Mecanica de los Medios Continuos Unidad 1.docx

ÍNDICE.

FUNDAMENTOS MATEMÁTICOS. 1.1.

NOTACIÓN IND INDICIAL.

1.2. 1.2.

OPER OPERA ACION IONES DE DE TENS TENSO ORES RES.

1.3. 1.3.

MÉTO MÉTODO DOS S PA PARA EL CÁLC CÁLCUL ULO O DE VALORE LORES SY

VECTORES PROPIOS. 1.4. 1.4.

GRAD GR ADIE IENT NTE, E, DIVE DIVERG RGEN ENCI CIA A Y RO ROT TACION CIONAL AL..

1.5. 1.5.

TEOR TEOREM EMAS AS DE GR GREE EEN N Y STO STOKES. KES.

BIBLIOGRAFA.

1.1.

Notación indicial.

Fundamentos de la mecánica de los medios continuos.

1

E! conve convenio nio de sumaci sumación ón de Einste Einstein in, notac notación ión de Einste Einstein in " notación *#+"/(",, '% '! indicial # !# $"%&'%$()% *+(!(#-# #/# #0/'&(#/ !# '$/(+*/# -' *#+"/(" *' ' */( */(' ' '! 0 0"!" "!" -' *# *#+" +"//(" /' /'/' /''%+# '%+#-" -" $"% $"% !# !'+/ !'+/# # 6/('6# (6# (6# 7 7 Σ8. E! $"%&'%(" 9*' (%+/"-*$(-" "/ A!0'/+ "/ A!0'/+ E(%+'(% E(%+'(% '% '% 1:1;. S' #!($ #!($# # '% #+'<+($# '% '' '$(#! $(#! # !" !" $!$(+#'%+' !" (6%" -' *#+"/(". '>/'()% !(%'#! !(%'#! '% Deinición. D#-# *%# '>/'()%

'% !# *' ' '$/(0'% '$/(0'% +"-" +"-" *

+?/(%" -' 9"/# '>!$(+#@ u=u 1 x 1+ u2 x 2 + … + u n x n

E+# *'-' '>/'#/' $"%&'%$("%#!'%+' $"" '! *#+"/("@ n

u=

u x ∑ = i

i

i 1

L# %"+# %"+#$( $()% )% -' E(% E(%+' +'(% (% "0+( "0+('% '%' ' *%# *%# '>/ '>/' '() ()% % #% #% < < $"%$"%-'% '%# #-# -# '!((%#%-" '! (6%" -' *#+"/(" = '%+'%-('%-" *' '% !# '>/'()% /'*!+#%+' *% %-($' (%-($# *# "0/' +"-" !" "(0!' &#!"/' -'! (". u=u i x i

Índice. U% %-($' *+(!(#-" '% !# %"+#$()% -' E(%+'(% *'-' ##/'$'/ '% 9"/# -' /"-*$+" #+# -" &'$' '% (" +?/(%" -' *%# '>/'()%, "/ !" *' !# (6*('%+' '>/'("%' "% &
C ijk e i e j ek

Y !# (6*('%+' '>/'("%' %" ' '%$*'%+/#% -'9(%(-# " "% (%&
2

am x mj y mk

E% $/'()% '% μ

0

. 1

2

3

a b μ =a b0 + a b 1+ a b2+ a b3

U% %-($' *' ' /'(+' -" &'$' '% '! (" +?/(%" -' *%# '$*#$()% ' $"%"$' $"" !ndice mudo, "/ ''!"@ A = A i e i

U% %-($' *' ' /'(+' '% $#-# *%" -' !" +?/(%" -' *%# '>/'()% # '>$'$()% -' !" +?/(%" $"%+#%+' ' $"%"$' $"" !ndice li"#e@ S r =ar x i+ br x j + c r −1

L" %-($' !(0/' %" ' '>#%-'% '% 9"/# -' *#+"/(", (%" *' /'/''%+#% *% (+'# -' '$*#$("%' (%-''%-('%+'. (%-''%-('%+'.

$e%#es $e%#esent entaci ación ón vec vecto# to#ial ial.. S' '!'# '! $"%&'%(" -' E(%+'(% '% Á!6'0/# !(%'#! #/# #/# -(+(% -(+(%6*( 6*(// 9<$(! 9<$(!'%+ '%+' ' '%+/' '%+/' &'$+"/ &'$+"/' ' $"!*% $"!*%# # = &'$+" &'$+"/' /' 9(!#. 9(!#. S' *'-', "/ ''!", "%'/ %-($' "0/'7'$/(+" #/# /'/''%+#/ '!''%+" '% *%# $"!*%# ' %-($' *0$/(+" #/# /'/''%+#/ '!''%+" '% *%# 9(!#. S(6*('%-" '+# $"%&'%$()%, '%+"%$'@


3

u=u i=[ u1 ,u 2 … un ]

R'/''%+#

1 x n

P#/# i=1,2,3, … , n

&'$+"/ 9(!# = /'/''%+# n x 1 &'$+"/ $"!*%#.

[] 1

V = v

j

=

u u2 n u

P#/#

j = 1,2,3, … , n

E% #+'<+($# #+'<+($# = = '% 9($# +')/($# = +')/($# = '% #/+($*!#/ '% !# /'!#+(&(-#- 6'%'/#!, 6'%'/#! , !" &'$+"/' 9(!# /'/''%+#% &'$+"/' $"%+/#&#/(#%+' $"%+/#&#/(#%+' ('%+/# *' !" &'$+"/' $"!*%# /'/''%+#% &'$+"/' $"&#/(#%+' $"&#/(#%+'..

&. &.'. '.

O%e# O%e#ac acio ione ness de Tenso# nso#es es..

P/"-*$+" P/"-*$+" +'%"/(#! = /"-*$+" /"-*$+" '>+'/("/. '>+'/("/. D#-" D#-" -" +'%"/' ' *'-' *'-' -'9(%(/ -'9(%(/ '%+/' '!!" '! !!##-" /"-*$+" +'%"/(#! $*=" +'%"/(#! $*=" /'*!+#-" ' *% +'%"/ -' +(" < $"!'" $*=# $""%'%+' *'-'% "0+'%'/' # #/+(/ -' !" +'%"/' "/(6(%#!'. E! /"-*$+" -' -" +'%"/' ' *% +'%"/ $*=" /#%6" ' !# *# -' !" /#%6" -#-" "/ !" -" +'%"/'. E+' /"-*$+" (!($# !# *!+(!($#$()% "/-(%#/(# -' !" $""%'%+' -' *% +'%"/ = ' !!##-" /"-*$+" '>+'/("/ . P"/ ''!"@ jk

l

jkl

A i B m=C ℑ

S*0( S*0(// = 0## 0##// %-( %-($' $'. . E% *%# *%# &#/('-#- /('#%%(#%# '>(+' /('#%%(#%# '>(+' !# "(0(!(-#- -' -'9(%(/ *%# "'/#$()% "0/' +'%"/', *' '% 6'%'/#! %" *'-' /'#!(#/' '% *%# &#/('-#- $*#!*('/#. E# "'/#$()% '/(+' *0+(+*(/ '% !" $
"/ "+/" -' +("

$"% +#! *'

. E+#

"'/#$()% ' -'%"(%# **#!'%+' !'= -' *0(/ " 0##/ %-($'. %-($' . Fundamentos de la mecánica de los medios continuos.

4

E# "'/#$()% ' 0## '% !# '>(+'%$(# -' *% (""/9(" (""/9(" '%+/' '%+/' '#$(" -' +'%"/' $" $"&# &#/(# /(#%+ %+' '

=

$"%+ $" %+/# /#&# &#/( /(#% #%+' +' -'9(% '9(%((-" "

/('#%%(#%# " /('#%%(#%# " '*-"/('#%%(#%#

"0/ "0/'

*%# *%# &#/('-#-

. P"/ +#%+" #/# '!'#/, !# *0(-#

= 0##-# -' %-($' ' %'$'#/(" *#/ '! +'%"/ ?+/($"

= * (% (%&'/"

,

!!##-" $"7+'%"/ ?+/($"8. E+# "'/#$("%' /' /'*!+# !+#% *= +(!' '% !# +'"/# 6'%'/#! -' !# /'!#+(&(-#- -"%-' -"%-' $*#!* $*#!*('/ ('/ # #6% 6%(+(+**- 9 9( ($# $# *'*'-' ' '/ '/ /'/ /'/' ''% '%+# +#-# -# "/ "/ +'% +'%"/ "/' ' $"&# $"&#/( /(#% #%+' +' " $"%+ $"%+/# /#&# &#/( /(#% #%+' +' (%-( (%-(+ +(% (%+# +#' '%+ %+', ', = (% (% #!+' #!+'/# /#// '! (6%(9($#-" 9($", '6% !# %'$'(-#-' -'! /"0!'# !#%+'#-".

A #/# $*#!*('/ #6%(+*- 9($# /'/''%+#-# "/ *% +'%"/ -' +'/$'/ /#%6", / #%6", *'-' *'-' '/ /'/' /'/''%+ '%+#-" #-" "/ &#/(" &#/(" $"%*% $"%*%+" +" -' #6%(+ #6%(+*-' *-' /'!#$( /'!#$("%# "%#0!' 0!' 6/#$(# # !# "'/#$()% -' *0(/ = 0##/ %-($'@

C"%+/#$$()%. L# $"%+/#$$()% -' +'%"/' ' +'%"/' ' *%# "'/#$()% *' /'-*$' '! "/-'% +"+#! -' *% +'%"/. E+# "'/#$()% /'-*$' *% +'%"/ +("

( n , m ) # "+/" +("

( n− 1, m−1 ) . E% +?/(%" -' $""%'%+', '+# "'/#$()% ' !"6/# *#%-" '! %-($' -' *% +'%"/ $"%+/#&#/(#%+' = *% $"&#/(#%+'. P"/ ''!", *% +'%"/ 1,18 1,18

*'-' *'-' '/ $"%+/# $"%+/#-" -" # *% '$#!# '$#!#// # +/#&? +/#&? -'

 -"%-' -"%-' '! $"%&'% $"%&'%(" (" -'

*#$()% -' E(%+'(% ' '!'#-". C*#%-" '! +'%"/ 1,18 ' !" (%+'//'+# $"" *% #'" !(%'#!, '+# "'/#$()% ' $"%"$(-# $"" !# +/## +/##.. L# $"%+/#$$()% ' *+(!(# **#!'%+' $"% '! /"-*$+" +'%"/(#! #/# $"%+/#'/ '! %-($' -' $#-# +'%"/. L# $"%+/#$$()% *'-' +#0(?% '%+'%-'/' '% +?/(%" -' !# -'9(%($()% -' *% +'%"/ $"" *% '!''%+" -' *% /"-*$+" +'%"/(#! -' $"(# -'! '#$("

$"% '! '#$("

, -'$""%('%-" -'$""%('%-" /('/" '! +'%"/ '%


5

*%# $"0(%#$()% !(%'#! -' +'%"/' < (!', = "+'/("/'%+' #!($#%-" *% 9#$+"/ -'

# *% 9#$+"/ -'

. P"/ ''!"@

P*'-' '/ '$/(+" $"" !# $"0(%#$()% !(%'#! -'

L# $"%+/#$$()% -'

'% '! /('/" = !+(" '#$(" ' '%+"%$' '! &'$+"/@

P/"-*$ P/"-*$+" +" (%+'/% (%+'/%". ". E! /"-*$+" (%+'/%" -' -" +'%"/' ' /"-*$' #! $"%+/#'/ '! /"-*$+" /"-*$+" '>+'/("/ '>+'/("/ -' !" +'%"/'. +'%"/'. P"/ ''!", ''!", -#-" -#-" -" +'%"/' +'%"/' /"-*$+" '>+'/%" ' (%+'/%"@

I6*#!#%-" %-($',

,

=

*

' "0+('%' '! /"-*$+"

.

1.3. M(todos %a#a el c)lculo de valo#es * vecto#es %#o%ios. E%
=

&#!" &#!"/' /'

/" /"(" (". .

U% '# '#$( $(" "

/" /"(" (",, #*+" #*+"' '# #$( $(" " " '(6' '(6'%' %' #$ #$(" (" '

'! $"%*%+" -' &'$+"/' /"(" $"% *% &#!"/ /"(" $"%.

L# +/#%9"/#$("%' !(%'#!' -'! !(%'#!' -'! '#$(" $"" !# /"+#$()% /"+#$()%,, !# /'9!'>()% /'9!'>()%,, '! '%#%$ '%#%$# #('% ('%+", +", " $*#!* $*#!*('/ ('/ $"0(% $"0(%#$( #$()% )% -' !# #%+'/ #%+'/("/ ("/' ' '% '+# '+# !(+# !(+# "-/#% (%$!*(/' "+/# +/#%9"/#$("%' *'-'% (%+'//'+#/' '-(#%+' '! '9'$+" *' /"-*$'% '% !" &'$+"/' &'$+"/'.. L" &'$+"/' *'-'% &(*#!(#/' $""


6

9!'$#

-'

*%#

$('/+# !"%6(+*- #*%+#%-"

'%

*%#

-(/'$$()%

= '%+(-" '%+(-" -'+'/(%#-". -'+'/(%#-". 

L" vecto#es +/#%9"/ "/#$( #$("%' "%' !(%'#! !(%'#!' ' "% "% &'$+" &'$+"/' /' vecto#es %#o%ios %#o%ios -' !# +/#%9 *', " %" ' &'% #9'$+#-" "/ !# +/#%9"/#$()% " ' &'% *!+(!($#-" "/ *% '$#!#/ , = "/ +#%+" %" &#/#% * -(/'$$()%.



E! valo# %#o%io -' *% &'$+"/ /"(" ' '! 9#$+"/ -' '$#!# "/ '! *' # (-" *!+(!($#-".



U% es%aci 9"/##-" " "/ "/ +"-" +"-" !" !" &'$+ &'$+"/ "/' ' es%acio o %#o%io %#o%io ' *% '#$(" 9"/ /"(" -'! (" &#!"/ /"(", #-'< -'! &'$+"/ %*!", *' %" ' *% &'$+"/ /"(".



multi%licidad +eom(t#ica +eom(t#ica -' *% &#!"/ L# multi%licidad #!"/ /" /"(" ' !# -('%()% -('%()% -'! -'! '#$(" /"(" #"$(#-".



E! es%ect#o -' *%# +/#%9"/#$()% '% '#$(" &'$+"/(#!' 9(%(+" ' '! $"%*%+" -' +"-" * &#!"/' /"(".

P"/ ''!", *% vector propio -' *%# /"+#$()% '% +/' -('%("%' ' *% &'$+"/ (+*#(+*#-" " '% '! '' -' /"+#$()% "0/ "0/' ' '! $*#! $*#! ' /'#! /'#!( (# # !# /"+# /"+#$( $()% )%.. E! valor "%+(' '%' # +"+"-" !" propio $"//'"%-('%+' ' 1 = '! espacio propio $"%+( &'$+"/' #/#!'!" #! ''. C"" ' *% '#$(" -' *%# -('%()%, * multiplicidad geométrica ' *%". E '! %($" &#!"/ /"(" -'! espectro -' '+# /"+#$()%8 *' ' *% %'/" /'#!. /'#!. F"/#!'%+', ' -'9(%'% !" &'$+"/' /"(" = &#!"/' /"(" -' !# (6*('%+' #%' #%'/# /#@@

S( A@ V H H V '

*%

"'/#-"/

!(%'#!

'%

*%

$('/+"

'#$("

&'$+"/(#! V , v ' *% &'$+"/ -(9'/'%+' -' $'/" '% V = = c ' ' *% '$#!#/ +#!' *' A v = c v

E%+"%$' -'$(" *' v ' *% &'$+"/ /"(" -'! "'/#-"/ A, = * &#!"/ /"(" #"$ #"$(# (#-" -" ' c . O0'/&' *' ( v ' *% &'$+"/ /"(" $"% '! &#!"/ Fundamentos de la mecánica de los medios continuos.

7

/"(" c '%+"%$ '%+"%$' ' $*#!* $*#!*('/ ('/ !+( !+(!" !" -(9'/' -(9'/'%+' %+' -' $'/" $'/" -' v ' +#0 +#0(? (?% % *% &'$+"/ /"(" $"% '! &#!"/ /"(" c . D' '$", +"-" !" &'$+"/' /"(" $"% '! &#!"/ /"(" #"$(#-" c *%+" *%+" $"% ,, 9"/#% *% *0'#$(" -' V , '! es%acio

%#o%io #/# '! &#!"/ /"(" c . O0'/&' #-'< *' *' *% *% '#$(" /"(" - ' su"es% es%aci acio o inv inva#i a#ian ante te -' A, ' -'$(/ -#-"  *% &'$+"/ '% -, '! *% su" &'$+"/ A +#0(?% '/+'%'$' # -. '#$ '#$(" (" /"(" /"(" - ' *% su"es%acio

inva#iante -' A, ' -'$(/ -#-"  *% &'$+" $+"/ '% -, '! &'$+"/ A +#0(?% '/+'%'$' # -. !(%'#!' -'! Casos Cas os de inte#e inte#ess es%cia es%cial. l. I%+*(+(&#'%+', #/# !# +/#%9"/#$("%' !(%'#!' -'! '#$( '#$(" " -' -' -" -" -('% -('%(" ("%' %' •

, !" !" &'$+" &'$+"/' /' /"(" /"(" "%@ "%@

R"+#$()%@ %(%6% &'$+"/ /"(" -' &#!"/' /'#!' '>(+'% '% $#0(" #/' &#!"/ /"(", &'$+"/ /"(" $"!'"8.

•

R'9!'>()%@ !" &'$+"/' /"(" "% '/'%-($*!#/' = #/#!'!" #! '' -' ('+/#, !" &#!"/' /"(" "% 71 = 1, /''$+(&#'%+'.

•

E$#!#-" *%(9"/'@ +"-" !" &'$+"/' "% &'$+"/' /"(", = '! &#!"/ /"(" ' '! 9#$+"/ -' '$#!#.

•

P/"='$$()% "0/' *%# /'$+#@ !" &'$+"/' /"(" $"% '! &#!"/ /"(" 1 "% #/#!'!" # !# !%'#, &'$+"/' /"(" $"% '! &#!"/ /"("  "% #/#!'!" # !# -(/'$$()% -' !# /"='$$()%.

C+' '>+'% %# #,, $#" $#" '% '! *' *' ' -'0' -'0' *#/ *#/ *% ?+"-" %*?/($". %*?/($" .


8

C)lculo C)lculo sim"ólico. sim"ólico. U%# '//#('%+# ("/+#%+' #/# '%$"%+/#/ &#!"/' /"(" -' #+/( # +/($' $' $*# $*#-/# -/#-# -# ' '! "!(%"(" $#/#$+'/+($"@ $#/#$+'/+($" @ -' -'$(/ $(/ * *' '  ' *% &#! &#!"/ "/ /" /"( (" " -' A ' ' '*( *(&# &#!' !'%+ %+' ' # -' -'$( $(// * *' ' '! ( (+' +'# # -' '$* '$*#$( #$("%' "%' !(!(%'# %'#!' !' A A &   & 77  A & 7  &   F#$+"/(#%-" "/ & *'-#8 A 7 I8 &   -"%-' I ' !# #+/( (-'%+(-#-8 (-'%+(-#- 8 +('%' +(' %' *%# "! "!*$( *$()% )% %" %* %*!# !# & *% &'$ &'$+"/ +"/ /" /"(" ("8, 8, = -' '+ '+# # 9"/ 9"/# # ' '* '*(&# (&#!'% !'%+' +' #! -'+'/(%#%+' -'+'/(%#%+'@@ det ( A − λI ) )=0

L# 9*%$()% p λ "!(%"(" -' -' λ *' !" -'+'/(%#%+' ' λ8  det ( A − λI ) ' *% "!(%"(" -'9(%'% $"" *# -' /"-*$+". É+' ' '! %olinomio ca#acte#!stico -' A@ !" &#!"/' /"(" -' *%# #+/( "% !" $'/" -' * "!(%"(" $#/#$+'/+($". $#/#$+'/+($" . T"-" !" &#!"/' /"(" -' *%# #+/( A *'-'% $#!$*!#/' /'"!&('%-" !# '$*#$()%@ S( A ' *%# #+/( n × n , '%+"%$' '%+"%$'

+('%' 6/#-" 6/#-" n = A +('%' $""

<>(" n &#!"/' /"(". E! +' +'"/ "/' '# #

9*%9* %-# #'% '%+# +#!!

-'!! -'

*'

'+#

'$*#$()%

+('%'

'>#$+#'%+' n /#$' $'/"8, /#$' $'/"8, +'%('%-" '% $*'%+# * *!+(!($(-#*!+(!($(-#-.. T"-" !" "!(%"(" /'#!' -' 6/#-" (#/ +('%'% *% %'/" /'#! $"" /#, # *' #/# % (#/ +"-# #+/( /'#! +('%' #! '%" &#!"/ /"(" /'#!. E% '! $#" -' !# #+/($' /'#!', #/# n #/ ' (#/, !" &#!"/' /"(" %" /'#!' "% #/' $"%*6#-".. $"%*6#-"


9

Encont#ando vecto#es %#o%ios. U%# &' *' ' $"%"$'% !" &#!"/' /"(" , !" !" &'$+ &'$+"/ "/' ' /" /"(" (" ' *'*'-'% '% #!! #!!#/ #/ /'" /'"!& !&(' ('%%-" " '! (+ (+' '# # -' '$*# '$*#$( $("% "%' ' ""6?%'"@

U%# 9"/# < '%$(!!# -' "0+'%'/ &'$+"/' /"(" (% /'"!&'/ *% (+'# -' '$*#$("%' !(%'#!' ' 0## '% '! +'"/'# -' C#=!'=7#(!+"% *' C#=!'=7#(!+"% *' '+#0!'$' *' $#-# #+/( $*#-/#-# #+(9#$' * /"(" "!(%"(" $#/#$+'/+($". A, (

"% !" &#!"/' /"(" -' A ' $*!' *'

"/ "/ !" *' *' !" !" &'$+ &'$+"/ "/' ' $"!* $"!*% %# # -' /"(" -'

"% "% &'$+" &'$+"/' /'

.

E/em%lo de mat#i0 sin valo#es %#o%ios #eales. U% ''!" -' #+/( (% &#!"/' /"(" /'#!' ' !# /"+#$()% -' : 6/#-" '% '! '%+(-" -' !# #%'$(!!# -'! /'!"@

C*=" "!(%"(" $#/#$+'/+($" '

= * &#!"/' /"(" "% '! #/ -'

$"%*6#-" $"!'" i , 7i . L" &'$+"/' /"(" #"$(#-" +#"$" "% /'#!'.

E/em%lo C"%(-?/'' !# #+/(

*' /'/''%+# *% "'/#-"/ !(%'#! $ H $. S( ' -''# $"*+#/ +"-" !" &#!" &#!"/' /' /" /"(" (" -' A, ' ""-/# /# '' '# #/ -'+' -'+'/ /(% (%#% #%-" -" '! "!(%" !(%"( (" " $#/#$+'/+($"@


10

= "/*' p x 7 28 x 7 18 x  18 ' &' *' !" &#!"/' /"(" -' A "% 2, 1 x 8  7  x x 7 x 7 x  = 71. E! +'"/' +'"/'# # -' C#=!' C#=!'=7# =7#(!+"% (!+"% '+#0! '+#0!'$' '$' *' *' $#-# $#-# #+/( #+/( $*#-/ $*#-/##-# # #+(9#$' * /"(" "!(%"(" $#/#$+'/+($". E -'$(/@

E9'$+(&#'%+', #/# '! $#" -'! &#!"/ /"(" 2, ' *'-' $"/"0#/ *'

D' -"%-' 1, 1, 718 ' *% &'$+"/ /"(" #"$(#-" # 2.

C)lculo C)lculo num(#ico. num(#ico. E% !# /<$+($#, !" &#!"/' /"(" -' !# #+/($' '>+'%# %" ' $#!$*!#% *#%-" '! "!(%"(" $#/#$+'/+($". C#!$*!#/ '! "!(%"(" /'*!+# *= $"+"", = '>+/#'/ !# /#$' '>#$+# -' *% "!(%"(" -' 6/#-" #!+" *'-' '/ -(9$(! -' $#!$*!#/ = '>/'#/@ '! +'"/'# -' A0'!7R*99(%( (!($# A0'!7R*99(%( (!($# *' !# /#$' -' !" "!(%"(" -' 6/#-" #!+" 5 " *'/("/8 %" *'-'% '>/'#/' *<%-"' (!''%+' /#$' '%?(#. E>(+'% #!6"/(+" '9($('%+' #/# #/">(#/ /#$' -' "!(%"(", '/" '*'J" '//"/' '% !# '+(#$()% -' !" &#!"/' /"(" *'-'% -#/ !*6#/ # '//"/' 6/#%-' '% !" &'$+"/' /"(". E% $"%'$*'%$(#, !" #!6"/(+" 6'%'/#!' #/# '%$"%+/#/ &'$+"/' /"(" = &#!"/' /"(" "% (+'/#+(&" (+'/#+(&".. L# #%'/# < 9<$(! ' '! ?+"-" -' !# "+'%$(#@ ' '$"6' *% &'$+"/ #!'#+"/("

= ' $#!$*!# *%# '$*'%$(#

-' &'$+"/' *%(+#/("@ *%(+#/(" @

,

,

, ...


11

E+# *$'()% *$'()% $#( $#( ('/' $"%&'/6'/< # *% &'$+"/ /"(" $"//'"%-('%+' #! #="/ #="/ &#!" &#!"// /" /"(" (".. E+' E+' #!6" #!6"/(+ /(+" " ' '%$ '%$(!! (!!", ", '/" '/" %" -'# -'#( (##-" " +(! +(! #(!# (!#-#'%+'.

S(%

'0#/6",

#=

?+"-"

< <

"*!#/',

$""

!# -'$""($() -'$""($()% % R R,, *' ' 0##% '% ?!.

1.4.

1#adiente2 dive#+encia * #otacional.

$
,

-' *% $#" '$# '$#!#/ !#/ '

'&#!*#-" '% *% *%+" 6'%?/($"

 8, (%-($# (%-($# !# -(/'$$ -(/'$$()% ()% '% !# $*#! $*#! '! $#" $#"

/<(-#'%+' = * * )-*!" )-*!" /'/''%+# '! /(+" -' &#/(#$()% -'

&#/# &#/# <

'% !# -(/'$$()% -(/'$$()%

-' -($ -($" " &'$+ &'$+"/ "/ 6/#6/#-(' ('%+ %+'. '. E! 6/#6/#-(' ('%+ %+' ' ' /'/ /'/' ''% '%+# +# $"% $"% '! "'/ "'/##-"/ "/ -(9'/'%$(#! %#0!#

'6*(-" -' !# 9*%$()% $*(-#-" -' %" $"%9*%-(/ '! 6/#-('%+'

$"% !# -(&'/6'%$(# -(&'/6'%$(#,, ?+# !+(# ' -'%"+# $"% *% *%+" -' /"-*$+" '$#!#/ '%+/' '! "'/#-"/ %#0!# = '! $#"8. T#0(?% *'-' /'/''%+#/' '-(#%+' , " *#% *#%-" -" !# %"+# %"+#$( $()% )% 6/#-('%+' # $#"

. L# 6'%' 6'%'/# /#!(!(# #$( $()% )% -'! -'! $"%$ $"%$' '+" +" -'

&'$+"/(#!' ' '! $"%$'+" -' #+/( Q#$"0(#%#. Q#$"0(#%#.

S( ' +"# $"" $#" '$#!#/ '! *' ' #(6%# # $#-# *%+" -'! '#$(" *%# /'()% P $#" '$#!#/ -' 3 &#/(#0!'8, '%+"%$' '! &'$+"/ 6/#-('%+' '% *% *%+" 6'%?/($" -'! '#$(" (%-($#/< !# -(/'$$()% '% !# $*#! !# /'()% $#0(#/< < /<(-#'%+'. O+/" ''!" ' '! -' $"%(-'/#/ '! ## -' !%'# -' %(&'! -' *%# *%# "%+ "%+# #J# $"" "" $#" " '$# '$#!# !#// *' #(6 (6%# # $#-# #-# #/' #/'# # -' $""/-'%#-# !#+(+*-!"%6(+*- *% '$#!#/ #!+(+*- $#" '$#!#/ -' 2 &#/(#0!'8. E% '+' $#" '! &'$+"/ 6/#-('%+' '% *% *%+" 6'%?/($" (%-($#/< !# -(/'$$()% -' <>( <>(# # (%$! (%$!(% (%#$ #$() ()% % -' !# "%+ "%+#J #J#. #. N)+' N)+'' ' *' *' '! &'$+ &'$+"/ "/ 6/#6/#-(' ('%+ %+' ' '/< '/< '/'%-($*!#/ # !# !%'# -' $"%+"/%" !%'# '*('$#!#/'8 -'! ##. E! 6/#-('%+' ' -'9(%' $"" '! $#" &'$+"/(#! $*=# 9*%$("%' $""/-'%#-# "% !# -'/(&#-# #/$(#!' -'! $#" '$#!#/, '+" '@


12

E+# -'9(%($()% ' 0## '% *' '! 6/#-('%+' '/(+' $#!$*!#/ 9<$(!'%+' !# -'/(&#-# -(/'$$("%#!'. D'9(%('%-" '% /('/ !*6#/ !# -'/(&#-# -(/'$$("%#! '6% *% &'$+"/@

U%# 9"/# '*(&#!'%+' -' -'9(%(/ '! 6/#-('%+' ' $"" '! %($" &'$+"/ *', *!+(!($#-" "/ '! &'$+"/ *%(+#/(", -# !# -'/(&#-# -(/'$$("%#! -'! $#" '$#!#/@

C"% !# -'9(%($()% #%+'/("/, '! 6/#-('%+' '+< $#/#$+'/(#-" -' 9"/# *%&"$#. E! 6/#-('%+' ' '>/'# #!+'/%#+(&#'%+' '-(#%+' '! *" -'! "'/#-"/ %#0!# %#0!#@@

Inte#%#etación del +#adiente. D' 9"/# 6'"?+/($# '! 6/#-('%+' ' *% &'$+"/ *' ' '%$*'%+/# %"/#! '/'%-($*!#/8 # !# $*/&# -' %(&'! '% '! *%+" *' ' '+<

( x , y ) , ( x , y , z ) , +(' +('", ", +'' +''/#+ /#+*/# */#8, 8, '+$?+ '+$?+'/# '/#.. A!6*% A!6*%" "

'+* '+*-( -(#% #%-" -",, !!< !!<' '' ' ''!" "%@



C"%(-'/' *%# #0(+#$()% '% !# $*#! !# +''/#+*/# ' -'9(%' # +/#&? -' *% $#" '$#!#/, -' +#! #%'/# *' '% $*#!*('/ *%+" +''/#+*/# '

∅

∅

( x , y , z ) , !#

( x , y , z ) . A*(/'" *' !# +''/#+*/# %" &#/# $"%

/''$+" #! +('". S('%-" '+" #, #/# $#-# *%+" -' !# #0(+#$()%, '! 6/#-('%+' '% '' *%+" %" -#/< !# -(/'$$()% '% !# $*#! ' $#!('%+# < /<(-". L# #6%(+*- -'! 6/#-('%+' %" -(/< $*<% /<(-" ' $#!('%+# '% '# -(/'$$()%.


13

C"%(C"%(-'/' '/' *%# "%+#J# "%+#J# '% !# $*#! $*#! * #!+*/# #!+*/# '% '! *%+"



-'9( -'9(%' %' $"" $""

H ( x , y ) .

( x , y )

'

E! 6/#-('%+ 6/#-('%+' ' -' H '% '' *%+" '+#/< '% !#

-(/'$$()% #/# !# *' #= *% #="/ 6/#-" -' (%$!(%#$()%. L# #6%(+*- -'! 6/#-('%+' %" "+/#/< $*<% '(%#-# ' '%$*'%+/# !# '%-('%+' '%-('%+'..

3#o%iedades. E! 6/#-('%+' &'/(9($# *'@ 

E "/+"6"%#! # !# *'/9($(' '*('$#!#/', -'9(%(-# "/

$+'.



A*%+# '% !# -(/'$$()% '% '% *' !# -'/(&#-# -(/'$$("%#! ' -(/'$$("%#! ' <>(#.



S* )-*!" ' (6*#! # '+# -'/(&#-# -(/'$$("%#! <>(#.



S' #%*!# '% !" *%+" '+#$("%#/("  <>(" <>(",, %(" %(" = = *%+" -' (!!#8. (!!# 8. E! $# #" 9"/ 9"/# #-" "/ '! 6/#6/#-(' ('%+ %+' ' '% $#-# *%+ *%+" " ' (' ('/' /'



(//"+#$("%#!, '+" ',

Dive#+encia. L# dive#+encia -' *% $#" &'$+"/(#! (-' !# -(9'/'%$(# '%+/' '! 9!*" '%+/#%+' = '! 9!*" #!('%+' -' *% $#" &'$+"/(#! "0/' !# *'/9($(' *' /"-'# # *% &"!*'% -' $"%+/"!, "/ +#%+", ( '! $#" +('%' 9*'%+' " *(-'/" !# -(&'/6'%$(# -' -($" $#" '/< -(9'/'%+' -' $'/".

Dive#+encia de un cam%o vecto#ial. L# -(&'/6'%$(# -' *% $#" &'$+"/(#! '% &'$+"/(#! '% *% *%+" ' *% $#" '$#!#/ , = ' -'9(%' $"" '! 9!*" 9!*" -'! -'! $#" &'$+"/(#! "/ *%(-#- -' &"!*'% $"%9"/' '! &"!*'% #!/'-'-"/ -'! *%+" +('%-' # $'/"@

D"%-' D"%-' 0"!"

' *%# *%# *'/9($(' *'/9($(' $'//#-# $'//#-# *' ' /'-*$' /'-*$' # *% *%+" *%+" '% '! '! !(+'. !(+'. E! /'/''%+# '! "'/#-"/ %#0!# "'/#-"/ %#0!#..


14

E+# -'9(%($()% '+< -(/'$+#'%+' /'!#$("%#-# $"% '! $"%$'+" -' 9!*" -'! $#". C"" '% '! $#" -'! 9!*", ( !# -(&'/6'%$(# '% *% *%+" ' "(+(&#, ' -($' *' '! $#" "'' fuentes . S( !# -(&'/6'%$(# ' %'6#+(&#, ' -($' *' +('%' sumideros. E! ''!" < $#/#$+'/+($" !" -#% !# $#/6# '!?$+/($#, *' -#% !# -(&'/6'%$(# -'! $#" '!?$+/($", ('%-" !# $#/6# "(+(&# #%#%+(#!' = !# %'6#+(&# *(-'/" -'! $#" '!?$+/($". S' !!##% fuentes escalares -'! $#"

#! $#" '$#!#/ *' ' "0+('%' # #/+(/ -' !# -(&'/6'%$(# -'

L# -(&' -(&'/6 /6'% '%$( $(# # -' *% $# $#" " &'$+ &'$+"/ "/(# (#!! ' /'!# /'!#$( $("% "%# # $"% $"% '! 9!*" 9!*" # +/#& +/#&? ? -'! +'"/'# -' G#* " G#* " +'"/'# -' !# -(&'/6'%$(#. -(&'/6'%$(# . C""/-'%#-# $#/+'(#%#. C*#%-" !# -'9(%($()% -' -(&'/6'%$(# ' #!($# #! $#" -' *% $#" '>/'#-" '% $""/-'%#-# $#/+'(#%#,

E! /'*!+#-" ' '%$(!!"@

C""/-'%#-# "/+"6"%#!'. S(% '0#/6", #/# *% $#" < 6'%'/#! -' $""/-'%#-# "/+"6"%#!' $*/&(!%'#, "/+"6"%#!' $*/&(!%'#, $"" !# $(!%-/($# " !# '9?/($#, !# '>/'()% ' $"!($# -'0(-" # !# -''%-'%$(# -' !" &'$+"/' -' !# 0#' $"% !# "($()%. L# '>/'()% #/# *% (+'# -' $""/-'%#-# "/+"6"%#!' "/+"6"%#!' '@

D"%D"%-' ' !" !"

"% "% !" !" 9# 9#$+ $+"/ "/' ' -' ' '$# $#!# !# -'! -'! (+ (+' '# # -' $""/ $""/-' -'%# %#-# -#, ,

/'!#$("%#-" $"% !# 9"/# -'! +'%"/ ?+/($" '% ?+/($" '% -($" (+'# -' $""/-'%#-#. Fundamentos de la mecánica de los medios continuos.

15

E+# E+# 9)/ 9)/*! *!# # 6'%'/# %'/#!, !, #/# #/# '! $#" $#" -' $"" $""/-'% /-'%##-# # $#/+' #/+'(#% (#%#  8 ' /'-*$' # !# '>/'()% #%+'/("/. P#/# $""/-'%#-# $(!%-/($#  $(!%-/($# 

8 /'*!+#@

P#/# $""/-'%#-# '9?/($#  '9?/($# 

C""/-'%#-#

6'%'/#!'.

8 /'*!+#

E%

(+'#

-'

$""/-'%#-#

6'%'/#!',

%"

%'$'# %'$'#/(# /(#'% '%+' +' "/+"6 "/+"6"%# "%#!' !',, !# -(&'/ -(&'/6' 6'%$( %$(# # -' *% &'$+"/ &'$+"/ *'-' *'-' '>/' '>/'#/ #/' ' '% +?/(%" -' !# -'/(&#-# #/$(#!' /''$+" # !# $""/-'%#-# = '! -'+'/(%#%+' -'! +'%"/ ?+/($"@ ?+/($"@

&'$+"/(#! , '! #otacional " #otacional " #oto# ' ' *% "'/#-"/ &'$+"/(#! *' &'$+"/(#! *' $otacional. E% '! $
M#+'<+($#'%+', '+# (-'# ' '>/'# $"" '! !(+' -' !# $(/$*!#$()% $(/$*!#$()% -'! -'! $#" &'$+"/(#!, $*#%-" !# $*/&# "0/' !# *' ' (%+'6/# ' /'-*$' # *% *%+"@


16

A*,

' '!
*%+". E! /'*!+#-" -' '+' !(+' %" ' '! /"+#$("%#! $"!'+" *' ' *% &'$+"/8, &'$+"/8, (%" "!" * $""%'%+' $""%'%+' '6% '6% !# -(/'$$()% -(/'$$()% %"/#! %"/#! #

= "/('%+#-# "/('%+#-#

'6% !# /'6!# -' !# #%" -'/'$#. P#/# "0+'%'/ '! /"+#$("%#! $"!'+" -'0' '0'/<% /<%

$#!$* #!$*!# !#/ /' '

+/' +/' !( !(+' +',

$"% "%(-'/ (-'/# #%-" %-"

+/' +/' $*/& */&#

(+* (+*##-# #

'% !#%" '/'%-($*!#/' '/'%-($*!#/'.. F*'%+' F*'%+' &'$+"/ &'$+"/(#! (#! = '$#!# '$#!#/. /. A! $#" &'$+"/(#!, Q, *' ' "0+('%' $#!$*!#%-" '! /"+#$("%#! -' *% $#" F '% $#-# *%+",  = ∇ × !

S' $"%"$' $"" !# uentes vecto#iales -' F ('%-" !# 9*'%+' '$#!#/' !# *' ' "0+('%'% '-(#%+' !# -(&'/6'%$(# -(&'/6'%$(#8. 8. U% $# $#" " $*= $*=" /"+# /"+#$( $("% "%#! #! ' %*!" %*!" '% +"-" +"-" !" !" *%+ *%+" " -'! -'! '# '#$( $(" " ' -'%"(%# i##otacional " ' -($' *' $#/'$' -' 9*'%+' &'$+"/(#!'. Y ( '+< -'9(%(-" "0/' *% -"(%(" (!''%+' $"%'>" '%+"%$' $"%'>" '%+"%$' -($" $#" *'-' '>/'#/' $"" '! 6/#-('%+' 6/#-('%+' -' -' *%# 9*%$()% '$#!#/, " -($" -' "+/# 9"/#, '! $#" -'/(&# -' *% "+'%$(#!@

E>/'()% '% 9"/# $#/+'(#%#. P#/+(' P#/+('%-" %-" -' !# -'9(% -'9(%($( ($()% )% '-(#% '-(#%+' +' *% !(+', !(+', *'-' *'-' -'"+ -'"+/#/ /#/' ' *' !# '>/'()%, '% $""/-'%#-# $#/+'(#%#, $#/+'(#%# , -'! /"+#$("%#! '

*'

'

*'-'

'>/'#/

"'/#-"/ %#0!# $""

*%

-'

9"/#

/"-*$+"

<

$"%$(#

&'$+"/(#!,

$"%

$#!$*!#0!'

#=*-#

-'!

'-(#%+'

*% -'+'/(%#%+' -'+'/(%#%+'@@


17

D'0' +'%'/' *= /''%+' *' -($" -'+'/(%#%+' '% /'#!(-#- %" ' +#! *' !" '!''%+" -' !# '6*%-# 9(!# %" +('%'% #/6*'%+" = "/ +#%+" $#/'$'% -' '%+(-". A-'< -($" -'+'/(%#%+' )!" *'-' -'#//"!!#/' "/ !# /('/# 9(!#. E% -'9(%(+(&#, !# %"+#$()% '% 9"/# -' -'+'/(%#%+' (/&' #/# /'$"/-#/ 9<$(!'%+' !# '>/'()% -'! /"+#$("%#!. E% !# %"+#$()% -' E(%+'(%, E(%+'(% , $"% '! 0"!" -' L'&(7C(&(+# ' L'&(7C(&(+# ' '$/(0' $""@

3#o%iedades4 T"-" "-" $#" potencial '>/' '>/'#0 #0!' !' $"" $"" '! 6/#-('%+' -' *% "+'%$ "+'%$(#! (#! '$#!#/8 ' irrotacional = = &($'&'/#, '+" ',

•

T"-" $#" $'%+/#! /#-(#! $'%+/#! /#-(#! = -''%-('%+' )!" -' !# -(+#%$(# #! $'%+/"8 ' (//"+#$("%#!.

E% #/+($*!#/ #/+($*!#/,, '! $#" $#" '!'$+/"+< '!'$+/"+<+($" +($" -' *%# $#/6# *%+*#! *%+*#! = "/ *'/"($()%, $*#!*('/ $#" '!'$+/"+<+($"8 ' (//"+#$("%#!. •

E! /"+#$("%#! -' *% $#" &'$+"/(#! ' ('/' *% $#" solenoidal , '+" ', * -(&'/6'%$(# -(&'/6'%$(# ('/' ('/' ' %*!#@

E'!"@


18

E% *% +"/%#-" !" !" &('% &('%+" +" '+< '+<% % /"+# /"+#%%-" " "0/ "0/' ' '! "", "", = *% $# $#" "



&'$+" &'$+"/(#! /(#! *' *' *'+/ *'+/# # !# &'!"$ &'!"$(-# (-#-' -' -'! &('%+" &('%+" +'%-/ +'%-/# # *% /"+#$ /"+#$("% ("%#! #! -(9'/'%+' -' $'/" '% '! "", = "(0!''%+' '% "+/# #/+' &?#' &"/+($(-#&"/+($(-#-8. 8. E% *% $#" &'$+"/(#! *' -'$/(0# !# &'!"$(-#-' !(%'#!' -' $#-#



#/+' (%-(&(-*#! -' *% -($" *' /"+#, '! /"+#$("%#! +'%-/< *% &#!"/ $"%+#%+' '% +"-# !# #/+' -'! -($". S( *%# #*+"(+# 9*'/# -'$/(+# $"% *% $#" &'$+"/(#!, = !" $#//(!'



+*&('/#% -(&'/" !(+' -' &'!"$(-#-, '! /"+#$("%#! '% !# 9/"%+'/# '%+/' !" $#//(!' '/# -(9'/'%+' -' $'/". L# !'= -' F#/#-#= -' F#/#-#= -' !# (%-*$$()% = !# !'= -' A/'7M#>'!!, -" -'



!# '$*#$("%' -' M#>'!!, M#>'!! , ' *'-'% '>/'#/ *= (!''%+' *#%-" '! /"+#$("%#!. L# /('/# (%-($# *' '! /"+#$("%#! -' *% $#" '!?$+/($" ' (6*#! # !# +## -' &#/(#$()% -' !# -'%(-#- -'! 9!*" #6%?+($", $"% (6%" "*'+" -'0(-" # !# L'= -' L'% L'%  !# '6*%-# (%-($# *' '! /"+#$("%#! -' *% $#" #6%?+($" ' (6*#! # !# *# -' !# -'%(-#- -' $"//('%+' = !# -'/(&#-# +'"/#! -' !# -'%(-#- -' 9!*" '!?$+/($". '!?$+/($" .

&. &.5. 5.

Teo#em o#emas as de 1#ee 1#een n * de de Sto Sto6e 6es. s.

1#een -# !# /'!#$()% 9($# = = #+'<+($# #+'<+($#,, '! teo#ema de 1#een -# Teo#em Teo#emaa de 1#een. 1#een. E% 9($# '%+/' *%# (%+'6/#! -' !%'# !%'# #!/'-'-"/ -' *%# $*/&# $'//#-# (!' C = = *%# (%+'6/#! -"0!' "0/' -"0!' "0/' !# /'6()% !#%# D !((+#-# "/ C . E! +'"/'# -' G/''% ' !!## # "/ '! $('%+9($" 0/(+<%($" G'"/6' G/''% = G/''% = ' *% $#" ''$(#! -'! < 6'%'/#! +'"/'# -' S+"'.. E! +'"/'# #9(/#@ S+"'

S'# C *%# $*/&# $*/&# $'//#$'//#-# # (!' (!' "(+(& "(+(&#' #'%+' %+' "/('%+ "/('%+##-#, #, -(9'/' -(9'/'%$( %$(#0! #0!' ' "/ +/"", '% '! !#%" = '# D !# /'6()% !((+#-# "/ C . S( L = M +('%'% +('%'% -'/(&#-# #/$(#!' $"%+(%*# #/$(#!' $"%+(%*# '% *%# /'6()% #0('/+# *' $"%+('%' D,


19

A &'$' &'$' !# %"+#$()%

S' *+(!(# #/# '+#0!'$'/ *' !# (%+'6/#! -' !%'# '+< $#!$*!#-# *#%-" !# "/('%+#$()% "(+(&# #%+("/#/(#  #%+("/#/(#88 -' !# $*/&# $'//#-# C . R'!#$()% $"% '! +'"/'# -' !# -(&'/6'%$(#@ E! +'"/'# -' G/''% ' '*(&#!'%+' # !# (6*('%+' #%#!"6# 0(-('%("%#! -'! +'"/'# -' S+"'@ S+"' @

D"%-'

' '! &'$+"/ %"/#! %"/#! #!('%+' '% !# 9/"%+'/#.

P#/# &'/ '+", $"%(-'/' !# *%(-#- %"/#! '% !# #/+' -'/'$# -' !# '$*#$()%. C""

' *% &'$+"/ #*%+#%-" +#%6'%$(#!'%+' # +/#&? -' *%#

$*/&#, = !# $*/&# C '+< "/('%+#-# -' #%'/# "(+(&# ' -'$(/, '% $"%+/# -'! '%+(-" -' !# #6*# -'! /'!"8 # +/#&? -' !# 9/"%+'/#, *% &'$+"/ %"/#! #!('%+' '/# #*'! *' #*%+# '% : #$(# !# -'/'$#, '! $*#! "-/# '/

.

E! )-*!" -' '+' &'$+"/ '@ . P"/ !" +#%+"@ . T"#%-" !" $""%'%+' -'@

E! !#-" -'/'$" ' $"%&('/+' '%@


20

*' "/ '-(" -'! +'"/'# -' G/''% /'*!+#@

-(9'/'%$(#! ' *%# Teo#ema o#ema de Sto6es Sto6es.. E! +'"/'# -' S+"' '% 6'"'+/# -(9'/'%$(#! ' /" /"" "($() ($()% %

"0/' "0/'

!# (%+'6/#$()% (%+'6/#$()% -' -' 9"/ 9"/# #

-(9'/' -(9 '/'%$( %$(#!' #!' *' *'

6'%' 6'%'/# /#!( !(# #

&#/(" +'"/'# +'"/'# -'! -'! $
9*%9* %-# #'% '%+# +#!!

-'!! -'

'! (%+'/&#!" (%+'/&#!"  a, bZ

$
'/

*'

!# (%+'6/#! -'

$#!$*!#-#

"/

'-("

*%# -'

*%# #%+(-'/(&#-# F -' -' f @

E! +'"/'# -' S+"' ' *%# 6'%'/#!(#$()% -' '+' +'"/'# '% '! (6*('%+' '%+(-"@



P#/#

!# F '!'6(-#,

.

E%

'!

!'%6*#'

-'

!# 9"/#

-(9'/'%$(#!' ' -(9'/'%$(#!' ' -'$(/ *' f  x ' !# -'/(&#-# '>+'/("/ -' -' !# 79"/# $"" x 8 dx ' "/ ''!" *%# 9*%$()%8 F @ dF = f dx . E! +'"/'# 6'%'/#! -' S+"' #!($# #/# 9"/# -(9'/'%$(#!' #="/'

'% &' -' F .


21



E% *% !'%6*#' #+'<+($", '! (%+'/&#!" #0('/+"  a, b8 ' *%# &#/('-##+'<+($# *%(-('%("%#!. S* 9/"%+'/# ' '! $"%*%+" *' $"%(+' '% !" -" *%+" a = b. I%+'6/#/ f '% '% '' (%+'/&#!" *'-' '/ 6'%'/#!(#-" $"" (%+'6/#/ 9"/# '% *%# &#/('-#- #+'<+($# -' #="/ "/-'%. P#/# '+" ' %'$'( %'$'(+#% +#% -" -" $"%-($ $"%-($("% ("%' ' +?$%($ +?$%($#@ #@ !# &#/('&#/('-##- #+'< #+'<+($ +($# # -'0' -'0' '/ "/('%+#0!', = !# 9"/# +('%' *' '/ $"#$+# -' #%'/# *' "+"/6*' *%# (%+'6/#! 0('% -'9(%(-#.



L" -" *%+" a = b 9"/#% #/+' -' !# 9/"%+'/# -'! (%+'/&#!" #0('/+". M< M< 6'%? 6'%?/( /($# $#' '%+ %+', ', '! +'"/ +'"/' '# # -' S+" S+"' ' ' #!( #!($# $# # &#/( &#/(''-##-' ' "/('%+#-# M $"% $"% 9/"%+'/#. L# 9/"%+'/# [ M -' -' M ' ' *%# &#/('-#- '%  (# = '/'-# !# "/('%+#$()% %#+*/#! -' M . P"/ ''!", !# "/('%+#$()% %#+*/#! -'! (%+'/&#!"

-#

*%#

"/('%+#$()%

-'

!"

-"

*%+"

9/"%+'/#.

I%+*(+(&#'%+' a'/'-# !# "/('%+#$()% "*'+# # b, #! '/ '>+/'" "*'+" -'! (%+'/&#!". (%+'/&#!". E%+"%$', E%+"%$', (%+'6/#%-" (%+'6/#%-" F '% !" !" -" -" *%+ *%+" " 9/"% 9/"%+' +'/# /# a, b ' '*(&#!'%+' # +"#/ !# -(9'/'%$(# F b8 \ F a8. P"/ !" *' '! +'"/'# 9*%-#'%+#! /'!#$("%# !# (%+'6/#! -' *%# 9*%$()% "0/' *% (%+'/&#!", $"% *%# (%+'6/#! " *# -' !# /((+(&# -' !# 9*%$()% '% !" !(+' *' '%$('//#% -($" (%+'/&#!"@

P"/ "+/" !#-" '! +'"/'# -' G/''% #$' G/''% #$' #!6" ((!#/ '% -" -('%("%', /'!#$("%# !# (%+'6/#! # !" !#/6" -' *%# $*/&# (!' $"% !# (%+'6/#! -' *%# $"0(%#$()% -' -'/(&#-# "0/' *%
S((!#/'%+' '! +'"/'# -' !# -(&'/6'%$(# /'!#$("%# -(&'/6'%$(# /'!#$("%# !# (%+'6/#! -' *%# 9*%$()% "0/' *%# *'/9($(' $"% !# (%+'6/#! -' *%# $"0(%#$()% -' -'/(&#-# "0/' '! (%+'/("/ -'! $"%*%+"@


22

E! +'"/' +'"/'# # -' S+"' S+"' 6'%'/# 6'%'/#!( !(# # +"-" +"-" '+" '+" /'*!+ /'*!+#-" #-", , /'!#$( /'!#$("%# "%#%-" %-" !# (%+'6/#! "0/' *%# 9/"%+'/# $"% !# (%+'6/#! -' *%# 9*%$()% -'/(&#-# "0/' '! (%+'/("/ -' !# /'6()% !((+#-# "/ !# 9/"%+'/#.

Fo#mulación 1ene#al. S'# M *%# *%# &#/('-#- -' -('%()% n -(9'/'%$(#0!' "/ +/"" "/('%+#-# $"#$+# = '# ] *%# 9"/# -(9'/'%$(#! '% M -' -' 6/#-" n71 = -' $!#' C. S( [ M -'%"+# -'%"+# '! !(+' -' M $"% $"% * "/('%+#$()% (%-*$(-#, (%-*$(-#, '%+"%$'

#* d ' ' !# -'/(&#-# '>+'/("/ , *' ' -'9(%' *#%-" "!#'%+' !# '+/*$+*/# -' &#/('-#-. E! +'"/'# -'0' '/ $"%(-'/#-" $"" 6'%'/#!(#$()% -'! +'"/'# 9*%-# 9*% -#'%+ '%+#! #! -' -'!! $+( '>+('% '%-' -' 9<$( 9<$(! !'% '%+' +' # !# !# $"0 $"0(%#$( (%#$("%' "%' !(%'# !(%'#!' !' -' !# !# *0&#/('-#-' -(9'/'%$(#0!' "/ +/"", +/"" , !#, # !!##-#, $#-'%#. E! +'"/'# -'

S+"'

-'*'+/#

'%+"%$'

*'

!#

9"/#

$'//#-#

-'9(%(-# )-*!" )-*!" *%# *%# 9"/ 9"/# # '> '>#$+ #$+# # ' *'*'-'% '% (%+'6 (%+'6/# /#// "0/ "0/' ' !# !# $#-' $#-'%# %# -'9(%(-# )-*!" 0"/-' 0"/-'.. É+# ' !# 0#' #/# '! ##/'#('%+" '%+/' !" 6/*" -' ""!"6# = ""!"6# = !# $"""!"6# -' -' R#. R# .


23

El cl)sico teo#ema de 7elvin8Sto6es4

*' /'!#$("%# !# (%+'6/#! -' *'/9($(' -'! /"+#$("%#! /"+#$("%#! -'! -'! $#" &'$+"/(#! "0/' &'$+"/(#! "0/' *%# *'/9($(' ^ '% '! 37'#$(" '*$!(-(#%" # !# (%+'6/#! -' !%'# -'! $#" &'$+ &'$+"/ "/(# (#!! "0/ "0/' ' * 0"/0"/-', ', ' *% $#" $#" '' ''$( $(#! #! -'! -'! +'"/ +'"/' '# # -' S+" S+"' ' 6'%'/#!(#-" $"% n  28 *%# &' *' (-'%+(9(*'" '! $#" &'$+"/(#! $"% *%# 179"/# *#%-" !# ?+/($# '% '! 37'#$(" '*$!(-(#%". A((" '! +'"/'# -' O+/"6/#-=7G#* O+/"6/#-=7G#* " T'"/'# -' !# -(&'/6'%$(#@ -(&'/6'%$(# @

E *% $#" ''$(#! ( (-'%+(9($#" *% $#" &'$+"/(#! $"% !# n71 9"/# "0+'%(-# $"%+/#='%-" '! $#" &'$+"/(#! $"% !# 9"/# -' &"!*'% '*$!(-(#%". E! +'"/'# 9*%-#'%+#! -'! $
D"%-'

' *% $#" &'$+"/(#! $*#!*('/#.


24

E+#0!'$'

*'

!#

(%+'6/#!

-'

*'/9($(' -'! /"+#$("%#! -'

*% $#"

&'$+"/(#! "0/' &'$+"/(#! "0/' *%# *'/9($(' #0('/+# ' (6*#! # !# (%+'6/#! $*/&(!%'#8 $'//#-# -'! $#" &'$+"/(#! # &'$+"/(#! # !" !#/6" -'! $"%+"/%" *' !((+# !# *'/9($('.

9i"lio+#a!a

F*'%+' V(/+*#!'@ ++@'.(('-(#."/6((C_C3_A1!$*!"`+'%"/(#! ++@3.'$#%($#.*.'$7(6A*%+'(%-($'.-9 ++@'.(('-(#."/6((V'$+"/`/" ++@'.(('-(#."/6((V '$+"/`/"("`=`&#!"/`/"(" ("`=`&#!"/`/"(" ++@'.(('-(#."/6((D(&'/6'%$(#`#+'<+($#8 ++@'.(('-(#."/6((G/#-('%+' ++@'.(('-(#."/6((T'" ++@'.(('-(#."/6((T '"/'#`-'`G/''% /'#`-'`G/''%


25

++@'.(('-(#."/6((T'" ++@'.(('-(#."/6((T '"/'#`-'`S+"' /'#`-'`S+"'


26

Mecanica de los Medios Continuos Unidad 1.docx

Recommend Documents