Mathematics | Higher education » B3 jegyzet építőmérnököknek és környezetmérnök hallgatóknak

                               !"#$ # $ #$ %  &     "             (   )  ) % # #  (!  #) % #(#  * #     +! ,- %  %.       " $$# /  0/    1  2  ,3( / 4 %.)  #( (  !  !  2  & "   !  5 / ,(  !. ## 0/  # 6 ) 78  # !& $$    #   #    ## #  ### #  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   ) % 45  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  #; 06""    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" ; 06" ; 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 =!+& 45 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;   Z13 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