Aplicaciones de La Transformación Conforme.

APLICACIONES DE LA  TRANSFORMACIÓN CONFORME

CARLA MARÍA CANO DUEÑAS 04055 ALBA OLÍAS LÓPEZ 04282

ÍNDICE 1. INTRODUCCIÓN

PÁGINA 3

2. TRANSFORMACIONES CONFORMES

PÁGINA 3

3. APLICACIONES DE LAS TRANSFORMACIONES CONFORMES

PÁGINA 5

4. TEMPERATURAS ESTACIONARIAS

PÁGINA 5

5. TEMPERATURAS ESTACIONARIAS EN UN SEMIPLANO

PÁGINA 7

6. TEMPERATURAS EN UN CUADRANTE

PÁGINA 9

7. POTENCIAL ELECTROSTÁTICO

PÁGINA 11

8. POTENCIAL EN UN ESPACIO CILÍNDRICO

PÁGINA 12

9. FLUJO EN UN FLUIDO IDIMENSIONAL

PÁGINA 14

1!. LA FUNCIÓN DE CORRIENTE

PÁGINA 17

11. FLUJOS EN TORNO A UNA ES"UINA # A UN CILINDRO

PÁGINA 19

12. ILIOGRAFÍA

PÁGINA 19

APLICACIONES DE LA TRANSFORMACIÓN CONFORME

2

1. INTRODUCCIÓN El métoo ! l" t#"$%&o#m"'()$ 'o$&o#m! *" %(o + !% ,t(l(-"o !$ l" %ol,'()$ ! .#o/l!m"% ! l" &%('" m"t!m1t('" o/!#$"o% .o# l" !',"'()$ ! L".l"'!3 +" ,! é%t" !% ($"#("$t! ',"$o %! ".l('" l" t#"$%&o#m"'()$6 D('*"% ".l('"'(o$!% .,!!$ %!# !&($("% 'o$ 7,%o% t#"('(o$"l!% !l métoo ! t#"$%&o#m"'()$ 'o$&o#m!6 Po# ot#" ."#t!3 l" m!toolo" *" %(o ,t(l(-"" 'o$ é9(to + .o# '!#'" ! m!(o %(lo !$ l" %ol,'()$ ! '(!#to% .#o/l!m"% .l"$o% ! l" t!o#" m"t!m1t('" ! l" !l"%t('("6 P!#o !$ l"% o% :lt(m"% é'""% !l métoo ! t#"$%&o#m"'()$ 'o$&o#m!3 ,%"o .o# .#(m!#" !- .o# Ptolom!o *"'! ;800 ""$ !$ %, #1&('"6 P!#o ."#" w = f(z)3 'o$  z  + w 'om.l!>o%3 $o !% .o%(/l! *"'!# ,$" #1&('" "$1lo"3 .o#,! '"" ,$o ! !%o% $:m!#o% 'om.l!>o% !%t1 !$ ,$ .l"$o3 $o !$ ,$" #!'t"6 No o/%t"$t!3 %! .,!! #!.#!%!$t"# '(!#t" ($&o#m"'()$ ."#'("l ! l" &,$'()$ ($('"$o ."#!% ! .,$to% 'o##!%.o$(!$t!%  z = (x, y) + w = (u, v)6 A t"l &($3 %! (/,>"$ .o# %!."#"o lo% .l"$o% !  z  + w6 C,"$o %! .(!$%" ! !%! moo !$ ,$" &,$'()$3 %! *"/l" ! t#"$%&o#m"'()$6

2. TRANSFORMACIONES CONFORMES S!" C  ,$ "#'o %,"! "o .o#  z = z(t) (a ≤ t ≤ b) + %!"  f(z) ,$" &,$'()$ !&($(" !$ too% lo%  .,$to% z  ! C 6 L" !',"'()$ w = f[z(t)] (a ≤ t ≤ b) !% ,$" ."#"m!t#(-"'()$ ! l" (m"!$  Γ  ! C  /">o l" t#"$%&o#m"'()$ w = f(z)6 S,.o$"mo% ,! C   ."%" .o# ,$ .,$to  z 0 = z(t 0 ) (a ≤ t 0 ≤ b) !$ !l ,!  f  !% "$"lt('" ; +  f’ (z 0 ) ≠ 06 Po# l" #!l" ! l" '"!$"3 %( w = f[z(t)]  !$to$'!% w’(t 0 ) = f’[z(t 0 )]z’(t 0 ) + !%to %($(&('" ,! arg w’(t 0 ) = arg f’[z(t 0 )] + argz’(t 0 )

?;@

D!$ot!mo% .o# Ψ 0 ,$ "lo# ! argf’(z 0 ) + .o# θ 0 !l 1$,lo ! ($'l($"'()$ ! ,$" #!'t" t"$!$t! " C  !$  z 0 (#(("6 θ 0 !% ,$ "lo# ! argz’(t 0 ) + ! ?;@ %! %(,! ,! φ  0 = Ψ 0 + θ 0 !% ,$ "lo# ! arg  w’(t 0 ) + !%3 .o# t"$to3 !l 1$,lo ! ($'l($"'()$ ! ,$" #!'t" t"$!$t! "  Γ   !$ !l .,$to w0  = f(z 0 ) (#(("6 A% .,!%3 l" (&!#!$'(" !$t#! !l 1$,lo ! ($'l($"'()$ ! l" #!'t" (#((" !$ w0 + !l 1$,lo ! ($'l($"'()$ ! l" #!'t" (#((" !$  z 0 (!$! "" .o# !l 1$,lo ! #ot"'()$ Ψ 0 = argf’(z 0 )6

;

 S! ('! ,! ,$" &,$'()$ ! "#("/l! 'om.l!>" !% &,$'()$ "$"lt('" !$ ,$ .,$to "o %( %! .,!! !9.#!%"# 'omo ,$" %!#(! ! .ot!$'("% ,! 'o$!#>" "/%ol,t" + ,$(&o#m!m!$t! !$ !l !$to#$o ! !%! .,$to6 U$" &,$'()$  f  ! ,$" "#("/l! 'om.l!>"  z  !% analít!a !$ ,$ 'o$>,$to "/(!#to %( t(!$! !#("" !$ too% lo% .,$to% ! !%! 'o$>,$to6  %! ('! ,! ,$" &,$'()$ !% "$"lt('" !$ ,$ 'o$>,$to "  $o "/(!#to %(  f !% "$"lt('" !$ "l:$ 'o$>,$to "/(!#to ,! 'o$t(!$! " " 6 E$  ."#t(',l"#3 f !% analít!a #n un $unt%  z 0  %( !% "$"lt('" !$ "l:$ 'o$>,$to "/(!#to ,! 'o$t(!$! "  z 0



S!"$ "*o#" C &3 C ' o% "#'o% %,"!% ,! ."%"$ .o#  z 0 , + %!"$ θ & , θ ' lo% 1$,lo% ! ($'l($"'()$ ! %,% #!%.!'t("% #!'t"% t"$!$t!% (#(("% !$  z 0 Po# lo ,! "'"/"mo% ! !# φ  & = Ψ 0 + θ & + φ  ' = Ψ 0 + θ ' %o$ lo% 1$,lo% ! ($'l($"'()$ ! l"% #!'t"% t"$!$t!% (#(("% ! l"% ',#"% (m"!$  Γ & + Γ ' !$ !l .,$to w0 = f(z 0 )6 A% .,!%3 φ  '  φ  & = θ '  θ & E$ ot#"% ."l"/#"%3 !l 1$,lo ! φ  '  φ  & ! Γ & "  Γ ' !% (,"l3 !$ m"$(t, + %!$t(o3 !l 1$,lo θ '  θ & ! C & " C '6 E%! "lo# %! !$ot" .o#  !$ l" &(,#" %(,(!$t!=

D!/(o " !%t" .#o.(!" ! 'o$%!#"'()$ ! 1$,lo%3 ,$" t#"$%&o#m"'()$ w = f(z) !% 'o$&o#m! !$ ,$ .,$to z 0 %( f  !% "$"lt('" !$  z 0 +  f’(z 0 ) ≠ 06 U$" t"l t#"$%&o#m"'()$ !% 'o$&o#m!3 !$ #!"l("3 !$ too .,$to ! ,$ !$to#$o !  z 0 E$ !&!'to3 f  !/! %!# "$"lt('" !$ ,$ !$to#$o !  z 0 +3 'omo f’  !% 'o$t($," !$ z 0 (m.l('" ,! *"+ ,$ !$to#$o ! !%! .,$to !$ !l ,!  f’(z) ≠ 06 U$" ".l('"'()$ w = f(z), !&($(" !$ ,$ om($(o  *3 %! ('! ,! !% ,$" t#"$%&o#m"'()$ 'o$&o#m! %( !% 'o$&o#m! !$ '"" .,$to !  *6 A% .,!%3 l" t#"$%&o#m"'()$ !% 'o$&o#m! !$  * %( f  !% "$"lt('" !$  * + %, !#(""  f’  $o t(!$! '!#o% !$  *6

P$%&'()*)(+ )( ,* -$*+/%$0*' %/%$0( •

F*-%$(+ )( (+*,* S! !&($!

  f  D ? z 0 @

 'omo   f  D ? z 0 @

=

lí+

 z → z 0

  f  ? z @ −   f  ? z 0 @  z  −  z 0

=  z lí+ → z 

0

  f  ? z @ −   f  ? z 0 @  z  − z 0

A,$,! !l 1$,lo ! #ot"'()$ argf’(z) + !l &"'to# ! !%'"l"#   f   D ? z @  "#"$ .,$to " .,$to ! l" 'o$t($,(" !  f’  %! %(,! ,! %,% "lo#!% %o$ ".#o9(m""m!$t! argf’(z 0 ) +   f  D ? z 0 @ !$ .,$to% '!#'"$o% "l  z 0 Po# t"$to3 l" (m"!$ ! ,$" .!,!<" #!()$ !$ ,$ !$to#$o !  z 0 !% 'o$&o#m! 'o$ l" #!()$ o#(($"l !$ !l %!$t(o ! ,! t(!$! ".#o9(m""m!$t! l" m(%m" &o#m"6 S($ !m/"#o3 ,$" #!()$ #"$! .,!! %!# t#"$%&o#m"" !$ ,$" #!()$ %($ ."#!'(o 'o$ l" o#(($"l6 •

I($+*+ ,%*,(+

U$" t#"$%&o#m"'()$ w = f(z) 'o$&o#m! !$ ,$ .,$to  z 0 t(!$! ($!#%" lo'"l !$ él6 E%to !%3 %( w0 = f(z 0 )3 !$to$'!% !9(%t! ,$" :$('" t#"$%&o#m"'()$  z = g(w)  !&($(" + "$"lt('" !$ ,$ !$to#$o    ! w03 t"l ,! ? w0 ) = z 0 + f[g(w)] = w ."#" too lo% .,$to% w !  6

3. APLICACIONES DE LAS TRANSFORMACIONES CONFORMES 4

L"% t#"$%&o#m"'(o$!% 'o$&o#m!% %(#!$ ."#" #!%ol!# .#o/l!m"% &%('o% #!l"'(o$"o% 'o$ l" !',"'()$ ! L".l"'! 2 !$ o% "#("/l!% ($!.!$(!$t!%6 S, ,%o %! /"%" .#($'(."lm!$t! " l" ($"#("$-" ! "l,$"% 'o$('(o$!% ! 'o$to#$o /">o '"m/(o% ! "#("/l! !&($(o% .o# t#"$%&o#m"'(o$!% 'o$&o#m!%6 L" té'$('" &,$"m!$t"l ."#" #!%ol!# .#o/l!m"% ! 'o$to#$o !% t#"$%&o#m"# ,$ .#o/l!m" ! 'o$to#$o "o !$ !l .l"$o  xy !$ ,$o m1% %(m.l! !$ !l .l"$o uv + ,%"# !$to$'!% (!#%o% #!%,lt"o% 'o$ !l &($ ! !%'#(/(# l" %ol,'()$ !l .#o/l!m" o#(($"l !$ té#m($o% ! l" %ol,'()$ o/t!$(" ."#" !l  .#o/l!m" m1% %(m.l!6 A$"l(-"#!mo% .#o/l!m"% ! 'o$,''()$ ! '"lo#3 ! .ot!$'("l!% !l!'t#o%t1t('o% + ! &l,>o ! &l,(o%6

4. TEMPERATURAS ESTACIONARIAS E$ l" t!o#" ! l" 'o$,''()$ !l '"lo#3 !l &l,>o " t#"é% ! ,$" %,.!#&('(! ($t!#(o# " ,$ %)l(o !$ ,$  .,$to ! !%" %,.!#&('(! !% l" '"$t(" ! '"lo# ,! &l,+! !$ l" (#!''()$ $o#m"l " l" %,.!#&('(! .o#  ,$(" ! t(!m.o + .o# ,$(" ! 1#!" !$ !%! .,$to6 Po# t"$to3 !l &l,>o %! m(! !$ ,$("!% ! '"lo#   .o# %!,$o .o# '!$tm!t#o ',"#"o6 S! !$ot" .o# - + %, "#("'()$ !% .#o.o#'(o$"l " l" !#("" $o#m"l ! l" t!m.!#"t,#" .  !$ !%! .,$to ! l" %,.!#&('(!=

Φ = − 0 

/.  /, 

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L" #!l"'()$ ?;@ %! 'o$o'! 'omo l" l#y /# 1%ur#r  + l" 'o$%t"$t!    %! ll"m" 'o$,'t((" té#m('" !l m"t!#("l !l %)l(o3 ,! %,.o$#!mo% *omoé$!o6 Lo% .,$to% !l %)l(o %! (!$t(&('"$ m!("$t! 'oo#!$""% #!'t"$,l"#!% !$ !l !%."'(o t#((m!$%(o$"l6 R!%t#($(mo% $,!%t#" "t!$'()$ " ",!llo% '"%o% !$ lo% ,! l" t!m.!#"t,#" .   "#" %)lo 'o$ l"% 'oo#!$""%  x !  y6 Como . $o "#" 'o$ l" 'oo#!$"" !l !>! .!#.!$(',l"# "l .l"$o  xy3 !l &l,>o ! '"lo# !% /((m!$%(o$"l + ."#"l!lo " !%! .l"$o6 S,.o$!mo% ,! !$t#o !l %)l(o $o %! '#!" $( %! !%t#,+! !$!#" té#m('"3 !% !'(#3 $o *"+ !$ él &,!$t!% o %,m(!#o% ! '"lo#6 A!m1% %,.o$!mo% ,! l" &,$'()$ t!m.!#"t,#" .(x, y) + %,% !#(""%  ."#'("l!% ! .#(m!# + %!,$o o#!$ %o$ 'o$t($,"% !$ too .,$to !l ($t!#(o# !l %)l(o6 Co$%(!#!mo% "*o#" ,$ !l!m!$to ! ol,m!$ ($t!#(o# !l %)l(o !$ &o#m" ! .#(%m" #!'t"$,l"#  ! "lt,#" ,$("3 .!#.!$(',l"# "l .l"$o  xy3 'o$ /"%! ! l"o% 93 + !$ !%! .l"$o6 El #(tmo t!m.o#"l !l &l,>o ! '"lo# *"'(" l" !#!'*"3 " t#"é% ! l" '"#" (-,(!#"3 !% −  0.  x ? x 3  y @∆ y G !l #(tmo ! &l,>o *"'(" l" !#!'*" " t#"é% ! l" '"#" ! l" !#!'*" !% −  0.  x ? x + ∆ x3  y @∆ y 6 R!%t"$o !l .#(m!#o !l %!,$o3 o/t!$!mo% !l #(tmo $!to ! .é#(" ! '"lo# ! !%! !l!m!$to ! ol,m!$  .o# !%"% o% '"#"%6 El #(tmo #!%,lt"$t! %! .,!! !9.#!%"#  .  ? x + ∆ x3  y @ − . x ? x3  y @  −  0   x ∆ x∆ y  ∆  x  

o 'omo

−  0.  xx ? x3  y @∆ x∆ y

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%( 2x !% m,+ .!,!
 E',"'()$ ! L".l"'!= H0

5

A$1lo"m!$t!3 !l #(tmo #!%,lt"$t! ."#" l" .é#(" ! '"lo# " t#"é% ! l"% ot#"% o% '"#"%  .!#.!$(',l"#!% "l .l"$o xy (!$! "" .o# −  0.  yy ? x3  y @∆ x∆ y

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Como !l '"lo# !$t#" o %"l! !l !l!m!$to ! ol,m!$ :$('"m!$t! " t#"é% ! !%"% ',"t#o '"#"% + l"% t!m.!#"t,#"% !$t#o ! él %o$ !%t"'(o$"#("%3 l" %,m" ! l"% !9.#!%(o$!% ?;@ + ?2@ !/! %!# '!#o= .  xx ? x 3  y @ + .  yy ? x3 y @

=0

?@

L" &,$'()$ t!m.!#"t,#" %"t(%&"'!3 .o# t"$to3 l" !',"'()$ ! L".l"'! !$ too% lo% .,$to% ($t!#(o#!% !l %)l(o6 E$ (%t" ! ?@ + ! l" 'o$t($,(" ! l" &,$'()$ t!m.!#"t,#" + %,% !#(""% ."#'("l!%3 .   !% ,$"  fun!3n ar3n!a ! x ! y !$ !l ($t!#(o# !l %)l(o6 L"% %,.!#&('(!% .(x, y) = ! & , o$! !& !$ot" ,$" 'o$%t"$t! #!"l3 %o$ l"% 4%t#ra4 !l %)l(o6 S!  .,!!$ !# t"m/(é$ 'omo ',#"% !$ !l .l"$o  xy3 !$ ',+o '"%o .(x, y)  %! ($t!#.#!t" 'omo l" t!m.!#"t,#" !$ !l .,$to (x, y) ! ,$" &($" .l"'" ?o l1m($"@ ! m"t!#("l !$ !%! .l"$o3 'o$ %,% '"#"% "(%l""% té#m('"m!$t!6 L"% (%ot!#m"% %o$ l"% ',#"% ! $(!l ! l" &,$'()$ . 6 El #"(!$t! ! .  !% .!#.!$(',l"# " l" (%ot!#m" !$ '"" .,$to + !l &l,>o m19(mo !$ ,$ .,$to %!  .#o,'! !$ l" (#!''()$ !l #"(!$t! !$ él6 S( .(x, y) !$ot" l"% t!m.!#"t,#"% !$ ,$" &($" .l"'" + "  !% "#m)$('" 'o$>,"" ! l" &,$'()$ . 3 ,$" ',#" "(x, y) = ! ' t(!$! !l #"(!$t! ! .  'omo !'to#  t"$!$t! !$ too .,$to o$! l" &,$'()$ "$"lt('" .(x, y) +  "(x, y) %!" 'o$&o#m!6 L"% ',#"% "(x, y) = ! ' %! ll"m"$ lín#a4 /# flu5%6 S( l" !#("" $o#m"l /.6/  !% '!#o !$ ,$" .o#'()$ !l /o#! ! l" .l"'"3 !l &l,>o ! '"lo# " t#"é% ! !%" .o#'()$ !% $,lo6 E% !'(#3 !%" ."#t! !%t1 "(%l"" té#m('"m!$t! + !%3 !$ 'o$%!',!$'("3 ,$" l$!" ! &l,>o6 L" &,$'()$ .   .,!! !$ot"# t"m/(é$ l" 'o$'!$t#"'()$ ! ,$" %,%t"$'(" ,! %! (&,$! .o# ,$ %)l(o6 E$ t"l '"%o3    !% l" 'o$%t"$t! ! (&,%()$6

5. TEMPERATURAS ESTACIONARIAS EN UN SEMIPLANO J"mo% " *"ll"# ,$" !9.#!%()$ ."#" l"% t!m.!#"t,#"% !%t"'(o$"#("% .(x, y)  !$ ,$" &($" .l"'" %!m(($&($(t" y 7 0 ',+"% '"#"% !%t1$ "(%l""% + ',+o /o#!  y = 0  %! m"$t(!$! " t!m.!#"t,#" '!#o !9'!.to !$ !l %!m!$to − ; < x < ; 3 o$! %! m"$t(!$! " t!m.!#"t,#" ,$("6 L" &,$'()$  .(x, y) *" ! %!# "'ot""3 ,$" 'o$('()$ $"t,#"l %( 'o$%(!#"mo% l" .l"'" 'omo lm(t! ! ,$" .l"'" 0 ≤  y ≤ y 0 ',+o /o#! %,.!#(o# %! m"$t(!$! " t!m.!#"t,#" &(>" ',"$o  y0 '#!'!6 El .#o/l!m" ! 'o$to#$o ,! *!mo% ! #!%ol!# %! &o#m,l" "%= .  xx ? x3  y@ + .  yy ? x3  y @ = 0 . ? x30@ =

{

;  4  x 0  4  x

? −∞ <  x < ∞3  y > 0@

<; >;

K

'o$ . ? x3  y@ <  8  3 o$!  8   !% ,$" 'o$%t"$t! .o%(t("6 E% ,$ .#o/l!m" ! D(#('*l!t   !$ !l %!m(.l"$o %,.!#(o# !l .l"$o  xy6 N,!%t#o métoo ! %ol,'()$ 'o$%(%t(#1 !$ t#",'(#lo 'omo ,$  .#o/l!m" ! D(#('*l!t !$ ,$" #!()$ !l .l"$o uv6 L" #!()$ %!#1 l" (m"!$ !l %!m(.l"$o /">o ,$" t#"$%&o#m"'()$ w =   f  ? z @ "$"lt('" !$ !l om($(o  y > 0  + 'o$&o#m! !$ !l /o#!  y = 0 3 !9'!.to !$ lo% .,$to% ? ±;30@ 3 o$! $o !%t1 !&($("6 S!"$  z  − ; = r ; !9.?θ ; @

o$!

0 ≤ θ 9 

 y

 z  + ; = r 2 !9.?θ 2 @

≤ π  ?9  = ;3 2@ 6 L" t#"$%&o#m"'()$ w = lo

 z  − ;  z  + ;

= l$

r ; r 2

 r  Bπ    π  + ?θ ; − θ 2 @  ; > 03 − < θ ; − θ 2 <    2 2    r 2

?;@

!%t1 !&($(" !$ too !l %!m(.l"$o %,.!#(o#  y ≥ 0 3 !9'!.to !$ lo% o% .,$to%  z  = ±; 3 +" ,! 0 ≤ θ ; − θ 2 ≤ π  !$ l" #!()$6 El "lo# !l lo"#(tmo !% "*o#" !l "lo# .#($'(."l 'o$ 0 ≤ θ ; − θ 2 ≤ π  6 El %!m!$to !l !>!  x !$t#!  z  = −;  +  z  = ; 3 o$! θ ; − θ 2 = π  %! ".l('" %o/#! !l /o#! %,.!#(o# ! l" &#"$>"G !l #!%to !l !>!  x3 o$! θ ; − θ 2 = 0 3 %o/#! !l /o#! ($&!#(o#6 L"% 'o$('(o$!% #!,!#("% ."#" %!# "$"lt('" + 'o$&o#m! %! %"t(%&"'!$ !(!$t!m!$t! !$ l" t#"$%&o#m"'()$ ?;@6 U$" &,$'()$ "#m)$('" "'ot"" ! u3 v ,! !% $,l" !$ !l /o#!  /o#! v π   !% 'l"#"m!$t!

v

= 0  ! (,"l " l" ,$(" !$ !l

=

.  =

; π 

v

"#m)$('" .o# %!# ."#t! (m"($"#(" ! l" &,$'()$ !$t!#" m!("$t! l" !',"'()$ w = l$

?2@ ?; L π @ w 6

C"m/("$o " l"% 'oo#!$""%  x3 +

 z  − ;

 z  − ;   +  "#      z  + ;   z  + ;  

!mo% ,!

 ? z  − ;@? z  + ;@   x 2 +  y 2 − ; +  2 y  v = "#   = "#  x + 2 +  y 2   ? ;@   ? z  + ;@? z  + ;@  

 El .#o/l!m" ! D(#('*l!t !% ,$ .#o/l!m" ! '1l',lo (&!#!$'("l 'o$%(%t!$t! !$ !$'o$t#"# ,$" &,$'()$ "#m)$('" %o/#! ,$ om($(o ! ?o m1% !$!#"lm!$t! ,$" "#(!" (&!#!$'("/l!@ ,! tom! "lo#!% .#!%'#(to% %o/#! !l 'o$to#$o ! ('*o om($(o6



o %!" v

  2 y     = "#'t"$ 2 2     x +  y − ;  

L" &,$'()$ "#'ot"$!$t! "#" ", !$t#!  0 + :  .o#,!

  z  − ;  = θ  − θ    ; 2   z  + ;  

"#

+ 0 ≤ θ ; − θ 2 ≤ π  6 A*o#" ?2@ "o.t" l" &o#m" .  =

; π 

  2 y     2 2     x +  y − ;  

"#'t"$

?0 ≤ "#'t"$ t  ≤ π @

?@

Como l" &,$'()$ ?2@ !% "#m)$('" !$ l" &#"$>" 0 < v < π   + l" t#"$%&o#m"'()$ ?;@ !% "$"lt('" !$ !l %!m(.l"$o  y > 0 3 %! 'o$'l,+! ,! l" &,$'()$ ?@ !% "#m)$('" !$ !%! %!m(.l"$o6 L"% 'o$('(o$!% ! 'o$to#$o ."#" l"% o% &,$'(o$!% "#m)$('" %o$ l"% m(%m"% %o/#! l"% ."#t!% 'o##!%.o$(!$t!% ! lo%  /o#!%3 +" ,! %o$ !l t(.o ; = ;0 6 L" &,$'()$ "'ot"" ?@ !%3 .o# t"$to3 l" %ol,'()$ !%!"" !l  .#o/l!m" o#(($"l6 L"% (%ot!#m"%

. ? x3  y @

= !; ?0 < !; < ;@  %o$ "#'o% ! l"% '(#',$&!#!$'("%  x

2

+ ? y − 'ot π !; @ 2 = '%' 2 π !;

,! ."%"$ .o# lo% .,$to% ?±;30@  + !%t1$ '!$t#""% !$ !l !>!  y6 F($"lm!$t!3 $)t!%! ,! 'omo !l .#o,'to ! ,$" &,$'()$ "#m)$('" .o# ,$" 'o$%t"$t! !% t"m/(é$ "#m)$('"3 l" &,$'()$ .  =

. 0

π 

  2 y     2 2   + − ;  x  y    

"#'t"$

?0 ≤ "#'t"$ t  ≤ π @

#!.#!%!$t" l"% t!m.!#"t,#"% !%t"'(o$"#("% !$ !l %!m(.l"$o "o ',"$o !$ !l %!m!$to − ; < x < ; !l !>! x %! &(>" ,$ "lo# .  = . 0  ."#" l" t!m.!#"t,#"3 !$ l,"# !l "lo# .  = ; 6

6. TEMPERATURAS EN UN CUADRANTE J"mo% " *"ll"# l"% t!m.!#"t,#"% !%t"'(o$"#("% !$ ,$" .l"'" &($" ,! o',." ,$ ',"#"$t!3 'o$ ,$ %!m!$to !l /o#! ($&!#(o# "(%l"o m(!$t#"% !l #!%to ! %, /o#! %! m"$t(!$! " t!m.!#"t,#" &(>" + !l  /o#! !#t('"l %! m"$t(!$! " ot#" t!m.!#"t,#" &(>"6 S,% '"#"% !%t1$ "(%l""%3 ! m"$!#" ,! !l  .#o/l!m" !% /((m!$%(o$"l6 Po!mo% !l!(# l"% !%'"l"% ! t!m.!#"t,#"% + lo$(t, ! moo t"l ,! !l .#o/l!m" ! 'o$to#$o  ."#" l" &,$'()$ t!m.!#"t,#" .  %! &o#m,l! "%=

.  xx ? x3  y @ + .  yy ? x3  y @

=0

? x

> 03 y > 0@

?;@

8

=0 . ? x30@ = ; .  y ? x30@

 4

0 <  x

 4

 x

>;

?y

> 0@

. ?03  y @ = 0

<;

?2@ ?@

o$! . ? x3 y@ !% "'ot"" !$ !l ',"#"$t!6 L" .l"'" + %,% 'o$('(o$!% ! 'o$to#$o %! m,!%t#"$ !$ l" &(,#" ">,$t"6 L"% 'o$('(o$!% ?2@ .#!%'#(/!$ lo% "lo#!% ! l" !#("" $o#m"l ! .   %o/#! ,$"  ."#t! ! ,$" #!'t" !l /o#! + lo% "lo#!% ! l" .#o.(" &,$'()$ %o/#! !l #!%to ! !%" #!'t"6 El métoo ! %!."#"'()$ ! "#("/l!% '(t"o "l &($"l ! l" %!''()$ "$t!#(o# $o %! "".t" " !%t" 'l"%! !  .#o/l!m"% 'o$ (%t($to% t(.o% ! 'o$('(o$!% %o/#! ,$ m(%mo /o#! #!'to6

L" &,$'()$ t!m.!#"t,#" .  #!,!#(" .o# !%t! $,!o .#o/l!m" ! 'o$to#$o !% 'l"#"m!$t! .  =

l" ."#t! #!"l ! l" &,$'()$ !$t!#"

?2 L π @ w 6

2 π 

?4@

u

A*o#" !/!mo% !9.#!%"# .  !$ té#m($o% !  x ! y6

P"#" !9.#!%"# u !$ té#m($o% !  x ! y3 !m.!-"mo% o/%!#"$o ,!3 .o#  x

=  4#nu   'o%* v 3 +

 y

= 'o% u  4#n;v

 z 

=

 4#nw

3

?5@

C,"$o 0 < u < π  L 2 6 T"$to %!$ u 'omo 'o% u %o$ $o $,lo%3 l,!o  x

2 2

 4#n u

−

 y

2 2

'o% u

?K@

=;

A*o#" 'o$(!$! *"'!# $ot"# ,!3 ."#" '"" u &(>o3 l" *(.é#/ol" ?K@ t(!$! &o'o% !$ lo% .,$to%  z  =

±

 4#n 2 u

+ 'o% 2 u = ±;

+ l" lo$(t, ! %, !>! t#"$%!#%"l3 ,! !% !l %!m!$to #!'to ,! ,$! lo% é#t('!%3 !% 2%!$ u6 A%  .,!%3 !l "lo# "/%ol,to ! l" (&!#!$'(" ! l"% (%t"$'("% !$t#! lo% &o'o% + ,$ .,$to (x, y) ! l" ."#t! ! l" *(.é#/ol" %(t,"" !$ !l .#(m!# ',"#"$t! !% ? x + a @

2

+  y 2 −

? x − ;@

2

+  y 2 = 2  4#nu



D! ?5@ %! !,'! ,! !%t" #!l"'()$ !% "%(m(%mo 1l(" ',"$o l" &,$'()$ t!m.!#"t,#" #!,!#(" (!$! "" .o#   .  = ar!4#n  π   2

? x + ;@

2

+  y 2 −

? x − ;@

2

o$!3 "l %!# 0 ≤ u ≤ π  L 2 3 l" &,$'()$ "#'o%!$o "#" !$t#! 0 + :6'6

u

2

= 0  o

u

= π  L 2 6 E$ (%t" ! ?4@3

+  y 2   

S( !%!"mo% 'om.#o/"# ,! !%t" &,$'()$ %"t(%&"'! l"% 'o$('(o$!% ! 'o$to#$o ?2@3 *!mo% ! #!'o#"# ,! ? x − ;@  %($(&('"  x − ; %(  x > ; + ; −  x %( 0 <  x < ; 3 +" ,! l" #"- ',"#"" !%  .o%(t("6 N)t!%!3 "!m1%3 ,! l" t!m.!#"t,#" !$ ',"l,(!# .,$to ! l" .o#'()$ "(%l"" !l /o#! ($&!#(o# ! l" .l"'" !% 2 . ? x3 = 0@ = ar!4#nx ?0 < x < ;@ 2

π 

E$ ?4@ !mo% ,! l"% (%ot!#m"% . ? x3  y@ = !; ?0 < !; < ;@  %o$ l"% ."#t!% ! l"% *(.é#/ol"% 'o&o'"l!% ?K@3 o$! u = π !; L 2 3 ,! !%t1$ !$ !l .#(m!# ',"#"$t!6 Como l" &,$'()$ ?2 L π @v  !% "#m)$('" 'o$>,"" ! l" &,$'()$ ?4@3 l"% l$!"% ! &l,>o %o$ ',"#to% ! l"% !l(.%!% 'o&o'"l!% o/t!$("% "l m"$t!$!# 'o$%t"$t! v !$ l"% !',"'(o$!% ?5@6

7. POTENCIAL ELECTROSTÁTICO E$ ,$ '"m.o ! &,!#-"% !l!'t#o%t1t('o3 l" ($t!$%(" !l '"m.o !$ ,$ .,$to !% ,$ !'to# ,! #!.#!%!$t" l" &,!#-" !>!#'(" %o/#! ,$" '"#" .o%(t(" ,$(" 'olo'"" !$ !%! .,$to6 El  $%t#n!al  !l!'t#o%t1t('o !% ,$" &,$'()$ !%'"l"# ! l"% 'oo#!$""% !%."'("l!% t"l ,!3 !$ '"" .,$to3 %, !#("" (#!''(o$"l !$ ',"l,(!# (#!''()$ !% !l $!"t(o ! l" ($t!$%(" ! '"m.o !$ !%" (#!''()$6 P"#" o% ."#t',l"% '"#""% !%t"'(o$"#("%3 l" m"$(t, ! l" &,!#-" ! "t#"''()$ o #!.,l%()$ !>!#'(" .o# ,$" ! !ll"% %o/#! l" ot#" !% (#!'t"m!$t! .#o.o#'(o$"l "l .#o,'to ! l"% '"#"% ! ($!#%"m!$t! .#o.o#'(o$"l "l ',"#"o ! l" (%t"$'(" !$t#! !ll"%6 A ."#t(# ! !%t" l!+ !l ($!#%o !l ',"#"o %! .,!! !,'(# ,! !l .ot!$'("l !$ ,$ .,$to3 !/(o " ,$" :$('" ."#t',l" !$ !l !%."'(o3 !% ($!#%"m!$t! .#o.o#'(o$"l " l" (%t"$'(" !$t#! !l .,$to + l" ."#t',l"6 E$ ',"l,(!# #!()$ l(/#! ! '"#"%3 !l .ot!$'("l !/(o " ,$" (%t#(/,'()$ ! '"#"% !9t!#(o# " !%" #!()$ %! .,!! !mo%t#"# ,! %"t(%&"'! l" !',"'()$ ! L".l"'! !$ !l !%."'(o t#((m!$%(o$"l6 S( l"% 'o$('(o$!% %o$ t"l!% ,! !l .ot!$'("l <  !% !l m(%mo !$ too% lo% .l"$o% ."#"l!lo% "l .l"$o  xy3 !$to$'!%3 !$ #!(o$!% l(/#!% ! '"#"%3 <  !% ,$" &,$'()$ "#m)$('" ! %)lo o% "#("/l!%3  x ! y= <  xx ? x3  y@ + <  yy ? x3 y@ = 0

El !'to# ($t!$%(" !l '"m.o !$ '"" .,$to !% ."#"l!lo "l .l"$o  xy3 'o$ 'om.o$!$t!%  x !  y #!%.!'t("% − <  x ? x3 y@  + − <  y ? x3 y @ 6 Po# t"$to3 !% !l $!"t(o !l #"(!$t! ! <  ? x3 y @ 6 U$" %,.!#&('(! %o/#! l" ',"l < ? x3 y@ !% 'o$%t"$t! !% ,$" %,.!#&('(! !,(.ot!$'("l6 L" 'om.o$!$t! t"$!$'("l !l !'to# ($t!$%(" !l '"m.o !$ ,$ .,$to ! ,$" %,.!#&('(! 'o$,'to#" !% '!#o !$ !l '"%o !%t1t('o3 .o#,! l"% '"#"% .,!!$ mo!#%! l(/#!m!$t! %o/#! t"l %,.!#&('(!6 A% .,!%3 < ? x3 y @ !% 'o$%t"$t! %o/#! l" %,.!#&('(! ! ,$ 'o$,'to#3 ! moo ,! !%" %,.!#&('(! !% #u$%t#n!al 6 S( >  !% ,$" 'o$>,"" "#m)$('" ! < 3 l"% ',#"% > ? x3  y @ = ! 2  !$ !l .l"$o  xy %! ll"m"$ lín#a4 /#  flu5%6 C,"$o ,$" ! !%t"% ',#"% 'o#t" " ,$" ',#" !,(.ot!$'("l < ? x3  y @ = !;  !$ ,$ .,$to o$! ;0

l" !#("" ! l" &,$'()$ "$"lt('" < ? x3  y @ + > ? x3 y @  !% $o $,l"3 l"% o% ',#"% %o$ o#too$"l!% !$ !%! .,$to + l" ($t!$%(" !l '"m.o !% t"$!$t! " l" l$!" ! &l,>o "ll6 Lo% .#o/l!m"% ! 'o$to#$o ."#" !l .ot!$'("l <  %o$ (é$t('o% " lo% ! ,$" t!m.!#"t,#" !%t"'(o$"#(" . 3 + "l (,"l ,! !$ !l '"%o ! l" t!m.!#"t,#"3 lo% métoo% !l "$1l(%(% 'om.l!>o %! l(m(t"$ "  .#o/l!m"% !$ o% (m!$%(o$!%6

8. POTENCIAL EN UN ESPACIO CILÍNDRICO U$ l"#o '(l($#o '(#',l"# *,!'o3 &"/#('"o 'o$ ,$" &($" '"." ! m"t!#("l 'o$,'to#3 !%t1 'o#t"o " lo l"#o !$ o% m(t"!% (,"l!%6 E%t"% o% m(t"!%3 %!."#""% .o# ,$" '"." ! m"t!#("l "(%l"$t!3 %! ,%"$ 'omo !l!'t#oo%3 ,$o ! !llo% m"$t!$(o " .ot!$'("l '!#o + !l ot#o " ,$ .ot!$'("l &(>o (%t($to6 Tom"mo% lo% !>!% ! 'oo#!$""% + l"% ,$("!% ! lo$(t, + ! (&!#!$'(" ! .ot!$'("l 'omo ($('" l" &(,#" ! "/">o6 I$t!#.#!t"mo% "*o#" !l .ot!$'("l !l!'t#o%t1t('o < ? x3 y@  !$ ,$" %!''()$ ',"l,(!#" !l '(l($#o3 "l!>"" ! lo% !9t#!mo%3 'omo ,$" &,$'()$ "#m)$('" !$ !l ($t!#(o# ! l" '(#',$&!#!$'("  x 2 + y 2 = ; !$ !l .l"$o  xy6 N)t!%! ,! <  = 0 !$ l" m(t" %,.!#(o# ! l" '(#',$&!#!$'(" + <  = ;  !$ l" ($&!#(o#6

S! .,!! !#(&('"# ,$" t#"$%&o#m"'()$ #"'(o$"l l($!"l ,! ".l('" !l %!m(.l"$o %,.!#(o# %o/#! !l ($t!#(o# ! l" m(t" %,.!#(o# ! l" '(#',$&!#!$'(" ,$(" '!$t#"" !$ !l o#(!$3 !l %!m(!>! #!"l  .o%(t(o %o/#! l" m(t" %,.!#(o# ! l" '(#',$&!#!$'(" + !l %!m(!>! #!"l $!"t(o %o/#! l" m(t" ($&!#(o# ! l" '(#',$&!#!$'("6 I$t!#'"m/("$o  z  + w !$ !ll"3 *"ll"mo% ,! l" ($!#%" ! l" t#"$%&o#m"'()$  z  =

−w

?;@

+w

 .#o,'! ,$ .#o/l!m" $,!o ."#" <  !$ ,$ %!m(.l"$o3 ($('"o " l" !#!'*" !$ l" &(,#"6 L" ."#t! (m"($"#(" ! l" &,$'()$ ;

π 

 ?%g w =

;

π 

l$  $ +



π 

φ 

?$

> 03 0 ≤ φ  ⊆ π @

!% ,$" &,$'()$ "'ot"" ! u + v ,! tom" lo% o% "lo#!% 'o$%t"$t!% #!,!#(o% !$ l"% .o#'(o$!% φ  = 0  + φ  = π   !l !>! u6 Po# t"$to3 l" &,$'()$ "#m)$('" /,%'"" !$ !l %!m(.l"$o !% <  =  

; π 

 v      u  

"#'t"$ 

?2@ ;;

o$! lo% "lo#!% ! l" &,$'()$ "#'ot"$!$t! "#"$ !$t#! 0 +

π  6

L" ($!#%" ! l" t#"$%&o#m"'()$ ?;@ !% w=

; −  z 

?@

; +  z 

+ #"'("% " !ll" %! .,!!$ !9.#!%"# u + v !$ té#m($o% !  x ! y6 !$to$'!%3 l" !',"'()$ ?2@ %! 'o$(!#t! !$ <  =

; π 

 ; −  x 2 −  y 2       2  y    

"#'t"$

?0 ≤ "#'t"$ t  = π  @

?4@

L" &,$'()$ ?4@ !% l" &,$'()$ .ot!$'("l !$ !l !%."'(o !$'!##"o .o# lo% !l!'t#oo% '(l$#('o% +" ,! !% "#m)$('" !$ !l ($t!#(o# ! l" '(#',$&!#!$'(" + tom" lo% "lo#!% !%.!'(&('"o% !$ l"% o% %!m('(#',$&!#!$'("%6 P"#" 'om.#o/"# l" %ol,'()$ *"+ ,! *"'!# $ot"# ,! lí+ "#'t"$ t  = 0

"#'t"$ t  = π  + lí+ t →0

t →0 t > 0

L"% ',#"% !,(.ot!$'("l!% '(#',$&!#!$'("%

t < 0

< ? x3  y @

 x

2

= !; ?0 < !; < ;@ !$ l" #!()$ '(#',l"# %o$ "#'o% ! l"% + ? y + t"$π !; @ 2 = %!'2 π !;

,! ."%"$3 to"% !ll"%3 .o# lo% .,$to% ?±;30@ 6 A!m1%3 !l %!m!$to !l !>!  x !$t#! !%o% o% .,$to% !% !,(.ot!$'("l 'o$ < ? x3  y @ = ; L 2 6 U$" 'o$>,"" "#m)$('" >  ! <  !% − ?; L π @ l$ ρ  3 l" ."#t! (m"($"#(" ! l" &,$'()$ − ? L π @ ?%g  w 6 E$ (%t" ! ?@3 >  %! .,!! !%'#(/(#  >  = −

;

π 

l$

; −  z  ; +  z 

E%t" !',"'()$ .!#m(t! !# ,! l"% l$!"% ! &l,>o > ? x3  y @ = ! 2  %o$ "#'o% ! '(#',$&!#!$'("% '!$t#""% !$ !l !>! 96 El %!m!$to !l !>!  y  'om.#!$(o !$t#! lo% !l!'t#oo% !% t"m/(é$ ,$" l$!" ! &l,>o6

9. FLUJO DE UN FLUIDO IDIMENSIONAL S,.o$!mo% ,! !l mo(m(!$to !l &l,(o !% (é$t('o !$ too% lo% .l"$o% ."#"l!lo% "l .l"$o  xy3 %(!$o l" !lo'(" ."#"l!l" " !%! .l"$o ! ($!.!$(!$t! !l t(!m.o6 E$ t"l!% '(#',$%t"$'("%3 !% %,&('(!$t! !%t,("# !l mo(m(!$to ! ,$" '"." !l &l,(o !$ !l .l"$o  xy6 D!$ot"mo% m!("$t! !l !'to# #!.#!%!$t"$t! !l $:m!#o 'om.l!>o < 

=  $ + =

l" !lo'(" ! ,$" ."#t',l" !l &l,(o !$ ',"l,(!# .,$to (x, y)6 A% .,!%3 l"% 'om.o$!$t!%  x ! y ! l" !lo'(" (!$!$ ""%3 #!%.!'t("m!$t!3 .o#  $? x3 y@ + =? x3 y@ 6 E$ .,$to% ($t!#(o#!% " ,$" #!()$ !l &l,(o l(/#! ! &,!$t!% + %,m(!#o%3 l"% &,$'(o$!% #!"l!%  $? x3 y @ 3 =? x3 y@   + %,% !#(""% ."#'("l!% ! .#(m!# o#!$ %! %,.o$!$ 'o$t($,"%6 L" !r!ula!3n !l &l,(o " lo l"#o ! ,$ '"m($o C  %! !&($! 'omo l" ($t!#"l3 'o$ #!%.!'to " l" lo$(t, ! "#'o σ  3 ! l" 'om.o$!$t! t"$!$'("l < .  ? x3 y@  !l !'to# !lo'(" " lo l"#o ! C = ;2

∫  < 

. 

C 

? x3  y @ / σ  

El 'o'(!$t! ! l" '(#',l"'()$ %o/#! C  .o# l" lo$(t, ! C  !%3 .o# t"$to3 ,$" !lo'(" m!(" !l &l,(o " lo l"#o ! C 6 S! %"/! ,! !%t" ($t!#"l %! .,!! !%'#(/(# 'omo=

∫  <  C 

. 

? x3  y @/ σ  =

∫   $? x3  y@/x + =? x3  y@/y C 

S( C  !% ,$" ',#" '!##"" %(m.l!3 o#(!$t"" .o%(t("m!$t!3 !$ ,$ om($(o %(m.l!m!$t! 'o$!9o ! &l,(o l(/#! ! &,!$t!% + %,m(!#o%3 !l t!o#!m" ! #!!$ 4 $o% .!#m(t! !%'#(/(#=

∫   $? x3  y@/x + =? x3  y@/y = ∫∫  [ = C 

 x

 @

]

? x3  y @ −  $ y ? x3  y@ /A

o$! R !% l" #!()$ '!##"" ,! &o#m"$ !l ($t!#(o# ! C  + lo% .,$to% ! C 6 A% .,!%3 ."#" ,$" ',#" ! !%" 'l"%!3

∫

C

< .  ? x3  y @/ σ  =

∫∫  [ =

 x

@

]

? x3  y @ −  $ y ? x3  y@ /A

?;@

E% &1'(l "# ,$" ($t!#.#!t"'()$ &%('" !l ($t!#"$o ! l" !#!'*" !$ ?;@ ."#" l" '(#',l"'()$ " lo l"#o ! l" ',#" '!##"" %(m.l! C 6 S!" C   ,$" '(#',$&!#!$'(" ! #"(o r   '!$t#"" !$ ,$ .,$to ? x 0 3 y 0 @ 3 #!'o##(" !$ %!$t(o .o%(t(o6 L" !lo'(" m!(" " lo l"#o ! C %! '"l',l" (((!$o l" '(#',l"'()$ .o# l" lo$(t, 2π r  ! l" '(#',$&!#!$'("6 L" 'o##!%.o$(!$t! !lo'(" "$,l"# m!(" !l &l,(o !$ to#$o "l '!$t#o ! l" '(#',$&!#!$'(" %! o/t(!$! (((!$o !%" m!(" .o# r = l 

;

∫∫  2 [=

π r 2

 @

 x

]

? x3  y @ −  $ y ? x3  y @ /A

A*o#" /(!$3 é%t" !% t"m/(é$ l" !9.#!%()$ !l "lo# m!(o ! l" &,$'()$ w? x3  y @

= ; [ = x ? x3  y @ −  $ y ? x3 y @]

?2@

2

%o/#! l" #!()$ '(#',l"#  @ "'ot"" .o# C 6 S, lm(t! ',"$o # t(!$! " '!#o !% !l "lo# ! w !$ !l  .,$to ( x0 3 y 0 ) 6 Po# t"$to3 l" &,$'()$ w? x3 y@ 3 ,! %! 'o$o'! 'omo r%ta!3n  !l &l,(o3 #!.#!%!$t" l" !lo'(" "$,l"# lm(t! ! ,$ !l!m!$to '(#',l"# !l &l,(o ',"$o %, '(#',$&!#!$'(" %! 'o$t#"! *"'(" %, '!$t#o ? x3 y @ 3 !l .,$to o$! w !% !"l,""6 S( w? x3  y @ = 0 !$ too .,$to ! "l:$ om($(o %(m.l!m!$t! 'o$!9o3 !l &l,>o !% rr%ta!%nal  !$ !%! om($(o6 A, 'o$%(!#"#!mo% %ol"m!$t! &l,(o% (##ot"'(o$"l!%6 S,.o$#!mo% "!m1% ,! !l &l,(o !% n!%$r#4bl# +  4n v4!%4/a/ 6 B">o $,!%t#" *(.)t!%(% ! &l,>o (##ot"'(o$"l !%t"'(o$"#(o ! ,$ &l,(o3 'o$ !$%(" ,$(&o#m!  ρ  3 %! .,!! !mo%t#"# ,! l" .#!%()$ !l &l,(o  B ? x3 y@ %"t(%&"'! !l %(,(!$t! '"%o ."#t(',l"# ! l" !',"'()$ ! B!#$o,ll( 5= 4

El t!o#!m" ! #!!$ " l" #!l"'()$ !$t#! ,$" ($t!#"l ! l$!" "l#!!o# ! ,$" ',#" '!##"" %(m.l! C  + ,$" ($t!#"l o/l! %o/#! l" #!()$ .l"$"  * l(m(t"" .o# C 6 El t!o#!m" ! #!!$ !% ,$ '"%o !%.!'("l !l m1% !$!#"l t!o#!m" ! Sto!%6 El t!o#!m" "&(#m"= S!" C  ,$" ',#" '!##"" %(m.l! .o%(t("m!$t! o#(!$t""3 (&!#!$'("/l! .o# t#o-o%3 !$ !l .l"$o + %!"  * l" #!()$ l(m(t"" .o# C 6 S( ? + 8  t(!$!$ !#(""% ."#'("l!% 'o$t($,"% !$ ,$" #!()$ "/(!#t" ,! 'o$t(!$!  *3 !$to$'!%

 δ  8 

∫  ?/x +  8/y = ∫∫    δ  x C 

 *

  − δ  ?    /A δ  y  

5

 L" QE',"'()$ ! B!#$o,ll(Q ,! !%'#(/! !l 'om.o#t"m(!$to ! ,$ &l,(o mo(é$o%! " lo l"#o ! ,$" l$!" ! 'o##(!$t! 'o$%t" ! !%to% té#m($o%6

;

 B   ρ 

+

; 2

< 

2

= !%n4 t"$ t#

 N)t!%! ,! l" .#!%()$ !% m19(m" "ll o$! !l m),lo ! l" !lo'("

<  

 !% m$(mo6

S!" * ,$ om($(o %(m.l!m!$t! 'o$!9o !$ !l ,! !l &l,>o !% (##ot"'(o$"l6 S!:$ ?2@3 E%t" #!l"'()$ !$t#! l"% !#(""% ."#'("l!% (m.l('" ,! l" ($t!#"l ! l$!"

 $ y

= = x  !$ *6

∫   $? 43 t @/4 + =? 43 t @/t  C 

" lo l"#o ! ,$ '"m($o C 3 'o$t!$(o !$  *3 ,! ,$" o% .,$to% ( x0 3 y 0 )  + (  x3  y )  ',"l!%,(!#" !  * !% ($!.!$(!$t! !l '"m($o6 A% .,!%3 %( m"$t!$!mo% ( x0 3 y 0 )  &(>o3 l" &,$'()$ φ ? x3  y @

? x 3 y @

= ∫ ? x 3 y 0

0

@

 $ ? 4 3 t @/4

+ = ? 43 t @/t 

?@

!%t1 /(!$ !&($(" !$  *6 Tom"$o !#(""% ."#'("l!% !$ "m/o% l"o% ! !%t" !',"'()$ #!%,lt" φ  x ? x3  y @

=  $ ? x3  y @3

φ  y ? x 3  y @

= = ? x 3 y @

?4@

E$ ?4@ !mo% ,! !l !'to# !lo'(" <  =  $ + =  !% !l #"(!$t! ! φ  6 A%(m(%mo3 l" !#("" (#!''(o$"l ! φ   !$ ',"l,(!# (#!''()$ #!.#!%!$t" l" 'om.o$!$t! ! l" !lo'(" !l &l,>o !$ !%" (#!''()$6 L" &,$'()$ φ ? x3 y@  %! ll"m" $%t#n!al 6 E$ ?@ !% !(!$t! ,! φ ? x3 y@  '"m/(" !$ ,$" 'o$%t"$t! "(t(" ',"$o %! '"m/(" !l .,$to ? x 0 3 y 0 @ ! #!&!#!$'("6 L"% ',#"% ! $(!l φ ? x3  y @ = !; %! ll"m"$ ',#"%  #u$%t#n!al#46 P,!%to ,! !% !l #"(!$t! ! φ ? x3 y@ 3 !l !'to# !lo'(" <  !% $o#m"l " ,$" ',#" !,(.ot!$'("l !$ ',"l,(!# .,$to o$! <  $o %! "$,l!6 I,"l ,! !$ !l &l,>o ! '"lo#3 l" 'o$('()$ ! ,! !l &l,(o ($'om.#!%(/l! !$t#! o %"l" ! ,$ !l!m!$to ! ol,m!$ :$('"m!$t! &l,+!$o " t#"é% !l /o#! ! !%! !l!m!$to !9(! ,! φ ? x3 y@ !/! %"t(%&"'!# l" !',"'()$ ! L".l"'! φ  xx ? x 3  y @ + φ  yy ? x 3 y @

=0

!$ ,$ om($(o !$ !l ',"l $o *"+" &,!$t!% $( %,m(!#o%6 D! ?4@ + ! l" 'o$t($,(" ! l"% &,$'(o$!%  $3  + ! %,% !#(""% ."#'("l!% ! .#(m!# o#!$3 %! %(,! ,! l"% !#(""% ."#'("l!% ! .#(m!# + %!,$o o#!$ ! φ   %o$ 'o$t($,"% !$ !%! om($(o6 E$ 'o$%!',!$'("3 !l .ot!$'("l φ   !% ,$" &,$'()$ "#m)$('"6 v

2

2 g 

+  y +

 B   ρ  g 

= !%n4 t"$ t#

o$!= • • • • •

v  !lo'(" !l &l,(o !$ l" %!''()$ 'o$%(!#""6  g   "'!l!#"'()$ ! l" #"!"  y  "lt,#" !omét#('" !$ l" (#!''()$ ! l" #"!"  B   .#!%()$ " lo l"#o ! l" l$!" ! 'o##(!$t!

  !$%(" !l &l,(o

;4

1!. LA FUNCIÓN DE CORRIENTE Po# !l "."#t"o "$t!#(o# %"/!mo% ,! !l !'to# !lo'(" < 

=  $ ? x3  y @ + = ? x3 y @

 ."#" ,$ om($(o %(m.l!m!$t! 'o$!9o !$ !l ,! !l &l,(o !% (##ot"'(o$"l %! .,!! !%'#(/(# 'omo < 

= φ  x ? x3  y @ + φ  y ? x3  y @ =  gra/ φ ? x3 y @

?;@

o$! φ  !% !l .ot!$'("l6 C,"$o !l !'to# !lo'(" $o !% $,lo3 !% $o#m"l " ,$" ',#" !,(.ot!$'("l ,! ."%" .o# !l .,$to (  x3  y ) 6 S( ψ  ? x3 y@   !$ot" ,$" 'o$>,"" "#m)$('" ! ψ  ? x3 y@ 3 !l !'to# !lo'(" !% t"$!$t! " ,$" ',#" ψ  ? x3  y @ = ! 2 6 L"% ',#"% ψ  ? x3  y @ = ! 2 %! ll"m"$ lín#a4 /# !%rr#nt# !l &l,>o + ψ   %! ll"m" fun!3n /# !%rr#nt# 6 E$ ."#t(',l"#3 ,$ 'o$to#$o " t#"é% !l ',"l $o .,!! ."%"# !l &l,(o !% ,$" l$!" ! 'o##(!$t!6 L" &,$'()$ "$"lt('"  1 ? z @

= φ ? x3  y @ + ψ  ? x3 y @

%! ll"m" $%t#n!al !%$l#5% !l &l,>o6 N)t!%! ,!  1 D ? z @ =φ  x ? x3  y @ + ψ   x ? x3 y @

o %!"3 t!$(!$o !$ ',!$t" l"% !',"'(o$!% ! C",'*+R(!m"$$ K3  1 D ? z @ =φ  x ? x3  y @ − φ  y ? x3 y@

L" !9.#!%()$ ?;@ ."#" l" !lo'(" %! 'o$(!#t!3 .o# t"$to3 !$ <  =  1 D ? z @

S( φ   !% "#m)$('" !$ ,$ om($(o %(m.l!m!$t! 'o$!9o  *3 ,$" 'o$>,"" "#m)$('" ! l" !9.#!%()$ ψ  ? x3  y @

φ   "m(t!

? x 3 y @

= ∫ ? x 3 y @ − φ t  ? 4 3 t @/4 + φ 4 ? 43 t @/t  0

0

o$! l" ($t!#"l !% ($!.!$(!$t! !l '"m($o6 A*o#"3 #"'("% " l"% !',"'(o$!% ?4@ !l "."#t"o "$t!#(o#3 .o!mo% !%'#(/(#  ψ  ? x3  y @

= ∫ C  − = ? 43 t @/4 +  $ ? 43 t @/t 

?2@

K

S!" ,$" &,$'()$ 'om.l!>"  f ? z @3 'o$ z    x  y +  f ? z @ %! .,!! !%'om.o$!# !$ %,m" ! o% &,$'(o$!% #!"l!% ! o% "#("/l!% u + v3 ! m"$!#" ,!  f ? z @   f ? x3 y@   f ? x  y@  u? x3 y@  v? x3 y@6 S( l" &,$'()$  f ? z @ %!" !#("/l! !$ ,$ .,$to  z 0   x0  y0 !$to$'!% !/!$ !#(&('"#%! l"% 'o$('(o$!% ! C",'*+R(!m"$$= u D x ? x0 3  y 0 @

= vD y ? x0 3  y 0 @

v D x ? x0 3  y 0 @ =

−uD y ? x0 3 y 0 @

A!m1% %! ',m.l! ,! !l "lo# ! l" !#("" !$ !l .,$to3 ! !9(%t(#3 !/! %!#=  f ? z 0@  u x? x0 3y0@  v x? x03 y0@  v y? x03 y0@  u y? x03 y0@

;5

o$! C  !% ',"l,(!# '"m($o !$  * ,! " !

? x 0 3 y 0 @  " ? x3 y @ 6

S! .,!! !mo%t#"# ,! !l m(!m/#o ! l" !#!'*" !$ ?2@ #!.#!%!$t" l" ($t!#"l3 'o$ #!%.!'to " l" lo$(t, ! "#'o σ  3 %o/#! C  ! l" 'om.o$!$t! $o#m"l <  ,  ? x3 y@  !l !'to# ',+"% 'om.o$!$t!%  x ! y %o$  $? x3 y@  + =? x3 y@ 3 #!%.!'t("m!$t!6 Po# 'o$%(,(!$t!3 ?2@ %! .,!! !%'#(/(# 'omo ψ  ? x3  y @ =

∫  <  C 

, 

? 43 t @/ σ 

F%('"m!$t!3 ψ  ? x3 y@ #!.#!%!$t" !l #(tmo !l &l,>o " t#"é% ! C 6 Co$ m"+o# .#!'(%()$3 ψ  ? x3 y@ !% !l #(tmo ! &l,>o3 .o# ,$(" ! ol,m!$3 " t#"é% ! ,$" %,.!#&('(! ! "lt,#" ,$(" l!"$t"" %o/#! C  .!#.!$(',l"# "l .l"$o  xy6

11. FLUJOS EN TORNO A UNA ES"UINA # A UN CILINDRO Al "$"l(-"# ,$ &l,>o !$ !l .l"$o  xy3 o .l"$o  z 3 %,!l! %!# m1% %(m.l! 'o$%(!#"# ,$ &l,>o 'o##!%.o$(!$t! !$ !l .l"$o uv3 o .l"$o w6 E$to$'!%3 %( φ   !% ,$ .ot!$'("l + ψ   ,$" &,$'()$ ! 'o##(!$t! ."#" !l &l,>o !$ !l .l"$o uv3 %! .,!! ".l('"# ,! %( !l om($(o  *w  !l .l"$o uv !% l" (m"!$ ! ,$ om($(o  * z   /">o ,$" t#"$%&o#m"'()$ w =   f  ? z @ = u ? x3  y @ + v? x3 y @

o$! f  !% "$"lt('"3 l"% &,$'(o$!% φ [ u ? x 3  y @3 v ? x 3 y @ ]

+ ψ  [ u ? x3  y @3 v ? x3 y @]

%o$ "#m)$('"% !$  * z  6 E%t"% $,!"% &,$'(o$!% .,!!$ ($t!#.#!t"#%! 'omo .ot!$'("l + &,$'()$ ! 'o##(!$t! !$ !l .l"$o  xy6 U$" l$!" ! 'o##(!$t! o /o#! $"t,#"l ψ ?u3 v@ = ! 2 !$ !l .l"$o uv 'o##!%.o$! " ,$" l$!" ! 'o##(!$t! o /o#! $"t,#"l ψ [ u ? x3  y @3 v? x3  y @] = ! 2  !$ !l .l"$o  xy6 U%"$o !%t" té'$('"3 %,!l! %!# m1% !&('"- !%'#(/(# .#(m!#o l" &,$'()$ .ot!$'("l 'om.l!>" ."#" l" #!()$ !l .l"$o w + o/t!$!# ! !ll" !l .ot!$'("l + l" &,$'()$ ! 'o##(!$t! ."#" l" 'o##!%.o$(!$t! #!()$ !$ !l .l"$o  xy6 M"% .#!'(%"m!$t!3 %( l" &,$'()$ .ot!$'("l !$ !l .l"$o uv !%  1 ? w@

= φ ?u 3 v@ + ψ  ?u 3 v@

!$to$'!% l" &,$'()$ 'om.,!%t"  1 [  f  ? z @]

= φ [ u ? x3  y @3 v? x3  y @] + ψ  [ u ? x3  y @3 v? x3 y @]

!% !l .ot!$'("l 'om.l!>o /,%'"o !$ !l .l"$o  xy6

12. ILIOGRAFÍA •

VVV6V((.!("6o#

•

*tt.="'"6,$'o#6!,.,/l('"'(o$!%

J"#("/l! Com.l!>" + A.l('"'(o$!% ! W"m!% X"# B#oV$ + R,!l J6 C*,#'*(ll ?E(to#("l M' #"V Y(ll3 Sé.t(m" E('()$@ •

;K

;