G + & ( / /* ( & t2( $ ) /(#2 x (t) # y (t)2( 9
/ D @+ 2 E x (t) = Dx (t) # y (t) = Dy (t) . 9 ! # ) % (D − 5) x ##C) % x (t) − 5x (t) . @+ $* + C ) # $(" # -.!/ T $(" @ 2x + y − 4x − y = et x + 3x + y = 0 @" D 2 (D − 2) x + (D − 1) y = et (D + 3) x + y = 0 2 # C # ( ) 2 (D − 2) D − 1 = − D2 + 1 = 0. Δ = D+3 1 & $(" # " t e D−1 2 (D − 2) et # Δy = . Δx = 0 1 D+3 0 $ % H /( − D2 + 1 x = et # − D2 + 1 y = −4et . x (t) = c1 cos t + c2 sin t − 12 et # y (t) = c3 cos t + c4 sin t + 2et . # # %) # c3 ,
c4 /%" c1 , c2 2( x(t), y (t) / # # % !( ⎛ ⎞ ⎛ ⎞ ⎝c2 + 3c1 + c3 ⎠ cos t + ⎝3c2 − c1 + c4 ⎠ sin t = 0 0 0 $(" −3c1 − c2 = c3 c1 − 3c2 = c4 E t x (t) = c1 cos t + c2 sin t − e2 . y (t) = − (3c1 + c2 ) cos t + (c1 − 3c2 ) sin t + 2et > -.!/ T x y $(" +% / − y + 3x = 0 x (0) = 1, y (0) = 3 + 2x = 0 @" K ! ,> B. D % ( (D + 3) x − y = 0 2x + Dy = 0 D + 3 −1 = D2 +3D +2 = 0. :% / & ! Δ = 2 D % & 2 & 0 −1 D+3 0 = 0 # Δy = =0 Δx = 0 D 2 0 !( x + 3x + 2x = 0 # y + 3y + 2y = 0. ,> D. B ( ) % x (t)
= c1 e−t + c2 e−2t # x (t) = c3 e−t + c4 e−2t . ( + # % # % ) ( % !2 ( ) # " &) % ⎛ ⎞ ⎞ ⎛ ⎝2c1 − c3 ⎠ e−t + ⎝2c2 − 2c4 ⎠ e−2t ≡ 0. 0 0 :% c3 = 2c1 , # c4 = c2 . ( % !( ,> D ) ) % @ x (t) = c1 e−t + c2 e−2t y (t) = 2c1 e−t + c2 e−2t . ,> 7F. +%2/ I ) % t = 02 % !* 2 # # / % y (0) = 3, x (0) = 1 / # 1 = x (0) = c1 + c2 3 = y (0) = 2c1 + c2 . # ) % c1 = 2, c2 = −1. ,> 7F.2 ( !( +%2/ x (t) = 2e−t − e−2t y (t) = 4e−t − e−2t . ,> 7>. / 66 # :* / ) % ! C $(" / #
t = 0 "2 ! A = (1, 3) % & (x, y) ) # (y − 3x, −2x) ( K ! @" ( A & # # (x , y ) = (y − 3x, −2x) ) % x # y " /*## !& x (t) , y (t) /(# / ,> B. +% / ) ( F # # x (t) = 2e−t − e−2t y (t) = 4e−t − e−2t . /*(# ! ,> 7 . & ) Y = F (X, Y ) % * % y = f (x, y) / y = y (x) /*(# # x /2 ( & ( T ⊂ R2 ! % # f : T R /*(# / y = f (x, y) ,> 77. I "C @ (
## * (2 ) % y = f (x, y) y (x0 ) = y0 ,> 75. +% / ,(x0 , y0 ) ∈ T . ) f /*(# / # ( , + ##. # % ( ) % ( ,+ ##. ( / % (* +%2/ # % ! ()*+(, ) % f : T R /(# y ( & #! . 03 <? ) % #) L ) % T & (x, y1 ) # (x, y2 ) # |f (x, y1 ) − f (x, y2 )| ≤ L |y1 − y2 | ,> 7?. * L2 +% %!( +%2/ # #! / @+ % & f (x, y) /*(# ,% T . :%) % f (x, y) # + 2 ( / T 2) f (x, y)2 * +%2/ # +%2/ # y ( & /*(# #!%) 2 # # + ( 0# 2 f (x, y) = x + |y| /*(#
@+ % & y 2 2 ) y ( & #! +%2/ # ) % |f (x, y1 ) − f (x, y2 )| = |x + |y1 | − (x + |y2 |)| = ||y1 | − |y2 || ≤ |y1 − y2 | . :2 % f (x, y) = x+|y| /*(# y ( & #! +%2/ # R # L = 1 +% $ 1-1 . , # ,(x0 , y0 ) ∈ T . . 9 y = f (x, y) f (x, y) /*(# / ) 2 y (x0 ) = y0 +% / D 7F $ % ) % / $ 2 $( %%) % # C##) + 2 2 3 ) % f (x, y) = y 3 , f / !) # 2 y = y 3 ) y (0) = 0 +% / y ≡ 0 3 y (x) = x3 /*(# 7 1-1 . ,F 9 f (x, y) /*(# / # & #! . 03 <? ) # # # C + y = f (x,
y) y ( 2 2 y (x0 ) = y0 +% / ,(x0 , y0 ) ∈ T . ,*# # % y = f (x, y) %% y = f (x, y) y (x0 ) = y0 ,> 7A. +% / ) / ) # C % &) % f / # y ( & +% / # 2 "/ % ) % /*(#) # % " ! ) ( + # ! ") " + % % $( ) % # +% / = 9 " ! ) $ !" % ) 2 " " $ ! +% / $* 2 # 1 & ; # y = f (x, y) , y (x0) = y0 +% / 2 xn = x0 + nh +% # y = y (x) 2 & &
$( $ !*) (xn , yn ) $ $) % xn = x0 + nh # yn+1 = yn + f (xn , yn ) h. $ !# % / h2 C G ## ,h %II. y = y (x) /*(# #"#( % !*) % y = f (x, y) , # (x0 , x1 ) y = y (x) % #") ( y = yn +y (x0 ) (x − x0 ) = yn +f (x0 , y0 ) (x − x0 ) /*(# ( f (x0 ,y0 ) ( 7> y y2 y1 y0 y−1 x−1 x0 x1 x2 x ) G H; <; # y = f (x, y) # y (x0 ) = y0 9 & /%) + ) ## % / h5 C G & & $$ ( $ !* y = y (x) /*(#) (xn , yn ) $ $) % xn = x0 + nh, # yn+1 = yn + 16 (k1 + 2k2 + 2k3 + k4 ) , % k1 = hf (xn , yn ) k2 = hf xn + 12 h, yn + 12 k1 . k3 = hf xn + 12 h, yn + 12 k2 k4 = hf (xn+1 ,
yn + k3 ) . /%0 / /* ( & t ∈ R) # /(# x = x (t) . $ # "" / ( , y = f (x, y).) @+ $ # ẋ = f (x, t) I) % ẋ = dx . dt / ẋ = f (x, t) @ + # %") ẋ = v (x) I @+ / E + x2" /*) Q + # / t2" , t ( & " . ẋ = v (x) I %!(2 ) % v : R R # / @+2 7 % & , / * ) (* -.!/ / v 2" . /*(# # ) ẋ = (a − bx) x. I % x = 0 # x = ab v(x) # (* I % ) % ẋ = v (x) " /
"## ! ) # % t0 " # x0 % ( ) v (x0 ) = 0, ẋ (t0 ) = v (x0 ) = 0. :% ( # " ; E ( ) % x0 % " (## & &) % x (t) ≡ x0 ẋ = v (x)) x (t0 ) = x0 +% # ) # ( v (x) # ) ! "" / # + % ,+.) % ! ()*+(, x (t) ≡ x0 I % / % (*) % ∀ε > 0 ∃δ > 0, %% ϕ (t) ẋ = v (x)2 ) |ϕ (0) − x0 | < δ, |ϕ (t) − x0 | < ε ∀t2 E % / # + ( ) %% & + + # I % " " I ( -.!/ # x : R R2 ) % x (t) :% x (t) = (x1 (t) , x2 (t)) . !
ẋ (t) = (−x2 (t) , x1 (t)) # x (0) = (u, 0) , u ∈ R v(x)=(−x2 ,x1 ) E ) v(x) = ẋ (t) # x (t) " # %I % % v (x) = 0) x = 0 ,% &. N I % x (t) ≡ 0. / +% / ẋ1 = −x2 ⇒ ẍ = −x1 ⇒ x1 = a cos t + b sin t ẋ2 = x1 u = x1 (0) = a, % x1 = u cos t + b sin t ) −x2 = ẋ1 = −u sin t + b cos t. x2 (0) = u sin 0 − b cos 0 = 0, ! b = 0, ( x1 = u cos t. ! x2 = −ẋ1 = u sin t. E +% / x (t) = (u cos t, u sin t) . ( 77 y (u, 0) x ( |x (t)| = u, % x (t) & $# I u I $$ ( G & )% x (t) ≡ 0 I % (2 =!+& " δ 2 ( % δ = ε2 $ ! ()*+(, 2 x (t) ≡ x0 I % % 0 (2 2 lim ϕ (t) = x0 * t∞
ẋ = v (x) ϕ (t) ) ϕ (0) x0 2% # $ ( -.!/ x (t) & % ) % & 2 % % ( ,+ I & / # *. < () % & % *) $$ ) % I % 9 ( + !) I & I % % / ) % I % % -.!/ ẋ = x v (x) = x, 75 :% I x ≡ 02 ( @+ 2 x (t) = cet . :% ẋ = x, x (0) = u +% / x (t) = uet . E % u +) F ) t ∞ lim uet = ∞, ( x ≡ 0 I % t∞ (2 ,# ! . U ẋ = kx @+ , ! !
&* ) % % k > 0, x ≡ 0 I ẋ = kx # /) % ( / $(## , &) 2 #%) #.) +& t " I ( cekt ) # / @+ 9 ( +& ) ẋ = (a − bx) x +& t " I ( a, b > 0, ! (a − bx) & Q +$" /*(#) # / ) % % x ab , I ( # % # ) ! ẋ < 0 ) ( +$ 9 ẋ = (a − bx) x @+ " # x ##2 ## #*) ẏ = (1 − y) y @+ % G I y ≡ 1, y ≡ 0. ẋ = (1 − x) x # x (0) = u +% / et−c et−c x (t) =
et−c , % u < 1 # x (t) = et−c ,% u>1 +1 −1 y I % x(t) = et−c et−c −1 x(t) = et−c et−c +1 t ! & ) % x ≡ 1 ) # x ≡ 0 ,# # . -.!/ +& % $(" # # ( ) % # % & ) % 2 ( 7? ,. ( # # # ,. % c # % ẋ = x (1 − x) − c . "$ I ) ( v (x) = 1 2 1 x (1 − x) − c /*(# $ ( v (x) = − x − 2 + 4 − c, # % * $$ y 1 4 −c v(x) A 1 2 x B , . c < 14 , ( ) v (x)2 # !( $ ( ) % c > 0. A # B E A # B I 9 x (0) < A ( x (0) > B, x (t) = B + B−A B−A . , % K = 1 +
1−4ct x (0) − B Ke −1 √ 9 0 < x (0) < A. 9 B < x (0) . :% % x (0) + A, % +& 9 x (0) > B, lim x (t) = B. t∞ 9 A < x (0) < B, x (t) = B − E B−A √ Ke 1−4ct +1 / % ) % A ) , % K = x(0)−A . B−x(0) B 2 7A x x(t) B A ln 2B−A A t x x(t) B A t y B x(t) A x , . c = 14 , A = B = 12 % = # 2 % ) ## ) /) ( I ,+ . c > 14 , + I +& (# " * ( 7H % # x x(t) t 7B 1 2- u u(x, t) 0 x L x :* l %II $# %I 9 %I x2 [0, l]
# ! x− " 2 %I x " (x, u) ! ) % %I x +I % ( t " I ( u (x, t) /*(# " %I % % ) % u (x, t) /*(# = / & &) u (x, t) /*(# #! 2 ∂ 2u 2∂ u = c ∂t2 ∂x2 7D , >. 5F ) C + @+ ) % c > 0 / , >. @+ " %I @+ # %!( %%) % " %I % # !& u (x, t)2 ) # %I t = 0 " ,K / *) % x2 [0, l] ( ) % % f (x) /*(# . :( *# # ) % t = 02 %I 2 ∂2u E = c2 ∂∂xu2 # ! u ∂t2 @+
u(x, 0) = f (x) / # ∂u | = g (x) ∂t (x,0) * / # $ # u (0, t) = u (l, t) = 0 ∀t2 K % ∂u | ∂t (x,0) 7 ! ()*+(, U ) % F /(# # H # # 2 , *. #% ) F |H F ! H 2 E F |H /*(#) # # H # F |H /*(# H % F /(# 9 # H C % ) H = {P }) F |P = F (P ) :% 2 ∂2u 2∂ u = c ∂t2 ∂x2 @+ u (x, 0) = f (x) , ut (x, 0) = g (x) # u (0, t) = u (l, t) = 0 / # & ! ( 2 %* u (x, t) = X (x) T (t) ∂∂xu2 = X (x) T (t) 2 # ∂∂t2u = X (x) T (t) . ( !( ) % X (x) T (t) = C 2 X (x) T (t) , ( T (t) = c2 T (t) X (x) = −α2 . X (x) "* 5> 3
+ t2" ) + x2" /*) " &) % " &( ") = & 2 !( / # * N % $$# @+ (t) (x) ,>. TT (t) = −α2 ; # , . c2 XX(x) = −α2 . E 2 ,> . T (t) + α2 T (t) = 0, # , X (x) + αc X (x) = 0, T (t) = A cos(αt) + B sin(αt), # X (x) = C cos Cα x + D sin Cα x G α α x + D sin x u (x, t) = X (x) T (t) = (A cos (αt) + B sin (αt)) C cos c c T (t) X(x) :)% u (0, t) = (A cos αt + B sin αt) C ≡ 0. G C = 0, # α (l, t) ≡ 0 = (A cos αt + B sin αt) D sin αc l , ! sin ( l) = 0, " c kπ αk = kcπ , l k = 1, 2, . " % &) % u(x, t) = ∞ ! sin k=1 kcπ kcπ Ak · cos t + Bk · sin t l l kπ x l {Ak }k=1 # {Bk }k=1 *% & / # " ) 2
f (x) # g(x) /*(# K2 !##2 ∞ ( !% " ( u (x, 0) = f (x) # u (x, 0) = Σ Ak sin( kπ x) ! Ak l k=1 f (x) /*(# [0, l] ( ( K2 k 2 *% & #) ∞ ut (x, t) = Σ (−Ak αk sin αk t + Bk αk cos αk t) sin k=1 α k c x . 9 !( t = 02 # / % ( ) % sin(0) = 0; cos(0) = 1 2 ) % ) 5 ∞ ut (x, 0) = Σ αk Bk sin k=1 kπ x l = g (x) . :% % g (x) f üggvény K2 [0, l] ( ∞ g (x) = Σ bk sin k=1 ∞ Σ αk Bk sin k=1 kπ x l = g (x) # ) % αk Bk = bk . :% k = kπ x . l ∞ g (x) = Σ bk sin bk αk = k=1 lbk . kcπ kπ x l $ -.!/ " &2 utt = c2 uxx 2 " c = 1. F + %I # (## $! %I /!* I) %
) # * 9 %I !& u (x, t) /(#P y y = 0.05x y = 1 − 0.05x x "* 57 . ( %I % , #.) # %I # F :% g (x) ≡ 0. G ∀k ≥ 12 Bk = 0, % g (x) ≡ 0 /*(# K2 bk % & F {Ak } *% & % % % 0 ≤ x ≤ 10 0, 05x, f (x) = , . 1 − 0, 05x, % 10 ≤ x ≤ 20 /##P Ak = 40 (kπ)2 2 l 20 0 f (x) sin kπl xdx = 2 20 10 0 0, 05x sin kπ x 20 20 2 dx+ 20 (1 − 0, 05x) sin kπ xdx = 20 10 sin kπ 2 , %& # ! +% 7 B % . :% ∞ 40 kπ kπ kπ cos t sin x u (x, t) = Σ 2 2 sin k=1 k π 2 20 20 E π π 40 1 t sin x − ( cos π 2 12 20 20 1 3π 1 5π 3π 5π − 2 cos t sin x + 2 cos t sin x − .) 3 20 20 5 20 20 u(x, t) =
&& )* +# 2 2 = g (x) ; u (0, t) = ∂∂t2u = c2 ∂∂xu2 u (x, 0) = f (x) ; ∂u ∂t (x,0) u (l, t) = 0 / # J #(# ) % % u1, u2 : R R # @+ % & " /*(#2 ) u (x, t) = u1 (x + ct) + u2 (x − ct) 2 ∂u #! " %I ∂∂t2u = c2 ∂x + @+ # 2 1 % u (x, t) = u1 (x + ct)+u2 (x − ct) , ∂u = cu1 (x + ct)− ∂t (x,t) cu2 (x − ct) . :( ∂ 2 u 2 2 (x,t) = c u1 (x + ct) + c u2 (x − ct) 2 ∂t ) 55 # ∂u ∂x = u1 (x + ct) + u2 (x − ct) # ∂ 2u = u1 (x + ct) + u2 (x − ct) 2 ∂x / # " & 2 ∂ 2 u 2 2∂ u (x,t) = c [u1 (x + ct) + u2 (x − ct)] = c (x,t) ∂t2 ∂x2 % / # " % u1, u2 /*(# . u (x, 0) =
f (x) # ut (x, 0) = g (x) / # # u (x, t) = u1 (x + ct) + u2 (x − ct) t = 02 !( u (x, 0) = u1 (x) + u2 (x) . 9 u (x, t) = u1 (x + ct) + u2 (x − ct) /# t ( ) t = 02 % !*) ∂u (x,0) = cu1 (x) − cu2 (x) . ∂t :% ) % f (x) = u (x, 0) = u1 (x) + u2 (x) # g (x) = ut (x, 0) = c (u1 (x) − u2 (x)) ! u1 (x) + u2 (x) = f (x) u1 (x) − u2 (x) = g(x) c $$# @+ ) ,> A /. ( & &) % ⎧ ⎪ ⎪ ⎨ u1 (s) = 12 f (s) + 1 2c ⎪ 1 ⎪ ⎩ u2 (s) = 2 f (s) − 1 2c s 0 s 0 ⎫ ⎪ g (τ ) dτ + A ⎪ ⎬ ⎪ g (τ ) dτ + B ⎪ ⎭ "* " 5? ) % A + B = 0 N % u (x, t) = u1 (x + ct) + u2 (x − ct) = 1 2c x+ct 1 [f (x + ct) + f (x − ct)] + 2 g (τ ) dτ x−ct E* #) % (# %
* / # ) % u (0, t) = u (l, t) = 0, % / (# M(# %IM %I # C$ 7 -.!/ :* "" # ) # J &#( %I ( %I %I !& u (x, t) /*(# y y = 0.05x y = 1 − 0.05x x . 1 2 u (x, t) = [f (x + t) + f (x − t)] , % f (x) = # 0.05x % 1 − 0, 05x % y f (x) x 0 ≤ x ≤ 10 10 ≤ x ≤ 20 2 ) 5A :% ) % 10 "# I ( ## I % ) % x = 2 cm2 ⎡ u (2, 10) = %I ⎞⎤ ⎛ 1 1 ⎝ [f (12) − f (−8)] = ⎣1 − 0.05 05 · 8⎠⎦ · 12 − −0, 2 2 f (12) f (−8) 1 [1 + 0, 4 − 0, 6] = 0, 4 = 2 * ) +, ! -+ . ( 2 ! ! u (x, 0) = f (x) , ut
(x, 0) = g (x) , ! u (0, t) = u (l, t) ≡ 0 / % 2 3( % ! u (x, t) = 12 [f (x + ct) + f (x − ct)]+ 1 2c x+ct g (τ ) dτ / # = (#2 x−ct #( 9 * / # f # g /*(# # ## [0, l]2" R2 $(" # = % & /*(#) [0, l] ( % 2 & & *$ ! [−l, l]2 # /*(# 2l y −3l −2l −l l 2l 3l 4l x J & C$ (# %I %I #) # + u (0, t) = u (l, t) ≡ 0 * / # ) # f, g /*(# # ( 3( % ! / ) % $2( ("## # %
,x = 0 # x = l.2# E # (x0 , t0 ) ) t = − 1c (x − x0 )+t0 # t = 1c (x − x0 ) + t0 / "* 5H t (x0 , t0 ) (0, t0 − x0 c (l, t0 + x0 c (l, t0 − (0, t0 + x0 c − 2l c x0 c − cl ) t0 − 1c (x − x0 ) + t0 t + 1c (x − x0 ) + t0 (l, t0 + −2l − cl ) −l l 2l x0 c − 3l ) cx 3l x00 + ct0 x0 − ct0 x0 − ct0 + 2l 4l − x0 − ct # /) x + ct = const1 # x − ct = const2 I /*(# "$ # ) % ( " f (x0 + ct0 ) & &) % x0 − ct0 −f (−x) −f (−x + 2l)−f (−x + 4l) f (x) −2l −l f (x + 2l) l 2l 3l 4l x f (x − 2l) x0 + ct0 :% f (x0 + ct0 ) = −f (4l − x0 − ct0 ) # ! f (x0 − ct0 ) = f (x0 − ct0 + 2l) . , 4l −x0 −ct0 # x0 −ct0 +2l :% [0, l]2" ( . ) 5B u (x0 , t0
) = 1 1 f (x0 − ct0 + 2l) − f (4l − x0 − ct0 ) 2 ⎡ 2 ⎢ −l 0 l 1 ⎢ ⎢ g (y + 2l) dy + −g (−y) dy + g (y) dy + ⎢ 2c ⎢ ⎣x−ct 0 −l 0 ⎤ x+ct ⎥ 2l 3l ⎥ + −g (−y + 2l) dy + g (y − 2l) dy + −g (−y + 4l) dy ⎥ ⎥ ⎦ l 2l 3l 0 x+ct % ( ) % g (y + 2l) dy = g (s) ds # −g (−y + 4l) dy = x−ct 2l+x−ct 3l 4l−x0 −ct0 s = −y + 4l, y 3l x0 + ct0 = g (s) ds. ds = −dy, s l 4l − x0 − ct0 l :% u (x0 , t0 ) = 1 (f (x0 − ct0 + 2l) − f (4l − x0 − ct0 )) + 2 ⎡ ⎤ 4l−x l 0 −ct0 1 g (s) ds + g (s) ds⎦ + ⎣ 2c 2l+x0 −ct0 l ⎡4l−x −ct ⎤ 0 0 1 1 = (f (x0 − ct0 + 2l) − f (4l − x0 − ct0 )) + ⎣ g (s) ds⎦ . 2 2c 2l+x0 −ct0 # % ) (x0 − ct0 + 2l) , (4l − x0 − ct0 ) , (4l − x0 − ct0 ) # (2l + x0 − ct0 ) [0, l] ( & ( < /) % / (x0 , t0 )2 ( ) &
& /*") % % M( ("# M % (x, t) & [0, l] ( ) / {(x, y) : 0 ≤ x ≤ l, 0 ≤ y} ( x + ct = kl I x − ct = kl "* 5D t 4 cl 3 cl 2 cl l c x l "" / (x0 , t0 ) ! (2, 3) ( ) x − ct = kl ) x + ct = kl 2 3"# *$"# ( :% (2, 3) #(" 2 (x, t)2 u (x, t) 12 (f (x0 − ct0 + 2l) − f (4l − x0 − ct0 )) + 1 2c 4l−x0 −ct0 2l+x0 −ct0 g (s) ds Q + / ( % &) 2 Q + / * $$" /* utt = c2 uxx , u (x, 0) = f (x) , ut (x, 0) = g (x) # u (0, t) = u (l, t) ≡ 0 * / # C ( / " 0 0 u (0, t) = u (l, t) ≡ 0 * / #
% % 2 utt = c uxx , u (x, 0) = f (x) , ut (x, 0) = g (x) , x ∈ R. & ) ) ) ( I / /% ) % ) ?F J &# ) 1 1 u (x, t) = (f (x + ct) + f (x − ct)) + 2 2c = -.!/ x+ct g (s) ds x−ct 9 f (x) ≡ 0 # (x) = cos x, u (x, t) = ,9 1 1 (sin (x + ct) − sin (x − ct)) = cos x sin ct. 2c c sin (α + β) − sin (α − β) = 2 cos α sin β $/*# . -.!/ :* (# %I %I) % 2 / (−a, 0) ; (a, 0) ; (0, b) ) * ! / %I !& u (x, t) /*(# u t=0 B −A A x ha |x| ≤ a b − b|x| a # g (x) ≡ 0, % 0 ha |x| > a %I % u (x, t) /*(# . f (x) = u (x, t) = 1 [f (x + ct) + f (x − ct)] , 2 > 5 # > -.!/
+*# "* ?> u t=0 B b 2 b 2 a 2c x t= a c x 2a t= c b 2 −3a t= −2a −a a 2a x x 3a u −a a x !" %I x /) # # ( 2 ++ *# $# x = 0 +*# t = 02 |x| < a g (x) = 1 # ) $ 2 # N / %I !& u (x, t) /*(#P . f (x) ≡ 0; g (x) = 1 u (x, t) = 2c 1, 0, x+ct g (s) ds = x−ct ha |x| < a . ha |x| ≥ a 1 {% 2c (x − ct, x + ct) ∩ (−a, a) 2 } . ? ) a 1 ) = u x, % 2c 2c * a a x − ,x + ∩ (−a, a) 2( . 2 2 * $$" |x| < a2 2R a2 < x < 3a 2 # x > 2 % 3a 2 1 1 u (x, t) = (f (x + ct) + f (x − ct)) + 2 2c x+ct g (s) ds x−ct & ) % (x0 , 0) / # % c # ) ( / c
# ! %# 0 0 0 # * ) # (## 00 C 2 (! % # I % l V # *) % I x ( ) # (# I %"## # f (x) /*(# u (x, t) /*(#) I %"## # ! x t " I ( %"## t = 02 > ( %" % " % / #) # # " I ( ( x I #% 2 ) #% + %"## " "% ( 9 x t " ( # u (x, t) %"## % % ) / % K" & #(# ) % %"# ut = kuxx , (0 < x < l; 0 < t) C + @+ ! , (## 2 % =
I$# +!C $(# . ! I %"## t = 02 ) ( u (x, 0) = f (x) . ! & &) % I # (# 00 C 2 ( & ( ) I& & %" " ?7 !"! !" " # $ 0◦ C 0◦ C % f (x) l / ! ,# ) % . :% ! u (0, t) = u (l, t) ≡ 0. E ⎧ 0 < x < l, 0 < t ⎨ ut = kuxx , u (0, t) = u (l, t) ≡ 0 ⎩ u (x, 0) = f (x) # K2 % ( , %"(# ) ?5 # # K ) # & ( # +# & ( >B 2 . . u(x, t) /*(#) + I " %I # u (x, t) = X (x) T (t) * (# $( "
%I @+ # (### # # E X T = = −λ = . kT X G :( T = −λkT ⇒ T (t) = Ae−λkT . −X (x) = λX (x) , 0<x<l 2 # X (0) = X (l) = 0. # ) # / " ) X (x) = sin :% ∞ nπx . l 2 nπ u (x, t) = Σ An e−( l ) kt n=1 / #() % " %I sin nπx l ∞ nπx n=1 l f (x) /*(# K2 ⎫ ⎧ (0 < x < l, 0 < t) ⎬ ⎨ ut = kuxx , u (0, t) = u (l, t) ≡ 0 ⎭ ⎩ u (x, 0) = f (x) # % I ) % f (x) /*(# K2 /* [0, l]2 , " %I # . G * {An }∞ n=1 *% & ) !##( / ! u (x, t) # # {An }2 ## % !##( f (x) = Σ An sin ∞ 2 nπ u (x, t) = Σ An e−( l ) n=1 kt sin nπx l " ??
$/*# 2 λn = nπ Xn (x) = sin nπx /*(#2 l l ! %!( ) % % A R R2 # " # / @+ % & /*(# # 2 I ) Ag = −g , $ # AXn = λn Xn . E* #) % A2 (# ## ( π 2 4π 2 9π 2 ; ; ;. l2 l2 l2 % ) % A2 ## ⎧ (0 < x < l, 0 < t) ⎨ ut = kuxx , -.!/ T ⎩ u (t, 0) = u (l, t) ≡ 0 / 2 u (x, 0) = f (x) 20x ha 0 ≤ x ≤ 5 ) % l = 10 # f (x) = 200 − 20x ha 5 ≤ x ≤ 10 , > . k = 1. E t = 02 I $ # %"## 1000 C 2) # (# / # +$ 02% . : K2 / f (x) /*(# ∞ nπx E {An }∞ n=1 ##) f (x) = Σ An sin l (0 ≤ x ≤ l) . & ) % An = 1 An = 5
= −200 5 0 1 nπ xdx + 20x sin 10 5 −2 sin( nπ ) 2 + nπ cos n2 π 2 nπ 2 5 (200− 20x) sin 5 +200 2 l n=1 l 0 f (x) sin nπ xdx. l nπ xdx = 10 + 2 sin( nπ ) −2 sin(nπ) + nπ cos nπ 2 2 = 2 2 nπ nπ 800 = 2 2 sin( ). , 7 nπ 2 ) ?A y y = 20x y = 200 − 20x ⎧ 0, ⎪ ⎪ ⎨ 1, ( sin nπ = 2 0, ⎪ ⎪ ⎩ −1, + xπ sin 10 f (x) = 800 − π2 12 9 ( ha ha ha ha n = 4k n = 4k + 1 , ! n = 4k + 2 n = 4k + 3 sin( xπ 3) 10 32 + sin( xπ 5) 10 52 − sin( xπ 7) 10 72 + ··· . ∞ 2 nπ u (x, t) = Σ An e−( l ) kt n=1 $/*# , sin nπx l u (x, t) = π )2 t −( 10 800 e [ π2 12 2 − 5π t 10 e ( ) sin xπ 10 − sin( xπ 5) 10 2 5 2 t − 3π 10 e ( ) − ···] sin( xπ 3) 10 32 + x " ?H 1+ , $(" # % K2 / % ) #
$/ =!+&) # f : R R, / , *#) + # / ) # % .) ( ∞. f (x) 4 ! ∞ F [f ] (λ) = g (λ) = ∞ −∞ |f (x)| dx < f (t) e−iλt dt. −∞ #*) % i # ,i2 = −1. > % &) % g (λ) # / R2) # lim g (λ) = 0. |λ|∞ ∞ 9 f / ) # f (x) dx < ∞, F [f ] (λ) = iλF [f ] (λ) , .−∞ # ! ∀n2 F f (n) (λ) = (iλ)n F [f ] (λ) * ) % f (n) / ) # ∞ f (n) (x) dx < ∞. −∞ %# 0 0 C# k = 1 %"(# ut = kuxx , # N $(" # ( ut = u”xx −∞ < x < ∞ # t ≥ 0 . u (x, 0) = f (x) :* / ) % ∞ −∞ |f (x)| dx; ∞ −∞ 2 |f (x)| dx; # ∞ −∞ |f (x)| dx
# (f ∈ C (R)) . / u (x, t) /*(# $# ) u ∈ C 2 ,( # / @+ % &. # ,. ∀x ∈ R2 ∞ −∞ |u (x, t)| dx; ∞ −∞ |ux (x, t)| dx; ∞ −∞ ” uxx (x, t) dx < ∞. ) ?B ,. ut (x, t) /*(# [0, r] ( ( (t2" /* ) % & g (x) |ut (x, t)| ≤ g (x) # ∞ g (x) dx < ∞. −∞ . E* ut = uxx # x2 K2 / ∞ v (λ, t) = u (x, t) e−iλx dx = F [u] . −∞ F [ut ] = ∞ ∂ ut (x, t) e−iλx dx = ∂t −∞ ∞ u (x, t) e−iλx dx = vt (λ, t) , −∞ %% / ,. / # u (x, t)2 ( F [uxx ] = −λ2 v (λ, t) , ! ∀λ2 t ( & & ⎫ vt (λ, t) = −λ2 v (λ, t) ⎬ ∞ )
$ ! f (x) e−iλx dx ⎭ v (λ, 0) = −∞ t ( & ( +% / −λ2 t ∞ v (λ, t) = e 2 f (x) e−iλx dx = e−λ t F [f ] (λ) . −∞ , + / π − λ2 2 "" & ) % F e−ax = e 4a . a = a % !# $(" x2 e− 4t 2 (λ) = e−λ t . F √ 2 πt / % ( & 2t v (λ, t) = e−λ $(" ∞ 2 f (x) e−iλx dx = e−λ t F [f ] (λ) 2 −∞ x2 x2 e− 4t e− 4t (λ) F [f ] (λ) = F √ ∗ f (x) , F [u] (λ) = v (λ, t) = F √ 2 πt 2 πt 1 4a " , % ) [h ∗ g] (x) = :% ∞ ?D h (u) g (x − u) du. −∞ 1 u (x, t) = √ 2 πt ∞ x2 e− 4t f (x − v) dv. −∞ :% K2 / !##( + @+ $$# @+ (* ( AF )
30 % 0 5 C $!( % ! & 0 = ! ()*+(, r : [a, b] R3 @+ % & /(# < C $# (*) % $(" , . r ) ( t1 = t2 2 r (t1 ) = r (t2 )R ,. r ! R ,+. ṙ = 0 r $ " " ## # $ # ,. r (t) = (x (t) , y (t) , z (t)) = ix (t) + jy (t) + kz (t) ,. / *) % r (t) /(# x (t) , y (t) , z (t) , $$ . @+ % & ,. [a, b] ( ( -.!/ A> A , . C $ 0 % ) P0 (x0 , y0 , z0 )) # ( ) v = (v1 , v2 , v3 ) . 2 , ) .) ## ⎞ ⎛ r (t) = ⎝x0 + tv1 , y0 + tv2 , z0 +
tv3 ⎠ x(t) ,. $ :* y(t) z C # a z(t) I $%2 z a a y x % / * # % " r (t) = (a cos t, a sin t, ct) , % 0 ≤ t, 0 < a, 0 < c $# (* ,+. - ) & ) % R I $ / * # % # $("# P R I $/ * ) ( P0 P xy ! ( & (* Ψ OP /# # z $ (0 ≤ Ψ ≤ π), # ϕ OP0 /# # x $ (0 ≤ ϕ ≤ 2π) . % P =(x0 , y0 , z0 ) ) % y0 = R sin Ψ sin ϕ z0 = R cos Ψ. :( . x0 = R sin Ψ cos ϕ E $/ * A7 ## r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ) . ) #!& ) % ϕ2 $!*) # Ψ2 2 # (0 ≤ Ψ ≤ π).
I ) % Ψ2 $!*) # ϕ2 2 # ,0 ≤ ϕ ≤ 2π . E R I $/ * (ϕ0 , Ψ0 ) #% & P0 % & ) ! * r (Ψ) = (R sin Ψ cos ϕ0 , R sin Ψ sin ϕ0 , R cos Ψ) . P0 2 % & r (ϕ) = (R sin Ψ0 cos ϕ, R sin Ψ0 sin ϕ, R cos Ψ0 ) . 6 % √ √ N / R = 4 I $ P0 = 6, 6, 2 % & %I # # # $$P √ 3 π ( 0 ≤ Ψ ≤ π # 4 cos Ψ = 2, ! Ψ = . :% sin Ψ = , ! 0 0 3 2 √ √ √ √ 3 3 π 4 2 cos ϕ0 = 6 # 4 2 sin ϕ0 = 6. # ϕ0 = 4 , % P0 ϕ0 = π4 # Ψ0 = π3 #% N ) ! 1 0 √ √ 2 2 sin Ψ, 4 sin Ψ, 4 cos Ψ , (0 ≤ Ψ ≤ π) , r (Ψ) = 4 2 2 r (ϕ) = 1 √ 1 3 3 cos ϕ, 4 sin ϕ, 4 , (0 ≤ ϕ ≤ 2π) 4 2 2 2 0 √ # # $ ( # C $!() r (0) = r (2π) , % # ( [0, 2π] (
#( # ( ) C $# ! ()*+(, # $/*" % % (*) % # $) % C $!( > ! ()*+(, @+ % & $# (*) % # (# ,0 $ ( . " A5 & -.!/ $* @+ % & $#W ) # ) @+ % & $ ) # B ) + ) C # ) $ $) % C $!( / ) +I+ 2 # @+ % & + *3" ! ()*+(, γ C $!( ## r (t) ; a ≤ t ≤ b. a = t0 < t1 < < tn−1 < tn = b, A = r (t0 ) , B = r (tn ) # Pi = r (ti ) , % 0 < i < n. $ / P1 A P2 γ P3 P4 Pn−1 B r(t) t0 = a a1 a2 a3 an−1 tn = b
L , / " . $ %# AP1 P2 . Pn−1 B % 9 L (# ) γ $ 7 0 # !(% L. A? = 1-1 . 9 γ C $!( ,/ *) % / @+2 % &.) γ = % &) # !(% b |ṙ (t)| dt L= a * γ ### % -.!/ 9 r (t) = (cos t, sin t, t) !( % / ## . 0 ≤ t ≤ 2π, ṙ (t) = (− sin t, cos t, 1) , |ṙ (t)| = :% L = 2π 0 |ṙ (t)| dt = 2π √ 0 / sin2 t + cos2 t + 1 ≡ √ 2. √ 2dt = 2 2π. 2- . 3 r (t) , a ≤ t ≤ b " !( s (t) = t ## |ṙ (τ )| dτ ) s (t) γ C $2 t " / a !( % ( s (t) I ) ! # ( /*(#) # I # / @+ % &
t (s) s (t) ( /*(# % L $ γ %) t : [0, L] [a, b] , r : [a, b] γ, (r ◦ t) : [0, L] γ :% 2 r (s) = (r◦t) (s) = r (t (s)) γ $ ## 2 r (s) ## γ $ 0 ( %!( -.!/ !(% r (t) = (cos t, sin t, t) , 0 ≤ t ≤ 2π + ( ( ###P . t s (t) = 0 |ṙ (τ )| dτ = t √ 2dτ = 0 (/*(# t (s) = √12 s. E s s √s √ √ # 2 r (s) = cos 2 , sin 2 , 2 , √ 2t. :% s (t) = √ 2t /*(# + ( ( # #2 √ 0 ≤ s ≤ 2 2π $C ) % AA 2 r (s)| ≡ 1. r (s) #" ( &) |2 $# % 1-1 . γ C $!() # 2 r (s) , (0 ≤ s ≤ L) r (s)| ≡ 1. ## |2 γ !(% I " 2 r (s) = ṙ (t (s)) dr (t
(s)) ṙ (t (s)) = ṙ (t (s)) t (s) = = ds s (t (s)) |ṙ (t (s))| #( & # # / % ) % s (t) = t |ṙ (τ ) dτ | , a ! s (t) = |ṙ (t)| . E !(% ## ##) % #" ,#. ( % > > 1-1 . 2 r (s) , 0 ≤ s ≤ L γ $ # # 2 r (s) * 0 ≤ s ≤ L2 r (s) ⊥ 2 #2 I " :) % |2 r (s)| ≡ 1, % 2 r (s) ≡ 1 ( ( r (s) 2 r (s) + 2 r (s) 2 r (s) ≡ 0, ! 22 r (s) 2 r (s) ≡ 0. # 2 r (s) 2 E 2 r (s) ∀s2 r (s) ⊥ 2 E #; γ C $ !(% ## 2 r (s) , 0 ≤ s ≤ L. :* / ) % 2 r (s0 ) % (I P0 2 r (s0 ) = 0. ) % P1, P2 $# P0 2% $ ) P1 P0 P2 * KP1 P0 P2 $ % & KP1 P0 P2 γ P1 P0 P2 P1 , P2 P0 KP1 P0 P2 $$
# % % ) 1 ) P0 2 #" " |r̃ (s I $ $ γ $ 0 )| P0 2 (* &$ $# % ( r̃ (s0 ) + |r̃r̃ (s(s0)|)2 . 0 AH γ r(s0 ) + r (s0 ) |r (s0 )| r(s0 ) r (s0 ) |r (s0 )| P0 r (s0 ) ! ()*+(, &$ + γ $ P0 = r̃ (s0 )2 ( % %!( O κ (s0 ) , % κ (s0 ) = |r̃ (s0 )| . % ) % r̃ (s) ⊥ r̃ (s)2) ) % r̃ (s) #(2 ) # r̃ (s) I #() n (s) = |r̃r̃ (s) (s)| ! (* #" I #( t (s) = r̃ (s)) ! ( 2 b (s) = t (s) × n (s) , % t (s) , n (s) , b (s) #(& & I P = r̃ (s) %!( #" (t (s)) # /"
#( (n (s)) ! * (u − r (s0 )) b (s0 ) = 0. /" (n (s)) # (b (s)) ! 0 (u − r (s0 )) t (s0 ) = 0. #" # ! 7 ) (u − r (s0 )) · n (s0 ) = 0. A 6 3 3 0 ! I $ = & ! %2 ) ! # #"! ( P0 = r̃ (s0 ) $* $ ( 2 &! % 2 = &! ! ) ! 2 ! 8 ! % / ! P0 ( ) % & ,=!+& . # $* ) AB b n t γ ! ( $! n b b t n t b 1-1 . $* #"( !(% $# 2 γ $ # ## r̃ (s) , a ≤ s ≤ b. I
" Δα s0 # s0 + s ( t (s) # t (s0 + s) #" $ t(s0 ) t(s0 + δs) δα γ #% & t(s0 + δs) r(s0 + δs) r(s0 ) s0 s0 + δs #"( !(% $# r̃ (s0 )2 =!+& 2 lim α = lim sinα , ( !( ) % % ) % sin α = s s s0 s0 AD |r̃ (s0 ) × r̃ (s0 + s)| = |r̃ (s0 ) × [r̃ (s0 + s) − r̃ (s0 )]| , ) % α sin α = lim = s0 s s0 s r̃ (s + s) − r̃ (s ) 0 0 = lim r̃ (s0 ) × = lim r̃ (s0 ) × r̃ (s0 ) = s0 s0 s = r̃ (s0 ) = χ (s0 ) . ,7 > lim C $!( * $$" % 2 I 2 &! % &! #( # ( #( $) 2 # $ !$#" ###
* ! ()*+(, r̃ (s) (% % & r̃ (s) = ##C γ $ s #2 τ (s) := − ḃ (s) , n (s) E* #) % #( ( ) ḃ (s) , % /" #( (n (s) 2 ) . 3 b (s) b (s) ≡ 1, ( 2ḃ (s) b (s) ≡ 0, ḃ (s) ⊥ b (s) . # b (s) t (s) ≡ 0, ( ḃ (s) t (s) + b (s) ṫ (s) = 0, χ(s)n(s) 0 ! ḃ (s) ⊥ t (s) . ḃ (s) ⊥ b (s) # ḃ (s) ⊥ t (s) " #" $() % ḃ (s) n (s) . :% ḃ (s) = −τ (s) n (s)) ) % ḃ (s) = |τ (s)| . 1-1 . |τ (s)| . τ (s) =!+& " $(2 #( !(% $# I " HF Δβ s0 és s0 + Δs #% & b (s0 ) # b (s0 + Δs)
#( $ 2 #( !(% $# s0 #% & r̃ (s0 ) " Δβ sin Δβ |b (s0 ) × b (s0 + Δs)| = lim = lim = Δs0 Δs Δs0 Δs Δs0 Δs |b (s0 + Δs) − b (s0 )| = lim b (s0 ) × = Δs0 Δs · · = b (s0 ) × b (s0 ) = b (s0 ) = |τ (s0 )| . ,7 lim <?"#; K2/ t = n = b = −κt κn +τ b −τ n ⎤ ⎡ ⎡ ⎤⎡ ⎤ t 0 κ 0 t τ ⎦⎣ n ⎦ K2/ ⎣ n ⎦ = ⎣ −κ 0 b 0 −τ 0 b I " " # % / I / n (s) = b (s) × t (s) # ( ( n (s) = b (s) × t (s) + b (s) × t (s) = −τ (s) n (s) × t (s) + b (s) × −b(s) × (κ (s) n (s)) = τ (s) b (s) − κ (s) t (s) , b (s) × n (s) = −t (s) . $(" # $* # & # 2 1-1 . 4A 6 0
5 $# # C 2 % !(% ## κ (s) # τ (s) /*(# :() % κ (s) > 0 # τ (s) / /*(# [a, b] ( & R2 # ) # $) !(% #2 # # $* κ (s) , & τ (s) 1" 62 " 0 # % 1-1 . r (t) a ≤ t ≤ b γ $ " # $* t #% & r (t) κ (t) = |ṙ (t) × r̈ (t)| |ṙ (t)|3 #2 + H> I " :) % !(% t " s (t) = t |ṙ (τ )| dτ. a r (t) = r̃ (s (t)) , ! 2 ṙ (t) = r̃ (s (t)) s (t) # r̈ (t) = r̃ (s (t)) s (t) + r̃ (s (t)) s (t) . |ṙ (t) × r̈ (t)| = |ṙ (t)|3 |r̃ (s (t)) × r̃ (s (t))| |s (t)|3 = |r̃ (s (t))| = κ (r̃ (s (t))) = κ (r (t)) . = 3 |r̃ (s (t))| |s (t)|3 1 ,7 7. 1-1 .
r (t) , a ≤ t ≤ b $ r (t) & τ (t) = γ " ## ṙ (t) r̈ (t) r (t) |ṙ (t) × r̈ (t)|2 45 / * #% (" * 2 # % z = f (x, y) z = x2 − y 2 F (x, y, z) = 0 0# x2 + y 2 + z 2 = 1 x2 a2 + x2 a2 + x2 a2 − y2 =1 ) − z2 c2 =1 $ C % ) − z2 =1 #$ C % + y2 b2 b2 $) c2 b2 y2 z2 c2 H + r (u, v) = (x (u, v) , y (u, v) z (u, v)) , 0# r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ) R I $) r (u, v) = ((a + b cos v) cos u, (a + b cos v) sin u, b sin v) * & /2 "?2 > , . xz ! z = f (x) /*(#) % / 2 x # xy ! && z $* ) / ( ( ) ( x2+ y 2
( E z C / // * z = f x2 + y 2 , % f " /*(# (f : R R) . z = x2 + y 2 ( z = sin (x2 + y 2 ) . ,. xz ! $ (x (u) , z (u)) , a ≤ u ≤ b + H7 z b y a x $ z $* / ( /2 a≤u≤b . / * r (u, v) = (x (u) cos v, x (u) sin v, z (u)) , 0 ≤ v ≤ 2π , . K % & K $ z $* z (a + b cos u, b sin u) b u a x #* & / ) K $ u (a + b cos u, b sin u) , 0 ≤ u ≤ 2π. N & / * r (u, v) = ((a + b cos u) cos v, (a + b cos u) sin v, b sin u) , 0 ≤ v ≤ 2π. ,. K % & /# $ z $* H5 z R cos ψ R P ψ R x R sin ψ R /# $ Ψ (R sin Ψ, R cos Ψ) , 0 ≤ Ψ ≤ π $ # r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ) , 0 ≤ ϕ ≤ 2π, $/ !
## E " :* / ) % / z = f (x, y) ) z = x + y . ∀c ∈ R2 # * c2% & % z = x2 + y 2 $("# {(x, y) |f (x, y) = c} . # c < 0 % & % 23 # "J c = 02% & % = {(0, 0)} . c > 02% & % " 00" : √c ; : % ) $#) ! #% " %!( " 2 2 y 0.251 √ 2 x :* / ) % z = f (x, y) / ( r(t) = (x (t) , y (t)) , a ≤ t ≤ b $ f (r (t)) ≡ c, ( c ∈ R # a ≤ t ≤ b2 f (r (t)) ≡ c # t2 ( ( ) + H? % f (r (t)) ṙ (t) ≡ 0. E ∀t ∈ [a, b]2 grad|r(t) ⊥ ṙ (t) :% ) % ( # #"# ( % ## ) % z = f (x, y) / *
%# / ) # ( $#) #(" % / $ 1-1 . z = f (x, y) / * # / * $(# #) xy ! " (* r (t) $ r (t)2 ( & f 2 / # # "## r (t) 2 % + ) % r (t) # % & ( z z = f (x, y) $!" y r(t) x $ -.!/ z = f (x, y) ( # r (t) $# 2 / # # "## A # B % 9 / * r (u, v) = (x (u, v) , y (u, v) , z (u, v)) , (u, v) ∈ D ⊂ R2 , ) % / * #! "" " < # !##( U + / * HA !! y 10 20 30 40 50 B A r(t) x # % /# # & / *
9 ( ( ) / * r (u, v) = (x (u, v) , y (u, v) , z (u, v)) 9 z = f (x, y) ( ) # ) I) % u = x, v = y # 2 !% r (u, v) r (u, v) = (u, v, f (u, v)) . :% & ) u, v r (u, v) I / * # ( & z = f (x, y) 2 / * (%" % /# & I + 2 2 2 2 2 2 x2 +y 2 +z 2 = 1 ( xa2 + yb2 − zc2 = 1 ( xa2 − yb2 − zc2 = 1 / / * & ! r (u, v) " *) % $/ * $ C %2 y2 x2 z2 a2 + b2 − c2 = 1 r (u, v) = (a cos uchv, b sin uchv, c · shv) , 0 ≤ u ≤ 2π, v ∈ R 2( #%* + / * ) r (u, v) # 2 / !) # / " # !##( 4.
! % ! ()*+(, r : T (⊂ R2) R3 # # F ### # # ?2 %!() % T ⊂ R2 $/*" ,+ . ) r : 2 1 − 1 # C) 2 (# @+ % &) + HH 2 ( / ) 2 ru # rv ( /* # ?2 0 # ! ()*+(, 2 ( " r # % ) $/*") # ( $$) % 2 / * 7 -.!/ r (u, v) = (u2 , v 2 , u3 + v 2 ) # ru = (2u, 0, 3u2 ) ; rv = (0, 2v, 2v) . :% ru (0, 0) = (0, 0, 0) # rv (0, 0) = (0, 0, 0) , ( r (u, v) # % / * T ) & P ") u) v T 2 %%) % r (u, v) / * % P ru E* #) % / # ) % ru # rv / ) × rv = 0 / #
(( $ ! ()*+(, ru (u0, v0) × rv (u0, v0)2 ?2 "# " %!( v z / * (u0 , v0 )2% & ru (u0 , v0 ) × rv (u0 , v0 ) rv (u0 , v0 ) v = v0 ru (u0 , v0 ) u = u0 y u x ( / * ( " u r (u, v0 ) $ r (u0 , v0 ) #"( ) v r (u0 , v) $ rv (u0 , v0 ) #"2 ( ) ! ru × rv / * ( " r (u0 , v0 )2 #"! ) % ( :% % r (u0 , v0 ) = (x0 , y0 , z0 ) # ru (u0 , v0 ) × rv (u0 , v0 ) = n = (n1 , n2 , n3 ) , ?2 r (u, v0 )<6 9 (n1 , n2 , n3 ) ((x, y, z) − (x0 , y0 , z0 )) = 0. HB ! n1 x + n2 y + n3 z − n1 x0 − n2 y0 − n3 z0 = 0. |ru × rv | / * % 2 2 2 2 |ru × rv | = (ru × rv ) (ru × rv ) = (ru ) (rv ) − (ru rv ) E |ru × rv | = F2 G
√ EG − F 2 / E = (ru )2 ; F = ru rv ; G = (rv )2 # G 0# %!( A ;<? < 1 5 , . "$ * # ,T 2 uv !. % ( / " 7 / * /& "" "< / * / !# !% + / ( % / !#) / ! $ / * / !# " !( z v δu · ru (u0 , v0 ) δv · rv (u0 , v0 ) r(u0 , v0 ) δv v0 δu y u0 u x ( /! * $ !2 " ) % Δuru # Δvrv % ΔuΔv |ru × rv | * # , $ !# ) # Δu # Δv. :% F / * / ! $ !" √ lim ΣΔuΔv |ru × rv | =
|ru × rv | dudv = EG − F 2 dudv, Δu,Δv0 T T + % 0 HD Σ2 $ # (* T ) % " =!+& :% ?2 ? √ EG − F 2 dudv. Δu, Δv T = -.!/ 9 R I $ / !#P . r (ϕ, Ψ) = (R sin Ψ cos ϕ, R sin Ψ sin ϕ, R cos Ψ)) T : (0 ≤ ϕ ≤ 2π; 0 ≤ Ψ ≤ π) rϕ = (−R sin Ψ sin ϕ, R sin Ψ cos ϕ, 0) rΨ = (R cos Ψ cos ϕ, R cos Ψ sin ϕ, −R sin Ψ) R2 E = R2 sin2 Ψ sin2 ϕ + R2 sin2 Ψ cos2 ϕ = R2 sin2 Ψ F = −R2 cos Ψ sin Ψ cos ϕ sin ϕ + R2 cos Ψ sin Ψ cos ϕ sin ϕ = 0 G = R2 cos2 Ψ cos2 ϕ+R2 cos2 Ψ sin2 ϕ+R2 sin2 Ψ = R2 cos2 Ψ + sin2 Ψ = 2 E = rϕ F = rϕ rΨ G = (rΨ )2 :%) R2 I $ / ! 2π π √ EG − F 2 dΨdϕ = ϕ=0Ψ=0 2π π / R4 sin2 ΨdΨdϕ = ϕ=0Ψ=0 = 2πR 2 π Ψ=0 sin ΨdΨ = 4R2 π ,7 5. 2 45 0 F / * ## r :T R3 .
( t (u (t) , v (t)) T 2 /& γ $ ## r (u (t) , v (t)) , a ≤ t ≤ b ) F / * /& γ $ ## -.!/ ! / γ $ !(%P BF r v z F γ t r(u(t), v(t)) T γ t (u(t), v(t)) y u x %> -.!/ 9 γ1 , γ2 # / & = / * $) P0 = r (u0 , v0 ) ) % γ1 # γ2 $# $#P . ! Φ (t) = r (u (t) , v (t)) , a ≤ t ≤ b. :) % γ !(% b L = Φ̇ (t) dt. a ( Φ̇ (t) = ru u̇ + rv v̇ ,$ /*(# ( .) # L= b / (ru u̇ + rv v̇) (ru u̇ + rv v̇)dt = a E u̇2 + 2F u̇v̇ + Gv̇ 2 dt. a 7 ! ()*+(, E u̇2 + 2F u̇v̇ + Gv̇ 2 /# 0?"# 0" K P ,9 EF FG X|ru × rv |2 > 0 # ru "3 % 0,
! E > 0. (* " b √ . / ! Q :* / ) % t (u1 (t) , v1 (t)) # t (u2 (t) , v2 (t)) 2 ##C γ1 # γ2 T 2 $# (u0 , v0 ) Φ1 (t) = r (u1 (t) , v1 (t)) # Φ2 (t) = r (u2 (t) , v2 (t)) ##C + B> a ≤ t ≤ b $# γ1 # γ2 r (u0 , v0 ) = P0 γ1 # γ2 $ P0 2 =!+& Φ̇1 # Φ̇2 $ P0 2 9!( % & $ α2 Φ̇1 Φ̇2 (r u̇1 + rv v̇1 ) (ru u̇2 + rv v̇2 ) = cos α = = u |ru u̇1 + rv v̇1 | |ru u̇2 + rv v̇2 | Φ̇1 Φ̇2 E u̇1 u̇2 + F (u̇1 v̇2 + u̇2 v̇1 ) + Gv̇1 v̇2 / =/ 2 . ,7 ? E u̇1 + 2F u̇1 v̇1 + Gv̇1 E u̇22 + 2F u̇2 v̇2 + Gv̇2 z F Φ̇2 P0 γ Φ̇1 γ α y x " &) % 7 1-1 . / * ( ( " F = 0. P0 = r (u0, v0 ) % & #2 (u0,
v0 ) " " ) % I " 0 #( # u̇1u̇2 = v̇1v̇2 = 0. :% / # & F 2 # u̇1 v̇2 # u̇2 v̇1 $* F) # / $ 0) % F = 0. " # (E, F, G) / * ,!(%) $) / !. % ( ) % / *2 % ! 9 / * # ( ) *# # L, M # N. B 9 O $ * / * #( m2 P E ru × rv . m= |ru × rv | 2 2 2 2 ∂ r ∂ r ∂ r ∂ r :( ruu := ∂u 2 ; ruv := ∂u∂v ; rvv := ∂v 2 ; rvu := ∂v∂u . 2 # mv := ∂m . & ) % ruv = rvu , ( mu := ∂m ∂u ∂v = ! ()*+(, L = ruum; M = ruv m; N = rvv m # #" G A ;<? ?# %!(
! ()*+(, (* Lu̇2 +2M u̇v̇ +N v̇ 2 /# #" 0?"#< 1 5 - - 5 = 1-1 . F / * ## r : T R3 . :( γ T 2 $) t (u (t) , v (t)) , a ≤ t ≤ b ## Φ (t) = r (u (t) , v (t)) ##C / * $ γ $* 1 Lu̇2 + 2M u̇v̇ + N v̇ 2 κ= 2 2 , mn E u̇ + 2F u̇ v̇ + G v̇ ) % γ $ /" #( ) n + / * #"! ) ( nm =0, ( / ( # ! !(% ## ## ) # $ /*(# ( & & $(# ) % 1-1 . 9 / * % & # $# &! ,# / * ( #"2 !( .) $* I " / * T γ1 : t (u1 (t) , v1 (t))
γ : t (u (t) , v (t)) $# F 2 2 2 # ) % (u0, v0 ) + B7 $$ &! P0 = r (u0 , v0 ) ! * $$ P0 2 #"!& P γ1 : Φ1 (t) = r (u1 (t) , v1 (t)) , a ≤ t ≤ b $# t1 (u0 , v0 ) = γ2 : Φ1 (t) = r (u2 (t) , v2 (t)) ±t2 (u0 , v0 ) # n1 (u0 , v0 ) = n2 (u0 , v0 ) ( #" # /" #2 ( Φ̇1 (t) = ru u̇1 + rv v̇1 |(u0 ,v0 ) % Φ̇2 (t) = ru u̇2 + rv v̇2 . :% (u0 , v0 )2 {ru , rv } /* # (u̇1 , v̇1 ) = λ (u̇2 , v̇2 ) ( 2 2 2 1 Lu̇ + 2M u̇v̇ + N v̇ λ = 02 E κ = mn 2 2 / P E u̇ + 2F u̇v̇ + Gv̇ n1 = n2 (u0 , v0 )2 :() % u̇1 = λu̇2 # v̇1 = λv̇2 % !# % ) λ2 2 && ("" ( # λ2 2 C!( )
% 1 Lu̇21 + 2M u̇1 v̇12 + N v̇12 1 Lu̇22 + 2M u̇2 v̇2 + N v̇22 κ1 = = = κ2 mn1 E u̇21 + 2F u̇1 v̇1 + Gv̇12 mn2 E u̇22 + 2F u̇2 v̇2 + Gv̇22 (u0 , v0 )2 > 1-1 . / * $ P0 2 $ $ &! / * " !$ $ #( ,K #() % &! #"!& P0 2 * $ . I " "" # " )"## J ?# & ( r : T R3 F / * P0 F ##) # > ! ()*+(, ! ) P0 ponton, # "2 P0 2 #"! ,% P0 " & / # m (. "#" J ! F / * " 2 $# "## (* ! ?# %!( 1-1 . 4@; < 4@ 55 9 $* κn , !& 0 < ϕ <
π2 $ / /# " ! / $* κn . κ= cos ϕ B5 z F m P0 y ! x I " / * $# $ # & 1 Lu̇ + 2M u̇v̇ + N v̇ 1 Q2 κ = mn 2 2 # " κ = mn Q1 , % Q2 , Q1 E u̇ + 2F u̇ v̇ + G v̇ 2 . ( " / # m n, % κn = Q Q1 ( ϕ = (m, n) ( ϕ = π − (m, n) , ! cos ϕ = |mn| . :% κ = 1 Q2 = cos1 ϕ κn G / % ) % κ # κn =!+&& && |mn| Q1 + !( %P :% / $* cos ϕ2( ( & , % ϕ 2 ! $. % % & " # + "## % ( / ) # $(# 2 (## κ 2 ! κn % $* # " ! ) % n = m (
n = −m, κ !( ( !( 9 $* % κ= κ 2 Q2 . Q1 " $* # $ ## % + B? 1# - 5 ,# 1-1 . t P0 " I #( F / * P0 2 #"! t|(u0 ,v0 ) = ξru + ηrv |(u0 ,v0 ) . t2% & $* κ = Lξ 2 + 2M ξη + N η 2 . z F S rv P0 ξ · ru t ru η · rv y x I " / * $ !(% ## ( Q1 ≡ 1 # t̄ = r = u ru + v rv , ! u = ξ, v = η. :% κ = Lξ 2 + 2M ξξ + N ξ 2 . 1-1 . 2# P0 2 S #"! % & (e) # ) + P0 2& /*" κ1 , κ2 ) % e2" ϕ $ / % & $* κ = κ1 cos2 ϕ + κ2 sin2 ϕ. I " : , "" # " .) t = ξru
+ ηrv , κ = Lξ 2 + 2M ξη + N η 2. E* / S 2 " {i, j} 2 ,i⊥j # i = j = 1.) :* / ) % t = xi + yj , t {i, j}2 x # y). 2 Q / !( L M ξ κ = [ξ, η] . M N η BA S j ru P t f rv e i φ 9 #* {ru, rv } " {i, j} ) x ξ =B ( 2 × 22 B Q ! [ξ, η] = [x, y] B T . y η L M T B 2 E A := B M N κ = [x, y] A x y = a11 x2 + 2a12 xy + a22 y 2 M B Q) a11 x2 +2a12 xy+a22 y 2 ( B T N ( /" %% & " "" /+&( ) ( % & {e, f } P E ∃C x h h =C . :% κ = [h, k] D Q) , % D = y k k ξ h κ1 0 T C AC Q) D = . ( = BC , 0 κ η k 2 h #
t ( {e, f } ( t = k 1, # % ϕ (t, e) , h = cos ϕ # k = sin ϕ. N % κ = cos ϕ κ1 0 , ( [cos ϕ, sin ϕ] 0 κ2 sin ϕ L M κ = κ1 cos2 ϕ + κ2 sin2 ϕ. ! : # ( cos2 ϕ + sin2 ϕ = 1, # κ κ1 # κ2 $# # Q # κ1 # κ2 ) % κ = κ1 + (κ2 − κ1 ) sin2 ϕ, # 0 ≤ sin2 ϕ ≤ 1. + BH f t sin φ p cos φ e ! ()*+(, κ1és κ2 $ F / P0 ? < 62 J % & #" ? " J /"2 % & ?# (* ( @ "L 62 : H := κ1 + κ2 . ! ()*+(, $ ( A ; 62 K := κ1 κ2 % 1-1 . 2 LN − M , EG − F 2 / % ) # K= " # H= EN − 2F M + GL . EG − F 2 " #3
" P0 2 #"! (ξ, η) % % &) % ξru + ηrv (ξ, η) ) $ , $(" . 1-1 . # /" & (ξ, η) 2 η −ξη ξ 2 E F G = 0 L M N /I $ 2 0" " " " : 2# $(# r (u, v) = r (u0 , v0 ) + (u − u0 ) ru (u0 , v0 ) + (v − v0 ) rv (u0 , v0 ) + . 1+ (u − u0 )2 ruu (u0 , v0 ) + 2 (u − u0 ) (v − v0 ) ruv (u0 , v0 ) (v − v0 )2 rvv (u0 , v0 ) + 2 + Ψ (u, v) , ,7 A. BB z F S rv P0 ξ · ru ξ · ru + η · rv ru η · rv y x 7 > (ξ, η) % S #"! % (u−u Ψ(u,v) 2 2 0. 0 ) +(v−v0 ) r̂ (u, v) = r (u, v) − Ψ (u, v) . r (u, v) # C r̂ (u, v) / * P0 2 $$ #2
"! 9 ( {ru , rv , m} , r̂ (u, v) I Z = 12 [Lx2 + 2M xy + ny 2 ] & , r̂ =!+& m2 ) # #(*) % {ru , rv , m}2 r (u0 , v0 ) = 0, # ru , rv ⊥m. " $() % r̂ (u, v) / * 2 )( 2 % 2 )( %) & /*") % Lx2 + 2M xy + N y 2 ( / K J K J ( # K ,6 Q (x, y) ( / 0" K ) % Q (x, y) > 0 ∀ (x, y) = 0, 0" # K J % Q (x, y) ≥ 0 ∀ (x, y)2 ) #2 (x, y) = 0, % Q (x, y) = 0. ( K J % ∃ (x1 , y1 ) # (x2 , y2 ) , % Q (x1 , y1 ) > 0; Q (x2 , y2 ) < 0. + BD #; 0 6"" r̂ (u, v) / * P0 %!( {ru , rv , m} * & z = 12 # z = − 12 !
, #"! P0 2 % ! . $# (!* " ,{ru , rv , m}2 2 . ru , rv ! , P0 2 #"! (* $# {ru , rv , m} Lx2 + 2M xy + N y 2 = 1 % 2 % & 2 ) 2 Lx2 + 2M xy + N y 2 = −1 # P0 0 ; ) % ) P0 3 06" ; 0 6" ; 9 P0 κ1 = κ2 = 0, $ 1-1 . P0 0 0" P0 ) % ) ( ,. κ , * $$" &) F2 / ( ) P0 . &) ( &) ( 2 ,. Q2 =) =) ( = ,. K > 0, K < 0, K = 0, κ12 + κ22 = 0 ,(. LN − M 2 > 0, LN − M 2 < 0, LN − M 2 = 0 DF & t O1 O2 P0 "?2
0" 9 " " K g $# t $* ! " F // g $# # 6 (* /" # $ %% & ! &$ $# (* O1 T 2 /" ! &$# $# O2 / * " # P0 0 ;J % O1 # O2 #"! " " " ( ) / " #( $ " P0 3 06" ;J % O1 # O2 #"! " ( ) ( $ / $ #( (Q P0 0 ) % $ $* # , YQ& #" " . % -.!/ % -.!/ H P E PL % r (u, v) = (u cos v, u sin v, v) , 0 ≤ u ≤ 1, 0 ≤ v ≤ 2π. / * I &) % + ( ( , x !
% . $$* z 2 . + D> H P E PL z 2π 1 y 1 x 4 ! /"$* P / * /" # . > # # / 0# #3 " ru (u, v) = (cos v, sin v, 0) ; rv (u, v) = (−u sin v, u cos v, 1) ruu (u, v) = (0, 0, 0) ; ruv (u, v) = (− sin v, cos v, 0) ; rvv (u, v) = (−u cos v, −u sin v, 0) 2 2 E = r2 u ≡ 1; F = ru rv ≡ 0; G = rv = u + 1 v m = |rru ×r = √u12 +1 (sin v, − cos v, u) u ×rv | i j k sin v 0 = (sin v, − cos v, u) ru × rv = cos v −u sin v u cos v 1 D |ru × rv | = √ 2 √ sin v + cos2 v + u2 = u2 + 1 L = ruu m = 0m ≡ 0; M = ruv m = = (− sin v, cos v, 0) (sin v, − cos v, u) √ 1 u2 +1 = −√ 1 u2 +1 ; ,7 H. N = rvv m = (−u cos v, −u sin v, 0) (sin v, − cos v, u) √u12 +1 ≡ 0 # # κ1 , κ2 #3 "
:) % − u21+1 1 LN − M 2 =− K = κ1 κ2 = = 2 . 2 2 EG − F u +1 (u + 1)2 # ) % 2 1 · 0 − 2 · 0 · √u−1 EN − 2F M + GL 2 +1 + (u + 1) · 0 ≡ 0. H = κ1 +κ2 = = EG − F 2 1 (u2 + 1) − 0 /"$* ) κ1 , κ2 % % /I # $ $/*#" ) κ1 , κ2 #! κ 2 − Hκ + K = 0 ) ( H = κ1 + κ2 # K = κ1 κ2 . :% κ1 , κ2 ### ) % κ 2 − Hκ + K = 0 1 1 1 H = 0 # K = − (u2 +1) # κ 2 − (u2 +1) 2 2 2 = 0 ⇒ κ1 = u2 +1 ; κ2 = − u21+1 . 7 # # η 2 " #3 " :) % /" t = ξru + ηrv () # 2 η E L −ξη F M ξ2 G N ξ, = 0. ξ ( % t #( ) η ! % ) ( " +
η 2 2 ( D7 ) % 1 E L % !( 1 1 0 E ξ1 = − ηξ 0 − √u12 +1 − ηξ F M G = 0. N ξ2 η2 u +1 =0⇒ 0 ξ2 η2 2 ξ η ξ η √ u2 + 1 ( √ . 2 =− u +1 = √ √ u2 + 1, η1 = 1 # ξ2 = − u2 + 1, η2 = 1. / ? √ u2 + 1 (cos v, sin v, 0) + (−u sin v, u cos v, 1) = √ √ u2 + 1 cos v − u sin v, u2 + 1 sin v + u cos v, 1 . ,7 B = ξ1 ru + η1 rv = / #" ? √ ξ2 ru + η2 rv = − u2 + 1 (cos v, sin v, 0) + (−u sin v, u cos v, 1) = √ √ = − u2 + 1 cos v − u sin v, − u2 + 1 sin v + u cos v, 1 . ,7 D D5 ! 6" % -.!/ K z $* ω $# P z F (P ) P y x 5 > z $* ω $#C / ω , 2π $* / +# . # P xy 2! % $ 9
F (P ) P #( ) = F (x, y, z) = (−ωy, ωx, 0)) % P (x, y, z) I( = E # (x, y, z) % % * ( (x, y, z) (−ωx, ωy, 0) D? ! DA z y x 5 E# # % ( * %% -.!/ f (x, y, z) = xyz ,( f R3 R. < (2 #) % f # ) (xyz)2 / /*(# %!( grad(f ) R3 R3 <( grad(f ) (x, y, z) = (#) % grad(f ) I ( (yz, xz, xy) . F = grad(f ) , % f : R3 R %!( ( % r = (x, y, z) ,5 >. $ # E r (x, y, z) % ( # (* ) % ( / ! C r % r2 $ * :% / r = |r| = x2 + y 2 + z 2 P
# $C ) (x, y, z) # & K$ $# ) ( 9 F (P ) ") M ( P 2 −− <L $(# F (P ) P M # F (P ) % /!( r2 2 " F (r) = −k rr3 , % k > 0 E 1 0 −ky −kz −kx . F (x, y, z) = 3 , 3 , 3 (x2 + y 2 + z 2 ) 2 (x2 + y 2 + z 2 ) 2 (x2 + y 2 + z 2 ) 2 ! DH z r (x, y, z) y x 5 7 r =!+& z y x 5 5 F (# ( " %!(% : R3 R3 (#) / /% ) 4 9 F # & / # ##) ( ( /# "2 ! # ) ( ! / # = ! I # ## / ( # # "#) F 9 % γ 2 /& "# # (# W ,E "# (#
W. ;*" γ ! $ ! DB F z F F F γ F B A y x 5 ? γ ! $ # # F (# 2## & / # ," <*" F ## F / * "# W &. & / # #/2 CC F x 5 A #/ K Q / * "# z FF F y % & / # " #/ =!+& F 2 F 2 ( & => %!( / ;* ( ) ! < / !##( ( 6" %# γ ! $) ## r (t) = (x (t) , y (t) , z (t)) , a ≤ t ≤ b # F : R3 R3 . , /*(# + ! DD ( #) # # / ( % & γ $# . / # ME M
/ % ! ()*+(, F "# γ 2 A2& B 2 & (# ( (* , F ( 2 F (r) dr γ 2 O γ F F z A F F B γ y x 5 H E :% = / / #) M" #"9; #; N X , " ".·, 6 :* / "$) % γ $ AB ) F "# F0 " − − AB F0 2 AB " " (* ( F0 − ) AB | | − " ") ( AB ) ) − AB − − W = F0 − AB = F0 AB AB . (# γ $) # F # # " # # ( ((
γ $ [a, b] # ( n #) a = t0 < t1 < · · · < tn−1 < tn = b I) % max {ti − ti−1 } − 0. 1≤i≤n n∞ ! >FF F0 F0 F0 B A 5 B (# ( 1 ≤ i ≤ n 2 Pi ) % ( r (ti ) . −−−− γ $# (Pi−1 , Pi ) !(# ri = Pi−1 Pi ( % !* ri ( " ## ri · F (r (ti−1 )) (# $ !" ) % F 2 * ri 2 :% n2 ti−1, ti #% & r (ti−1 ) , r (ti ) !( (# $2 !" F (r (ti−1 )) [r (ti ) − r (ti−1 )] ≈ F (r (ti−1 )) ṙ (ti−1 ) (ti − ti−1 ) . $( b n 3 W = lim F (r (ti−1 )) ṙ (ti−1 ) (ti − ti−1 ) = F (r (t)) ṙ (t) dt. n∞ i=1 a F (r) dr ( :% ) % # !%
$("2 γ 7 1-1 . b F (r) dr = γ % -.!/ F (r (t)) ṙ (t) dt a F (x, y, z) = (x, yz, y) # γ ## ! >F> # z r(t2 ) Δri r(ti−1 ) r(ti ) r(t1 ) A B y x t0 = a t1 t2 ti−1 ti 5 D tn = b $# r (t) = (t, t2 , t3 ) 0 ≤ t ≤ 1 , 5 >F. 1 F (r) dr = 0 γ = 5 2 t, t , t F (r(t)) 1, 2t, 3t 2 ṙ(t) t2 2 7 3 5 + t + t 2 7 5 1 = 0 1 dt = t + 2t6 + 3t4 dt = 0 ,5 . 97 1 2 3 + + = . 2 7 5 70 / "$ F 2 γ 2 ) F (r (t))2 2 I $#) % F (x, y, z) = (x, yz, y) # # ∀x % # r (t) " /*(##) t2 !) ∀y % # r (t) /*(##) t2 2 !) ∀z % # r (t) % /*(##) t3 2 ! F (r (t)) = (t, t2 · t3 , t2 )) ṙ (t)2 ! ṙ (t) = (1, 2t, 3t2 )
% F (t) · ṙ (t) ) t2" /*" ( & ##C /*(#) # & [a, b]2 / " ,5 0! ## ) % / # # ) % $! A # B # A # B 2 $$" # $# F (# ! >F z B = (1, 1, 1) γ A y x 5 >F γ $ ( ) ( /* I& ) + "#" # A # B (# & /*W γ1 B A γ2 γ3 5 >> E I& ( & /* # % ! ()*+(, ) % F (# ) % F (# E F = grad f ( f : H ⊂ R3 R /*(#) f /(# F (# ! ! ()*+(, ) % F ! ) ) % A # B #
$$" # γ1, γ2 ! C $!() " A, # (# B ,5 7. F (r) dr = F (r) dr γ1 γ2 ! >F7 * ) / #() % γ1 # γ2 (# F # # % @ AB ACD-E E* #) % F ( /2 I& ) ∀ γ $# F (r) dr = 0, % F (r) dr = γ F (r) dr − γ1 γ F (r) dr = 0 γ2 γ2 A A γ1 B γ γ γ γ B 5 > E I& ( & /* # # $# ( " ( / +# ) % / # ) ! $##) % (# ( /* 2 I& %% # * $% $(" / (# # $ ! ()*+(, curl F ( rot F =!+& 4 5 2 ! (#) % F = (F1 , F2 )) F 9 F : R2 R 2 +& rot F = ∂F − ∂F 1 <#%
!) % F ∂x ∂y rot F % curl F ( rot F , . ## % ! (# # ( & 465 3 3 9 F : R # (#) % F = (F1, F2, F3 ) R curl F = rot F ( j k i ∂ ∂ ∂ curl F = rot F = ∂x ∂y ∂z F1 F2 F3 = ∂F3 − ∂y ∂F2 ∂F1 ; ∂z ∂z − ∂F3 ∂F2 ; ∂x ∂x − ∂F1 ∂y ! >F5 %$ -.!/ z $* ω $# ( & / # +& F (x, y, z) = (−ωy, ωx, 0) . j k i ∂ ∂ ∂ rot F = ∂x ∂y ∂z = (0, 0, 2ω) −ωy ωx 0 :% z % # / / # & ω %I ( z ω 2ω rot(F) y x 5 >7 z $* ω $#C / +& %7 -.!/F (r) ## curl F = −kr r3 (+& (#) % k > 0 ( & ##
−ky −kx −kz F (x, y, z) = 3 , 3 , 3 2 2 2 2 2 2 2 2 2 (x +y +z ) 2 (x +y +z ) 2 (x +y +z ) 2 i j k ∂ ∂ ∂ rot F = ∂x ∂y ∂z −ky −kz −kx (x2 +y2 +z2 ) 32 (x2 +y2 +z2 ) 32 (x2 +y2 +z2 ) 32 5 2 2 2 −2 3 (x + y + z ) k (yz − yz, − xz + zx, xy − xy) ≡ (0, 0, 0) = 0 ( / # # % &) % +& $(# # + ) % (+& (## # ( & / # G + $(# #) # rot F = 0. ! /& !& " # (# +& C % ! >F? %= -.!/ F (x, y) = (−ωy, ωx)) ω $#C & $* /2 # xy ! curl F = ω + ω = 2ω. % -.!/ f : R3 R " # / @2 + % & /*(# # F = grad (f ) j k i ∂ ∂ ∂ rot F = ∂x ∂y ∂z f f
f x y z ⎛ ⎞ ⎠ = ⎝fzy = 0, − fyz , fxz − fzx , fyx − fxy 0 0 0 ( ( + ( ( / +# %" :2 % / ! *) % ( !( , . (# +& G< G; Z 13 E :T1P = 1-1 . F (# ( /* I& ) % F ( !( # + :% /" # ) % () ( /* I& &) % ( !( ( % $(" ## /" F " M" 92 J 3" " <N ( + 2 ( M # + M ! I) % ##) *# / # + 2# #2 " !) ( / ) ( C $/*" % / / 6
+ . F : R2 R2 ! (#) F = (F1 , F2 ) . ( (( curl F + #& * $$" ## ( / ) F < ( !( :* / ) % curl F ≡ 0 9 ! >FA / 1 F /*(## + ( ∂F , ∂x ∂F2 ∂F1 ∂F2 , ∂y , ∂y ! G< < = ( ( F ∂x * I E ) + ( = ) % + 2 ++ ( ,% ) . %) % F 2 ( !() %) % / 6 + F : R3 R3 (# ( (( # F = (F1, F2 , F3 ) curl F + #& * $$" ## ( / ) F < ( !( :* / ) % curl F ≡ 0 9 / 1 1 1 2 3 9 + + ( ∂F , ∂F , ∂F , ∂F , .
, ∂F ∂x ∂y ∂z ∂x ∂z # ( ( ) 0 F * I 9 / + + ( $* + # 2 ( (# ) + 2 ++ () % %) % F ( !() # ) % + 2 $ y x > -.!/ F (x, y) = x2+y2 , x2+y2 F ! (2 # curl F = 0, F + # ( & ) ! GG[ /) ( + 2 ++ ( -.!/ F : R3 R3 , F (r) = −k rr3 . ) % curl F ≡ 0 ) % F + # ( & $ $ F ) + + ( # ( ( ) ! F ( !( ! // ,5 >FH 9 F ( !( (#) ( + /*(#) ( # f : R3 R) F = grad (f ) + /*(# % # #
-.!/ F (x, y, z) = (1 + 4y + 5z, 2 + 4x, 3 + 5x) i j k ∂ ∂ ∂ ≡ (0, 0, 0) ≡ 0. curl F = ∂x ∂y ∂z 1 + 4y + 5z 2 + 4x 3 + 5z # $ + ( * # ) ! F 2 ( !( + 2 :) % fx = 1 + 4y + 5z ⇒ f (x, y, z) = (1 + 4y + 5z) dx + g (y, z) = x + 4xy + 5xz + g (y, z) # fy = 2 + 4x ⇒ ∂ (x + 4xy + 5xz + g (y, z)) = 4x + gy (x, y). ∂y E 4x + gy (y, z) = 2 + 4x ⇒ gy (y, z) = 2 ⇒ g (y, z) = 2y + h (z) . :% f (x, y, z) = x + 4xy + 5xz + 2y + h (z) . N) ( fz = 3 + 5x ⇒ 5x + h (z) = 3 + 5x ⇒ h (z) = 3 ⇒ h (z) = 3z + c. # f (x, y, z) = x + 4xy + 5xz + 2y + 3z + c. , "" / % ( . x -.!/ F (x, y, z) = x2−y+y2 ; x2+y ; −4 ## ) % ( !(2 2 F W ≡ 0, ( P F 2 . curl F = z
) % (# ) ! GG[ /) + 2 % !P γ x2 + y 2 = 1 # z = 12 ! >FB $) F (r) dr ###P 9 !( ! ! γ F (r) dr =0, " * ) % &) % F ( !( γ : , 7 #.) % ( !( ( $# ( ( # γ ## x (t) = cos t, y (t) = sin t, z (t) ≡ 12 # 0 ≤ t ≤ 2π; ( r (t) = (cos t, sin t, 12). :% t cos t , 1 , −4 , ṙ (t) = (− sin t, cos t, 0) . F (r (t)) = − sin 1 F (r(t))· ṙ (t) = sin2 t+cos2 t ≡ 1. F (r) dr = γ 2π 0 F (r (t))· ṙ (t) dt = 2π 0 1dt = F (r) dr = 0, ! F ( !( 2π = 0. :% γ $(" # ) % ( !( (# # # $$ (# + /*(# ( ## * $# 1-1 . F : R3 R3
# 2 ( # <(* F + /(## f 2 :% F = grad(f ) γ $) " A # (# B 2 F (r)dr = f (B) − f (A). γ I " γ ## r(t)) a ≤ t ≤ b :* g(t) = f (r(t)) /*(# ( R R / @+ % & /(#) % <L2 # 9 ( ) % g (t) = grad(f )(r(t)) · ṙ(t) = F (r(t)) · ṙ(t). :% b f (B) − f (A) = g(b) − g(a) = F (r(t)) · r(t) = g (t)dt = a b a F (r)dr. γ 2 ! + >FD % -.!/ γ $ ( r (t) = (t, 2t3 ) 3t2 ) , % 0 ≤ t ≤ 1, ## ! F (r) dr ###) % γ F (x, y, z) = grad(ex+y+z+1 )! & $ $ A (1, 2, 3) ) # F + /*(# f (x, y, z) = ex+y+z+1 . :% F (r) dr =f (1, 2, 3) − f (0, 0, 0) = e7 − e. . γ $ γ 45#
F / * ) # r :T R3 ## F 2 ( F : R3 R3 (#) # / @+ % & # * ) % / # I ) % # P F (P ) / # # " F / * !% ( ru × rv , ( −ru × rv ( / * ! / # ( CC z m x 5 >5 F y ( m = ru × rv ( m = −ru × rv / * ! I ) % ) % / * ! M/ / # & M , # . % ! F / * ) # % F 2 , ! . "# / & / # " #/ ! >>F 7 ! ()*+(, F 2 "# / &) F ## 2 & / # #/ F (# F 2 ( ! 2( $
* ) ( => %!() # F dA F :% 2% YQ !() / * ! $ / # / R 2% YQ !() ! # / $ / #R 2% YQ #) / # / ! " # # "# " 9 % YQW 0 :* / ) % F (#) F ≡ F0 # F ) a # b ( /! ) # a × b2 ( ! a × b z F0 · a×b |a×b| r0 F0 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx b xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx F xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx a F0 y x 5 >? K Q ! F / * a × b "# / & / # % & $ :% F 2 F 2 ! + >>> ( & YQ #/ ) alapterület × magasság. * F = a × b ) F0 2 b a × b2 ( " (* ) F0 · a× " a×b| | b #/ a × b · F0 · a× = F0 · (a × b) |a×b| 2 E ) % F = F0 ) # F a, b /! a × b2( = F0 · (a × b). ! ) F dA F 0 F 4> F v Δu z F xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx v0 + Δv v0 Δv F r(u0 , v0 ) u0 u0 + Δu 5 >A u x y K Q F 2 F / * ## r :T F
2( !2 ##( + ) # + / * # $ !# ) % C) % / * / % I =! ( $ !# ) % /#( ! / * % #" ! / " * ( $ !# ) % F 2 ) F ≡ F (r (u0 , v0 ))2 (*) ! % ; ## 9 " / * !) / ;* ## ( ) % $ !" F (r (u0 , v0 )) · (ru × rv ) uv ,5 5. m̄(P ) >> ! r̄u Δu F̄ (P ) r̄v Δv v0 + Δv u0 0 u0 + Δu 5 >H v0 K Q YQ # / * I ) % / ( YQ $*) # u, v 0 E = lim ΣF (r (u0 , v0 )) · (r × r ) uv F dA u v u, v0 F =
F (r (u, v)) · (ru × rv ) dudv, ,5 ?. T / #() % F ru × rv 2( ( !( 9 F 2 − ru × rv 2( !) F dA = − F (r (u, v)) · (ru × rv ) dudv. F T F (# F / * / * " " ( ) 4 9 F (p) · m (p) ≡ q, ! + >>7 = q · f elszı́n(F). F dA ,5 A. F % & # ) % F !) # F (# 2 -.!/ F / * ) T : 0 ≤ u ≤ 3, 0 ≤ v ≤ 1 ### r (u, v) = (u + 2v, −v, u2 + 3v) # F 2 / / # & ! F (x, y, z) = (xy, 2x + y, z) . =? F dA " F 0 F 2 F / * F (x, y, z) = (xy, 2x + y, z) # % & x % # u + 2v 2) y % # −v 2 # z % # u2 + 3v 2 ! N F (r (u,
v)) = ((u + 2v) (−v) , 2(u + 2v) − v, u2 + 3v) . G F (r (u, v)) = (−uv − 2v 2 , 2u + 3v, u2 + 3v) . 0 % ( %% ! ru ×rv = i j k 1 0 2u = 2ui + (4u − 3) j − k. ( k *% & !( 2 −1 3 # ru × rv / # ) % n = −ru × rv = (−2u, 3 − 4u, 1) . 0 ! F (r (u, v))·n F (r (u, v))·n =2u2 v+ 4uv + 6u + 3v − 8u2 − 4uv + u2 + 3v 2 %0 −21 F = F dA 3 1 4uv 2 + 2u2 v − 7u2 + 6v + 6u − 4uvdvdu = · · · = u=0 v=0 ) % "## 21 # $ / # F 2 * M / #M) M/ / #M $ -.!/ A (1, 0, 1) ; B (1, 1, 1) ; C (2, 0, 3) F ABC %$) # ! F 2 M / #M & 2 ( F ≡ (5, 4, 3) . ! >>5 " ( " F dA? F )
# F !) ! . ( F YQ F 2 = (0, 1, 0) ; b = AB " (* ( * #( c = AC = (1, 0, 2) i j k b × c = 0 1 0 = (2, 0, −1) , ( & 1 0 2 b × c / " ( c F (* b×c2 F · b× . %$ * |b×c| 5 4 3 b× c | | F ·( b× c) c 1 F bc = 1 5 0 1 0 = = = F · b× · 2 2 2 2 2 1 |b×c| 1 0 2 !() # |b×c| 2 . :% YQ 3 = 2 7 2 = F dA. F / 2+, = ! ()*+(, F : R3 R3 . F div F (x, y, z) = (x, y, z) ∂F1 ∂F2 ∂F3 + + , ∂x ∂y ∂z % F = (F1, F2, F3 ) . 7 -.!/ div F = F (x, y, z) = (xy, y 2 sin z, z 2 + y) . ∂ ∂ ∂ (xy) + y 2 sin z + z 2 + y = y + 2y sin z + 2z. ∂x ∂y ∂z (# (+ % ( & ##C /*(#) ( (2 +
) , # +&( ) (. (+ /"# ### % 9 #) $(" # ! ()*+(, 9 K #) ∂K 2( $ * K % G / *) % ∂K ! / ! ! + / 2 >>? K # ∂K 2) K % ,/ * #. ! = F dA div F dxdydz, / # & K ∂K % F K # ( ) # div F ( & ##C /*(# ( ρ + , ρ 0 .) # # P $* ρ I Bρ $$ z m̄ Bρ P y x 5 >B ρ (+ # ! K % / # G! Bρ 2 ∂Bρ / * 2 # / # & ) % ρ +) / %") % div F |Bρ ≡ div F (P ) . , + % (#*) ρ
02 P. ; # div F dxdydz ≈ div F (P ) · térf ogat(Bρ ). Bρ ! >>A :% ; # ≈ div F (P ) · térf ogat(Bρ ) F dA ∂Bρ F dA ( div F (P ) ≈ ∂Bρ térf ogat(Bρ ) . 0 F dA div F (P ) = lim ρ0 ∂Bρ térf ogat(Bρ ) . :% (+ P 2 # + $ / * # "2 # & / # #/ ) ( + $ #/ ( / div F (P ) < 0, $ / # / ) / P $# I + I $ # P I div F (P ) > 0, P $* + $" $ / # / ) / ) % P ! + div F (P ) = 0, / # / ) / P $* + $" = -.!/ T 5 >D % & $$ &) ∂T / * # / #
& ! F (x, y, z) = (x, y, z) = (2x + y 2 , y + sin z, z − x3 ) 9 (yz, xz, xy) # G 5 65 # F dA ∂T A ###P Gd ∂T ≡ 0, div G = 2 + 1 + 1 = 4. :% ( . div F # /*(#) # ; # " = div F dxdydz = 0, ,div F . F dA T ∂T ≡ 4. div Gdxdydz = 4 · térf ogat (T ) , ,div G A = Gd ∂T " T T $$ & #/ W (+ ! + >>H z m̄ ∂T T 1 3 y x 5 >D :& ?B 0# % :) % T 2 I ) % D = (x, z) : (x − 3)2 + z 2 ≤ 1 $ / z $* D2 % & (x − 3)2 + √ 2 z 2 = 1 $ z 2% $ √ /# $!( x = f1 (z) = 3− 1 − z , # ( x = f2 (z) = 3 + 1 − z 2 . & #/ I ) % #/ & ) x = f2 (z)2 z $* / & π x =
f1 (z) z $* / ( 1 T érf ogat(T ) = π −1 f22 (z) − f12 (z) dz = π √ 9 − 6 1 − z 2 + 1 − z 2 dz = π 1 −1 1 −1 f22 (z) dz ) ( #/ π 1 −1 1 −1 f12 (z) dz √ 9 + 6 1 − z2 + 1 − z2 − √ 12 1 − z 2 dz = 12π 1 −1 √ 1 − z 2 dz = 6π 2 / > ! ()*+(, ) % F / * ) # ∂F % % ( !( ) % (## ∂F 2 ∂F ! >>B z xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx 1 xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx D x = f (z) 5 F x = f2 (z) 4 x 3 :$$ & #/ I) % / % # F ! / " ( ) # / * # -.!/ F 5 % & % ) ∂F % # /" $ 9 F / # & !) % ! % % ) ∂F 2 & & # / " $$W . & $ ! & &( / " $ ! & &(
> 1-1 . 4E " 5 # :* / ) % F / # ∂F % % ! ) % F + ( F = ) F dr = curl F dA ∂F F / * % F ∂F ) $$ $ ! + >>D n̄ 5 > K * # % % ! %) % F % ) ∂F # $ $ & ) % # "/ )% F % *% 0# % F $ / * ) ( F / ∂F = ∅ ( F dr = 0, ! % F / * ) ∂F curl F dA = 0, $( # " F $> -.!/ γ 5 5 % & F (x, y, z) = (x, x, x) . ( " F dr? K; ! $) # γ . F / * ) γ % ) γ = ∂F. G!2 F 2 γ 2( % ( j k i ∂ ∂ ∂
= curl F = ∂x ∂y ∂z = (0, 0, 0) = 0 ) % curl F dA F x y z = 0. # curl F dA F dr = 0. 9 ( ∂F F $ -.!/ F = {(x, y, z) | x2 + y 2 ≤ 1, z = 0} , ( F xy ! # I $ ) / / # & !( 2 F (x, y, z) = √ 21 2 (y, −x, 3z) . ( " F dr? x +y ∂F ! > F ∂F F ∂F 5 9 # % % ! F dr2 . 5 ( ∂F / ( ∂F #) r (t) = (cos t, sin t, 0) , % 0 ≤ t ≤ 2π. F (r (t)) = (sin t, − cos t, 0) ; ṙ (t) = (− sin t, cos t, 0) :% F (r (t)) · ṙ (t) = − sin2 t − cos2 t = −1. E 2π F dr = ∂F 9 ( 2π F (r (t)) · ṙ (t) dt = 0 2# −1dt = −2π. 0 ( F dr2 ) ∂F ! + > > % 5 7 / * %
%) % # $ z H / * % n = (4, 1, 5) G F F E x 5 5 y K;9 % "$ i j k ∂ ∂ ∂ curl F = ∂x ∂y ∂z −x 3z √ y √ √ x2 +y2 x2 +y 2 x2 +y 2 −3/2 = −3zy x2 + y 2 , 3zx x2 + y 2 −3/2 ,0 ,5 H. ## % " ( & ) % F dr = 0) % ∂F xy ! , % z = 0.) curl F = 0 E % % ( ) % " F dr = −2π & ∂F % & &) % < % & # # ) 2# / # ) % F ! > + ( # F / * ( F z # // ) * ) * F 4 7 # ) # Sρ P $# I) + I $) P n. ∂Sρ # Sρ % ! z
xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxx xxxx
xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxx xxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx n̄ Kρ Sρ P ρ y x 5 ? + F = # /% Kρ = ∂Sρ . 9 ρ +) curl F $ !"
" curl F (P ) Sρ 2 :% |n| = 1. U curl F dA ≈ curl F (P ) ·ndA = curl F (P ) ·n · terület (Sρ ) . Sρ Sρ F dr = :% 2# " Kρ Sρ ∂Sρ n · terület (Sρ ) " $() % n · curl F (P ) ≈ F dr 0 Kρ ter ület(Sρ ) , % |n| = 1. ≈ curl F (P ) · curl F dA F dr = F dr Kρ terület (Sρ ) . ! + > 7 curl F 2 n2 " " (* " ,$ !"2 ) % ρ > 0, +. F Kρ 2 ( & (# ) ( Sρ * #( E $ -.!/ F (x, y, z) = (z 2 , x + x2 + z 2 , y 2 + x) / # 2 #) " & % * / # I # + P (1, 1, 1) , ! 2 # −n & ( n = (−2, 5, −1) . #* n̄ P ρ 5 A <
+ # K2 #) # % ) & & ( " ( # W + curl F = (2y − 2z, 2z − 1, 2x + 1) , ! curl F (P ) = (0, 1, 3) , % P = (1, 1, 1) ( curl F 2 n = (−2, 5, −1)2 √ ( & (* (0, 1, 3) · |nn| = (0, 1, 3) · (−2,5,−1) = √230 > 0. :% 4+25+1 F dr Kρ n ·curl F (P ) ≈ terület(Sρ ) $/*#" F !( (# |n| Kρ 2) % / # !() & & ( # / 9 # + P (1, 1, 1) ( $!() % ( n ( %%) % # ! > 5 /R 6 / & & ( " , !(. W # /) % F "# (# ) ( ) % F dr = 0. 9 ( ) % + ?? Kρ F dr
curl F (P ) · n ≈ Kρ terület (Sρ ) $( ) % curl F (P ) ⊥ n. :% # /) % n " () " curl F (P ) = (0, 1, 3)2 0# n = (0, −3, 1) # ∀n⊥ (0, 1, 3) . (P ) · n ≈ curl F F dr Kρ / 2 / # & & ( " ) % I ##C !( (# / # 2 $( ) % curl F (P ) · n %" I ##C !( :% n −curl F (P ) + ?%? ter ület(Sρ ) & #( E n = (0,−1,−3) √ 1+9 = √1 10 (0, −1, −3) . /8 2+ 2# + # ;2# P S xy ! ) % ∂S C $) # F S % @ % F xy ! (#) ( F (x, y, z) =
(P (x, y) , Q (x, y) , 0) , curl F = 0, 0, Qx − Py . # 2# " ) % = curl F dA n · curl F dxdy = Q − P dxdy. F dr = x ∂S S S S y ! + > ? z n̄ x x x x x S x x x x ∂S x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 5 H y S xy ! ( / ! F (#) S xy ! ( ) # / % ! !#
# 1-1 . 4A < 5 % D !) 2 C C $ y C = ∂D xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D 5 B x D ,D P. :( F ! (#) + ( D # ) % F (x, y) = ! > A (P (x, y) , Q (x, y)) : F dr = C Qx − Py dxdy D $ -.!/ F (x, y) = (2y, x) # C (x − 3)2 + 9 (y − 1)2 = 9 !( ! ( " F dr? C Qx . P (x, y) = 2y, Q (x, y) = x, ! Qx = 1 # Py = 2 :% − Py = −1. E Qx − Py dxdy = −terület (D) = −π · 3 · 1 = −3π. F dr = C D D 5 D C = ∂D (3, 1) A7 / 12 # 9 P (x, y) ≡ 0 # Q (x, y) = x) ( P (x, y) = −y # Q (x, y) ≡ 0 ⇒ Qx − Py ≡ 1. :% / terület (D) = xdy − ydx.
G % C C $ #) % F = (F1 , F2 ) (# (F1 dx + F2 dy) := F dr γ γ ! + G > H &) % 1 terület (D) = 2 xdy − ydx. ,5 B. C y C = ∂D xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D 5 7F :* x D + / C $ r = r(ϕ), % a ≤ ϕ ≤ b x (ϕ) = r (ϕ) cos ϕ, y (ϕ) = r (ϕ) sin ϕ. :% 5 B xdy − ydx = (x (ϕ) · ẏ (ϕ) − y (ϕ) · ẋ (ϕ)) dϕ . = r2 (ϕ) cos2 ϕ + sin2 ϕ dϕ = r2 (ϕ) dϕ E 5B 1 T erület (D) = 2 b a r2 (ϕ) dϕ. ,5 D. ! > B $% -.!/ r = sin 3ϕ I( M% ( C &%M , 5 7> . * # , sin 3ϕ < 0, r / !( ) % ) % r $ && ( ( I
/ ) % (|r| , ϕ + π) . . # " ! " R * % 0≤ϕ≤ π . 3 / # T erület (R) = 1 2 π 3 r2 (ϕ) dϕ = ϕ=0 1 2 π 3 T erület (R) = 1 2 · 1 2 ϕ=0 $(" ϕ=0 ! cos2 3ϕ + sin2 3ϕ = 1 # cos2 3ϕ − sin2 3ϕ = cos 6ϕ 2 sin2 3ϕ = 1 − cos 6ϕ ⇒ sin2 3ϕ = π 3 (sin 3ϕ)2 dϕ, (#( G 1 2 (1 − cos 6ϕ)) & .π (1 − cos 6ϕ) dϕ = 14 ϕ − 16 sin 6ϕ 03 = π . 12 π % ( C &% * 3 12 = π4 . $(" / # * ## # #" / % $ -.!/ (# C ) " (a, b) , (c, d) . xdy − ydx ##W C ## $(" 0 ≤ t ≤ 1, r (t) = (a, b) + t · (c − a, d − b) = (a + t (c − a) , b + t (d − b)) , F (x, y) = (−y, x) , % F (r (t)) = (−b −
t (d − b) , a + t (c − a)) , ṙ (t) = (c − a, d − b) . F (r (t)) · ṙ (t) = −b (c − a) − t (d − b) (c − a) + a (d − b) + t (c − a) (d − b) = ad − bc. . C F dr = E C $$ -.!/ * P 1 0 F (r (t)) · ṙ (t) dt = 57 1 0 ad − bcdt = ad − bc. % & . " / ## # n ;2# " ) % % C = ∪ Ci i=1 $ T ;2# % !! " T erület (t) = 1 2 xdy − ydx = C n 1 Σ 2 i=1 xdy − ydx = Ci > D n 1 Σxy 2 i=1 i i+1 − xi+1 yi, % xn+1 = x1 , # yn+1 = y1 . ,K % ( "" / #) ) % xdy − ydx = xi yi+1 − xi+1 yi . Ci + T erület (t) = 1 [(x1 y2 − x2 y1 ) + (x2 y3 − x3 y2 ) + . 2 + (xn−1 yn − xn yn−1 ) + (xn y1 − x1 yn )] # + ) % (x1 , y1 ) , (x2 , y2 ) , (xn, yn )
% > P % P # + !##( $ P ,2 7% % % "% % 0 $7 -.!/ F = rr3 , % r = (x, y, z) # r = |r| . S x2 + y 2 + z 2 = b2 $/ * / # & !( ## =? F dA S @" ( ( ( F %( ( $/ * F (r) ) # F 2 I (* |r| F (r) = r3 = r12 2 " ( b I $ / * # ( ) r = b % I ( ) % , ,5 A.) 1 b2 = 1 · $/ ! = 1 4b2 π = 4π F dA b2 b2 S /* $= -.!/ $# b2" div F dxdydz ) % K & K I) b I $) % F "" / P ! >7F ⎛ @" "$ div F 2
( F (x, y, z) = ⎞ r r3 = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x y z ⎜ ⎟ , , 3 3 3 ⎟ # / / ⎜ / 2 2 2 2 2 2 2 2 2 ⎜ ⎟ x +y +z x +y +z x +y +z ⎝ ⎠ F1 F2 ∂F1 = ∂x F3 / 3 / 2 2 2 x +y +z − x 32 x2 + y 2 + z 2 2x (x2 + y 2 + z 2 )3 / x2 + y 2 + z 2 (y 2 + z 2 − 2x2 ) . (x2 + y 2 + z 2 )3 √2 2 2 2 2 2 √2 2 2 2 2 2 x +y +z (x +z −2y ) x +y +z (y +z −2z ) ∂F2 ∂F3 3 ! ∂y = # ∂z = . G 3 (x2 +y 2 +z 2 )√ (x2 +y 2 +z 2 )3 2 2 2 2 2 2 2 2 2 2 2 2 x +y +z (y +z −2x x +z −2y +y +z −2z ) 1 2 3 div F = ∂F + ∂F + ∂F = = 0. ∂x ∂y ∂z (x2 +y 2 +z 2 )3 :% div F ≡ 0 # # div F dxdydz = 0. = K % @ AB ACD-E / # / " (2 + # # & * $$"# ) % (+ # + % ) % F ( # K # ( ( , / F = rr3 # K # ( * .
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" 6>8 O&/ K C# 6 8 31O 1 % 7 5) ( + ( + G 3( / 4 % ,>DDA. 678 31O 1 Q++ ( E+ + ) +; L29 ) G+ ,>DDF. 658 (+ K+ ! #$% & FF> & + 6?8 1 0## $$# @+ : $( &) 2 >DDF 6A8 1 O&/ GG[> : $( & 6H8 4 1 ! >DHB 6B8 +% E >DBD ) C $( & 2 @+ C $( & 2 >77
