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since (^) Gabed (^) is like arank- 4 covarianttensor^ ; the (^) usual (^) term : OnLand-F.... by
1.^2
Ambared = Ombabed - Tid (^) ......^ -^ W^. Tencased w (
San (X)^ = * God(x)
g
jg gabgb" = Sa Fj = ] (^59) Omg = Omg)
from
symmetric
matix
, Ombulg = g Jugga =>^ amFg = (^) E Fggagan
Gabed =
Fg Eabed (scalar^ density of (^) reight -
=> ImWand = (Gng) and = Eg g mSpz and (^) Tem = Egamgab
and =o
Im Eg
ile (^) F = (^) T but E-e (^). is (^) antisymmetro that (^) is to say :
- We^ +^ Temladed e: and So (^) Pubravad =^ Ounhabod + 0 = 0
do the^ same^ again to gabed & m (^) gabed = Om gabed (^) de Le Tem gabed 1
(^) 1)
= (^0) + 0 =
n.^8 Is" = ede-exdr-vido"-visited where (^) v = U (t, r) ,
X =^ x^ (t, v)
gr =^ ev ,
gov =^
,^900
= -^
= -Using
K =^ le-^ e-VErsint)^ by n^.^48 UK^ = (^0) ,
E
check (^) X=^ t, v , 0 , %^ que into^ -
t
: =^ et Vr^ V
r =^ &rv = (^) (
= V'(t
, v) I levi) =^ te't.^ V, +^ = so (^) eq : (evi-ev+ ev^ = 0 v (^) : = (^) tetYr-te *^ +r- rErso z =^ ej eg--ei)
- [vir-eid, -ro-remo 0 : E =^ Er 25000^ = (^) VSMG 2010
28 :v)^ -^ v'snoooo U = E
· E^ = -vsm % ef :^ ful-vism20^ %)^ =>
here (^) comes T =^ E v'(t, v) ↑= -red si Tee =^ +e^
v'(t, v) No = in =^ Ex(t,^ v)^ I Ti =^ T
- (^) X T =^ -re To% = (^) cot (^) O
2oX* +^ taX"^
wa +^ Wa = (^50) W =^ - Wha - is^ antisymmetric so (^) it has^3 independant components in (^) 3D (^3) constaus from Wab =^ rotations (^3) ta :^ translations Translations :^ T1.^ X")^ =^ Ex I^ translation about (^) X) T2. Y14^ = by [ 11 y) Ts.^ **
= Gz &^
- (^1) z) Rotations RI :x()^ = y7z
- z2y rotation^ about X)
R
: <
= (^) z(2x - XGz [^ -y) R3" (^) y(4) = x by
X (^) , ... Xo be^ constants so Xa^
= X , 0x + Xa
Gy
+ x30z +
xy(y8z
- zzy) +^ x5(z7x^ -X(z) + Xo(X(y - y(x) comput [ *** ) , <153] (^2). (^) [Ti
Tjj
= 0 for
is j^ = (^1).^2. 3
[Ri (^) , Rj J
WijkRk =>^ [R, RuJ = Ry = Y(x) [Rn (^) , Rs =^ Ri^ =^ **
& ERs , R,^ FRc = (15)
3 - (^) [Ri
, Tjj
=jaTk = [R,Tz] =^ Ts^ =^ **
S
[Rz
, i1)^
= (^) - T3 =^ -^ T()
[Rc , is]
= T
= (^) ** [Rs
, T2]^
= - T
= -^ *
I [Ps^ , T1]
= Tz = (^) **
/[m
, , Ts]^
= - Tz =^ -^ ** lothers = (^8)
Y" = 2x
** (^) = (^) y8z - zby
M = by
(b) =^ z7x^ = x7z T() = 0z (0) =^ X^ Jy
