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7/25/2019 Class 05 Handout
1/24
Fluid Mechanics AS102
Class Note No: 05
Tuesday, August 7, 2007
7/25/2019 Class 05 Handout
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Review: C. C. S. & Tensor Analysis - transformations
xi=xi(uj), xi=x
i(uj);
ui =ui(uj), ui =ui(uj) (1)
x1
x2
x3
r
P
u1u2
u3g1
g2g3
u1u2
u3
g1
g
2
g
3
Figure:representing two curvilinear coordinate systems
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Review: C. C. S. & Tensor Analysis - transformations
gk = um
ukgm, g
k =uk
um gm,
gkl= um
uku
n
ul gmn,
gkl =uk
umul
un gmn (2)
# 2nd order covariant tensor & 2nd order contravarianttensor
# understand the transformation rules
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Review: C. C. S. & Tensor Analysis - transformations
0th order tensors (scalars):
= (xi) = (ui) = (ui) (3)
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Review: C. C. S. & Tensor Analysis - transformations
1st order tensors (vectors):
a = aiii=ai
(u)gi=ai(u)gi
= ak(u)gk =a
i(u)gi (4)
# Hereurepresentsum andu representsum
# thepairingin thesummationfor the representations in a
cuvilinear c. s.:e.g. indexk, one up & one down
ai(u) = ui
um
am(u) =ui
xm
am
ai(u) =
um
uiam(u) =
xm
uiam
ai(u) =gijaj(u)
ai(u) =gijaj(u) (5)
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Review: C. C. S. & Tensor Analysis - transformations
1st order tensors (vectors):
# hints to derive the above transformations:
e.g. dot product both sides ofak(u)gk=ai(u)gi with
gm
e.g. dot product both sides ofak(u)gk=ai(u)gi withgm
# covariant tensor & contravariant tensor
# understand the transformation rules:positions of & correspondences between the indexes
# HOMEWORK Assignment derive the relations in (5)
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Review: C. C. S. & Tensor Analysis - transformations
other 2nd order tensors:
A=Aijiiij=Aij(u)gigj=Aij(u)g
igj =Aijgig
j
=Aij(u)gi g
j =A
ij(u) gigj =Ai
jg
igj (6)
# Hereurepresentsum andu representsum
# thepairingin thesummationfor the representations in acuvilinear c. s.:
e.g. indexi, one up & one down
Aij(u) =ui
umuj
unAmn(u) =
ui
xm
uj
xnAmn
Aij(u) =
um
u
i
un
u
j
Amn(u) = xm
u
i
xn
u
j
Amn
Aij(u
) =ui
umun
ujAm
n(u) =ui
xm
xn
ujAmn
Aij(u) =gimgjnAmn(u), Aij(u) =gimgjnAmn(u)
Aij(u) =g
im
Amj(u) =gjnAin
(u) ... (7)
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Review: C. C. S. & Tensor Analysis - transformations
other 2nd order tensors:
# hints to derive the above transformations:
e.g. dot product both sides ofAij(u)gig
j =aij(u)gigj
withgm andgn consecutively, each operating on its
immediate neighboring base vectore.g. dot product both sides ofAij(u)gigj=Aij(u)g
igj withgm andgn consecutively, each operating on its immediateneighboring base vector
# covariant tensor & contravariant tensor
# understand the transformation rules:positions of & correspondences between the indexes
# HOMEWORK Assignment derive the relations in (7)
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Review: C. C. S. & Tensor Analysis - transformations
2nd order tensors:
# working example 1
Aij(u) = ui
umu
j
unAmn(u) ux= u
i
xmu
j
xnAmn (8)
from
A= Aij(u)gi g
j =Aij(u)gigj (9)
(use the hints mentioned above).
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Review: C. C. S. & Tensor Analysis - transformations
2nd order tensors:
# working example 2
Aij(u) = ui
umu
n
ujAmn(u) u
x= ui
xmxnuj
Amn (10)
from
A= Aij(u
)gi gj =Ai
j(u)gigj (11)
(use the hints mentioned above).
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Review: C. C. S. & Tensor Analysis - transformations
2nd order tensors:
# working example 3
Aij(u) =gimgjnAmn(u) (12)
from
A= Aij(u)gigj=Aij(u)gigj (13)
(use the hints mentioned above).
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Curvilinear Coordinate Systems & Tensor Analysis
Todays topic:
# tensor differentiations
# example: a= aiii=ai(u)gi=ai(u)g
i
in a rectangular c.s.
xja=
xj(aiii) =
aixj
ii =ai,j ii (14)
in a curvilinear c.s.
uja =
uj
a
i(u)gi=
ai(u)
uj gi
Q?
uja = ai;j(u) gi ? a
i;j(u) = ?
form similar to the case of rectangular c.s. ai,j in(14)
(YES !! below ...)
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Curvilinear Coordinate Systems & Tensor Analysis
ui =ui(x) , xi=xi(u) ,
r=xiii=xi(u) ii=: r(u) (15)
O
x1
x2
x3
r
Pu1
u2
u3
i1i2
i3
g1 g2
g3
Figure:curvilinear coordinate system
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Curvilinear Coordinate Systems & Tensor Analysis
scalars:
= (x) = (u) {= (u)} (16)
def
= ik
xk
c. r.= ik
um
um
xk
def=
umgm =;mg
m
{u2 u
=
umgm} (17)
um =
u
n
um
un (18)
1st order covariant tensor ?
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Curvilinear Coordinate Systems & Tensor Analysisvectors:
a= ak(u)gk=ak(u)gk
{=ak(u)gk =ak(u)gk} (19)
a def
= ik
xka
def= gl
ula= ? (20)
ula=
ul
ak(u)gk
=
ak(u)
ul gk+ a
k(u)gk
ul
ula=
ul
ak(u)g
k
=ak(u)
ul gk + ak(u)
gk
ul
gk
ul = ?
gk
ul = ? the key to the diff
C ili C di S T A l i
7/25/2019 Class 05 Handout
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Curvilinear Coordinate Systems & Tensor Analysis
one can check (defns + chain rules + tricks)
gk
ul =
m
k l
gm
gkul
=
kl n
gn (21)
mk l
:= 12
gmn(gnk
ul +
gnl
uk
gkl
un) (22)
the christoffel symbols of the second kind
C ili C di S & T A l i
7/25/2019 Class 05 Handout
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Curvilinear Coordinate Systems & Tensor Analysis
some hints to show (21):
#
gkul
= 2
xiuluk
ii= um
xi
2
xiuluk
gm
#
um
xi=gmn
xi
un
C ili C di t S t & T A l i
7/25/2019 Class 05 Handout
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Curvilinear Coordinate Systems & Tensor Analysis
some hints to show (21):
#
um
xi
2xi
uluk =gmn
xi
un2xi
uluk
=gmn ul
xiun
xiuk
2
xiulun
xiuk
=gmngnkul
un
xiul
xiuk
+
xiul
2xiunuk
=gmngnk
ul
glk
un +
ukxiul
xi
un
2xi
ukul
xi
un
=gmngnk
ul
glk
un +
gln
uk
xi
un2xi
ukul
...
C ili C di t S t & T A l i
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Curvilinear Coordinate Systems & Tensor Analysis
some hints to show (21):
#
gk
ul =
ul(gkigi) = g
ki
ulgi+ gkigi
ul
gkj gjl=kl gkj
ungjl+ g
kjgjl
un =0
...
C r ilinear Coordinate S stems & Tensor Anal sis
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Curvilinear Coordinate Systems & Tensor Analysis
a=gl
ula= ?
case 1:
a = gl
ul(ak
gk) =glak
ul gk+ akgk
ul
= gl
am
ul gm+ a
k
m
k l
gm
=a
m
ul +m
k l
a
kg
l
gm=a
m
;lg
l
gm
am;l :=
am
ul +
m
k l
ak
2nd o.mixed t. Understand the rule (23)
Curvilinear Coordinate Systems & Tensor Analysis
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Curvilinear Coordinate Systems & Tensor Analysis
how to show thatam;lis a 2nd o. mixed t.?
am;lg
lgm= a=iia
xi
uu= am
;lglgm
am;l :=
am
ul +
m
k l
ak
a
m
;l=
um
up
uq
ul a
p
q (24)
Curvilinear Coordinate Systems & Tensor Analysis
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Curvilinear Coordinate Systems & Tensor Analysis
a=g
l
ula= ?
case 2:
a = gl
ul(akgk
) =glak
ul gk
+ akgk
ul
= gl
am
ul gmak
k
l n
gn
=am
ul k
m l
akg
l
gm=am;lgl
gm
am;l :=am
ul
k
m l
ak
2nd o.covariant t. understand the rule(25)
Curvilinear Coordinate Systems & Tensor Analysis
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Curvilinear Coordinate Systems & Tensor Analysis
how to show thatam;lis a 2nd o. covariant t.?
am;lglgm = a= ii
a
xi
uu= am;lg
lgm
am;l := amul
k
m l
ak
a
m;l=
up
um
uq
ul ap;q (26)
Curvilinear Coordinate Systems & Tensor Analysis
7/25/2019 Class 05 Handout
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Curvilinear Coordinate Systems & Tensor Analysisuseful formula:
diva := a= gk
uk[algl] =g
k
uk[algl]
= gk al;kgl=a
k;k (27)
2 := =gl
ul[gm
um] =gl
ul[,mg
m]
= gl(,m);lgm =glm(
2
ulum
k
m l
uk) (28)
curla := a=gk
uk(alg
l) =gk
uk(alg
l)
= gk(al;kgl) =al;kgkgl (29)
(b)a = (bmgm)gk
uka= bk
uk(algl)
= bk al;
k
gl (30)