notacion indical

8/19/2019 NOTACION INDICAL

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NOTACION INDICIAL, OPERADORES DE TENSORES, TEOREMA

DE GREEN Y TEOREMA DE STOKES.

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INDIC

NOTACION INDICIAL................................................................................................1

OPERADADORES DE TENSORES..........................................................................3TEOREMA DE GREEN.............................................................................................5

TEOREMA STOKES..................................................................................................7

BIBLIOGRAFIA..........................................................................................................9

PAGINAS CONSULTADAS:.......................................................................................9

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NOTACION INDICIAL

En l !"#$n%# &"l !"&%' #'n(%n)', l n'(#%*n %n&%#%l + l #'n"n#%*n &" -)!

&" E%n-("%n "-(l"#"n )n /'0! 0"%& &" 0"0"-"n(#%*n &" #!'- +'"0#%'n"-, "#('0%l"- + ("n-'0%l"-. Tl n'(#%*n '!%(" l'- -%n'- &" -)!('0%

-'0" #& n&%#" 0""(%&' "n )n "40"-%*n, &"n&' %!l#%( l -)! &" l'-

0'&)#('- "n(0" l'- #'!'n"n("- &"l "#('0 ' &"l ("n-'0.

L- #'!'n"n("- &" l- #n(%&&"- "#('0%l"- -" %n&%#n #'n l- l"(0- "n

!%n6-#)l #)0-% #'!&- &" )n -)n&%#", '0 ""!l' ai , b j , c k , etc. En

#!%', l- #'!'n"n("- &" l'- ("n-'0"- -" 0"0"-"n(n !"&%n(" l"(0- "n

!+6-#)l #)0-% #'!&- &" 0%'- -)n&%#"- &" #)"0&' #'n "l '0&"n

&"l ("n-'0. P'0 ""!l', l- #'!'n"n("- &" )n ("n-'0 &" -")n&' '0&"n -"

)"&"n "40"-0 #'!' Aij , Bkl, C rs, etc.

S% -" 0"8)%"0" 0"0"-"n(0 )n #'n)n(' &" 9 #n(%&&"-, -" )(%l% &'l" -)n&%#"

B%, "n &'n&" !'- -)n&%#"- i y j ('!n l'- l'0"- 1, 2 + 3. S% "l #'n)n(' (%"n"

27 #n(%&&"-, -" n"#"-%(n (0"- -)n&%#"- D %,. F%nl!"n(", -% -" (%"n"n 3n

#n(%&&"- -" )(%l%n n -)n&%#"- D%l;n.

E4%-("n &'- (%'- &" -)n&%#"-:

< =n&%#" l%0": "- 8)"l 8)" 0"#" -'l' )n " + ('! l'- l'0"- 1,2 + 3.

El n6!"0' &" (>0!%n'- 8)" "4%-("n "n l "40"-%*n "- 3 n "n &'n&" n "- "l

n6!"0' &" n&%#"- l%0"-.< =n&%#" !)&': 0"#" &'- "#"- " %n&%# )n -)! &" 1 3 l' 8)"

#'n-(%()+" "l #'n"n%' &" l -)! ' &" E%n-("%n.

pág. 1


5/13

En "l -%)%"n(" #)&0' -" 0"-)!"n l- '"0#%'n"- !$- #'!)n"- "n l"0 +

#l#)l' ("n-'0%l. En ('& l (l "- )n /)n#%*n "-#l0, , , # -'n "#('0"- +ᶲ

R,S, T -'n ("n-'0"- &" '0&"n &'-.

Tabla N° 1: Operaciones más comunes en algebra y calculotensorial y sus expresiones en notación inicial!

O"#$ADADO$#% D# T#N%O$#%

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P0 l'- ("n-'0"- -" &"/%n"n '"0#%'n"- &" &%#%*n, -)-(0##%*n + 0'&)#('. En "l

#-' &" l &%#%*n + -)-(0##%*n "l 0n' &" l'- ("n-'0"- %n'l)#0&'- "n l

'"0#%*n &""0$ -"0 "l !%-!' + "-(- '"0#%'n"- -" 0"l%n (>0!%n' (>0!%n'.

Al ?#"0 0"/"0"n#% l- 0'%"&&"- "- #'n"n%"n(" 0"#'0&0 l /#(%%l%&& &"

0"0"-"n(0 l'- ("n-'0"- &" 0%!"0 '0&"n #'!' "#('0"- @!(0%#"- 0"nl*n '

#'l)!n, l- &&- @("n-'0"- &" -")n&' '0&"n #'!' !(0%#"- &" 33 + l'-

("n-'0"- &" #)0(' 0n' #'!' !(0%#"- &" 99, "n('n#"- l- 0'%"&&"- #'n

0"-"#(' l- '"0#%'n"- -"0$n l- !%-!- 8)" l- &"-#0%(- 0 l- !(0%#"-.

1< C'n!)((%%&&

a b b a

a b b a

2< A-'#%(%%&& #'n 0"-"#(' l &%#%*n@a b c a @b c

3< A-'#%(%%&&, &%-(0%)(%%&& + #'n!)((%%&& #'n 0"-"#(' l

!)l(%l%##%*n '0 )n "-#l0. S"n + "-#l0"- @("n-'0"- &" 0n' #"0'

+ A, B ("n-'0"- &" 0n' -)"0%'0,

"n('n#"-:

@ A @ A @ A A

ϕ ⇒@ A ϕ A

P'0 '(0 0(":

@ A A A

@ A B A

H< A-'#%(%%&& &" l &%#%*n #'n 0"-"#(' l 0'&)#(' "n(0" ("n-'0"- &"

&%!"n-%*n -)"0%'0 l #"0'. Al %)l 8)" #'n l- !(0%#"- n' "4%-("

#'n!)((%%&& "n l '"0#%*n 0'&)#('. S"n T , S ("n-'0"- &" 0n' &'-

@&&- + a )n ("n-'0 &" 0n' )n', "n('n#"-:

(T + S )a =Ta+ Sa

(T + S )a ≠ a(T + S )

L &%#%*n &" ("n-'0"- -" 0"l% (>0!%n' (>0!%n', &" (l /'0! 8)":

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7/13

T + S = W

En n'(#%*n n&%#":

T ij + S ij = W ij

❑Tij=[

T 11

T 12

T 13

T 21

T 22

T 23

T 31

T 32

T 33

]&'n&", &"-&" l)"', "l ("n-'0 W (%"n" "l !%-!' 0n' &" -)- 0"&"#"-'0"-.

W ij=[T

11+S

11 T

12+S

12 T

13+S

13

T 21+S

21 T

22+S

22 T

23+S

23

T 31+S

31 T

32+S

32 T

33+S

33

]5< A-'#%(%%&& &" l '"0#%*n 0'&)#('. C'!' + n("- /)" !"n#%'n&' n'

"4%-(" #'n!)((%(%%&& "n "-( '"0#%*n.

(TS )a = T (Sa)TS ≠ ST

(T (SV ))a = T ((SV )a) = T (S (Va))(TS )(Va) = T (S (Va))T (SV ) = (TS )V

< O"0#%'n"- #'n l (0n-)"-( &"l ("n-'0

aTb = bT T a

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8/13

T#O$#&A D# '$##N

El ("'0"! &" G0""n "-(l"#" l 0"l#%*n "n(0" )n %n("0l &" ln" l0"&"&'0 &"

)n #)0 C #"00& + -%!l", + )n %n("0l &'l" -'0" l 0"%*n ln D l%!%(&

'0 C. J"0 /%)0 1.

(I')$A 1 $egión "lana D limitaa por C!

E-(" (%' &" ("'0"!- 0"-)l( !)+ 6(%l '08)", &&'- )n #!' "#('0%l + )n

#)0 #"00& -%!l" -'0" l #)l ?+ 8)" %n("00l', '&"!'- "l"%0 l

'-%%l%&& !$- -%!l" "n(0" %n("00 "l #!' &%0"#(!"n(" -'0" l #)0 ' %"n

%n("00 l &%/"0"n#% &" -)- &"0%&- 0#%l"- #0)&- -'0" "l 0"#%n(' 8)"

&"l%!%( l #)0. P'0 '(0' l&', l 0"l#%*n - "-(l"#%& "n(0" l %n("0l &" ln"

-'0" )n #)0 + l %n("0l &'l" -'0" l 0"%*n %n("0%'0 "-( "0!%(" "#"-'("n"0 %n/'0!#%*n -'0" )n /)n#%*n ' -) %n("0l "n )n 0"#%n(' 0(%0 &"l

#'!'0(!%"n(' &" l /)n#%*n -'0" l /0'n("0 &" &%#?' 0"#%n('.

T"'0"! &" G0""n: S" C )n #)0 -)" (0''-, #"00&, -%!l" +

'-%(%!"n(" '0%"n(& &"l ln', + -" D l 0"%*n %"0( 8)" #'n(%"n" D,

"n('n#"-

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∂Q

∂ x

(¿¿−∂ P

∂ y)dA

∫ D

¿

∫C

( Pdx+Qdy)=∫¿

L n'(#%*n

∮C

P dx+Q dy

-" )(%l% 0 %n&%#0 8)" l %n("0l &" ln" -" #l#)l )-n&' l '0%"n(#%*n

'-%(% &" l #)0 #"00& C. '(0 n'(#%*n 0 l #)0 /0'n("0 '-%(%!"n("

'0%"n(& &" D "- ∂ D , &" !'&' 8)" "l n)n#%&' &"l ("'0"! &" G0""n -"

)"&" "-#0%%0 #'!':

∂ Q

∂ x

(¿¿−∂ P

∂ y)dA=∫

∂ D

( Pdx+Qdy)

∫ D

¿

∫¿

D"!'-(0#%*n &"l ("'0"! &" G0""n 0 "l #-' "n 8)" D -" )n 0"%*n -%!l".

El ("'0"! &" G0""n 8)"&0 &"!'-(0&' -% -" 0)" 8)"

P dx=¿∫∫ D

∂ P

∂ y dA

∫C ¿ 1

Y

pág. 6


10/13

Q dy=¿∫∫ D

∂Q

∂ x dA

∫C

¿ 2

P0 &"!'-(00 l "#)#%*n 2 "40"-"!'- D #'!' )n 0"%*n (%' I:

D={( x , y )|a ≤ x ≤ b , g1( x)≤ y ≤ g 2( x)}

D'n&" g1 y g2 -'n /)n#%'n"- #'n(%n)-. E-(" "0!%(" #l#)l0 l &'l" %n("0l

&"l l&' &"0"#?' &" l "#)#%*n 2 #'!' -%)":

∫∫ D

∂ P

∂ y dA=∫

a

b

∫g1

( x )

g2

( x )∂ P

∂ y ( x , y ) dydx=∫

a

b

[ P ( x , g2 ( x ) )− P( x , g1( x)) ] dx

A?'0 #l#)l!'- "l l&' %8)%"0&' &" l "#)#%*n 2 &"-#'!'n%"n&' C #'!' l

)n%*n &" l- #)(0' #)0- C1, C2, C3, + CH. En C1 ('!!'- 4 #'!' "l 0$!"(0' +

"-#0%%!'- l- "#)#%'n"- 0!>(0%#- #'!' 44, +1@4, ≤ 4 ≤ .

En('n#"-

∫C 1

P ( x , y ) dx=∫a

b

P ( x , g1 ( x ) ) dx

O-"0" 8)" C3 &" &"0"#? %8)%"0& "0 C3 &" %8)%"0& &"0"#?, &"

!'&' 8)" '&"!'- "-#0%%0 l- "#)#%'n"- 0!>(0%#- &" C 3 #'!'

44,+2@4, ≤ 4 ≤ . P'0 (n'

∫C 3

P ( x , y ) dx=∫−C 3

P ( x , y ) dx=−∫a

b

P ( x , g2 ( x ) ) dx

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11/13

En C2 + CH 4 "- #'n-(n(", &" !'&' 8)" &4 +

∫C 2

P ( x , y ) dx=0=∫C 4

P ( x , y ) dx

P'0 (n(',

∫C

P ( x , y ) dx

P ( x , y )dx=¿∫a

b

P ( x , g1 ( x ) ) dx−∫a

b

P ( x , g2 ( x )) dx

∫C 1

P ( x , y ) dx+∫C 2

P ( x , y ) dx+∫C 3

P ( x , y )dx+∫C 4

¿

C'!0n&' "-( "40"-%*n, "!'- 8)"

∫C P ( x , y ) dx=−

∫∫ D∂ P

∂ y dA

T#O$#&A %TO*#%

El ("'0"! &" S('"- 0"l#%'n l %n("0l &" ln" &" )n #!' "#('0%ll0"&"&'0 &" )n #)0 #"00& -%!l" C "n R3, #'n l %n("0l -'0" )n

-)"0/%#%" S &" l #)l C "- l /0'n("0.

T"'0"! &" S('"- 0 0/%#-: S" S l -)"0/%#%" '0%"n(& &"/%n%& '0 )n

/)n#%*n C2, /@4,+, @4,+ D, + -" F )n #!' "#('0%l Cϵ 1 "n S. En('n#"-, -%

∂ S

&"n'( l #)0 /0'n("0 '0%"n(& &" S -"6n -" &"/%n%* n("-, ("n"!'-

∫S

∇× F ⋅ⅆ S=∮C

F ⋅ⅆ r

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A- #'!' "l teorema de Gauss 0"l#%'n )n %n("0l -'0" )n 'l)!"n #"00&'

#'n )n %n("0l -'0" -) -)"0/%#%" l!%(", "l teorema de Stokes 0"l#%'n )n

%n("0l &" ln" l0"&"&'0 &" l #)0 l!%(" &" l -)"0/%#%", &" (l /'0! 8)" l

%n("0l &" -)"0/%#%" &"l 0'(#%'nl &" )n /)n#%*n "#('0%l ('!& -'0"

#)l8)%"0 -)"0/%#%" "- %)l l %n("0l &" (0+"#('0% &" l /)n#%*n "#('0%l

-'0" "l '0&" &" l -)"0/%#%".

S" C )n #)0 #"00& "n )n "-#%' (0%&%!"n-%'nl + S )n -)"0/%#%" 0")l0

%"0( l%!%(& '0 C , "n('n#"-, 0 )n #!' "#('0%l u &"/%n%&' (n(' "n S

#'!' "n C , -" #)!l":

&'n&" t "- )n "#('0 (n"n(" )n%(0%' C , "l #)l -" -)!" 8)" "-($ '0%"n(&'

'-%(%!"n(" "n 0"l#%*n l "#('0 n'0!l )n%(0%' n &" S .

L "#)#%*n n("0%'0 "n n'(#%*n n&%#" -" "40"- #'!':

S% S "- )n -)"0/%#%" #"00&, "n('n#"- "l l&' %8)%"0&' -" 0"&)#" #"0', '0

(n(' -" #)!l%0$:

+I+LIO'$A(IA:

pág. 9


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1< LPE CELA, UAN OSQ @1999. "MECÁ!CA E #$S ME!$S

C$T!%$S" . ESPAA: EDICIONES DE LA UNIJERSIDAD DE

CASTILLA

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