alvarado j

7/23/2019 Alvarado j

1/82

U

T

E

N

U

AI D

S

RE

VI

N

D DE

OR

I

E

*V

*

EN

EL

EZ

A


2/82


3/82


4/82


5/82


6/82

a

1 0 0,5 0,6

a

a M= 1 A= 5


7/82


8/82


9/82


10/82


11/82


12/82

c= G = 1

G=R 12

R g= 8 T

R g R

T

T= ( +P) U U P g+ k k


13/82

P

U

U =dx/ds k kk = 0

= 0(1 + ) 0

R= gR

R= 8 T = 8 ( 3 P)

R= 4 [2 ( +P) U U ( P)g+ 2 k k]

T =T;= 0

x

( +P)UU; (g UU) P ;= 0


14/82

;U + ( +P) U;+ ; k

k U+

k; k +k k;

U= 0

( k k) ; U U k k; k= 0

ds2 =e2

V

rdu2 + 2du dr

r2 d2 + sen2 d2

V u r u = x0

r = x1 (g11 = 0) = x2

= x3

g=

V

re2 e2 0 0

e2 0 0 0

0 0 r2 0

0 0 0

r2 sen2()


15/82

g =

0 e2 0 0

e2

V

re2

0 0

0 0 r2 0

0 0 0 r2 sen2 ()

|g| = e4r4 sen2 ()

000 = 2 ,u 1

2 r(2 ,rV +V,r V /r) 022 = r e2

221 = 1/r 033 = r sen

2() e2

100

= V

2 r2 r V,uV 2 r ,u+V,r+ 2 ,rV Vr 111 = 2 ,r110 = (2 ,rV +V,r V /r) /(2 r) 122 = V e2

133 = V sen2() e2 233 = 1

2sen(2 )

332 = cot() 331 = 1/r

R


16/82

R = R

R= , ,+

R00 =2 V ,ur

r V V,r,r

r2 V

2 ,rr3

,rrV2

r2 V V,rr

2 r2 V,u 2 ,u V

r2

R01 = 2 ,ur 2 ,rV,r+ 2 ,rrV +V,rr2 r

V ,rr2

R11 = 4 ,rr

R22 = V,r e2 1

R33 =

V,r e2 1 sen2()

8 T00=V,u 2 ,u V

r2 V

r3

e2 V,r+ 2 ,rV

8 T01= 1

r2

e2 V,r+ 2 ,rV 8 T11=

4 ,rr


17/82

8T22 = 8T33 =e

2

2,ur 1

2 r2 [r V,rr 2,rV + 2r (,rr V +,rV,r)]

V

+

ds

2

=dt

2

dx2

dy2

dz2

T

t x y z


18/82

dt= e Vr1/2 du+ r

V1/2 dr

dx= e r

V

1/2dr

dy= r d

dz= r sen d

p

3

T=

+

+ 3

0 0

p++ 0 00 0 p+ 00 0 0 p+

= dx/dt 1<


19/82

>0


20/82

T

T=x

xx

xT

x t x y z

T00=e2V

r

+P 2

1 2 +

T01= e2

1 +( P )

T11=r e2

V1 1 +

( +P)

T33 = T44 = P

T00 =

e2r ( +P) ( 1)

V (1 +)

T01 = e2 ( +P)

1 +


21/82

T11 = [(1 2) + 2 +P] e2 V

(2 1) r

T22 = sen2() T33 = P

r2

V

U= 0 e

(1 2) 12

V

r +1 e

1 1 +

1

2

r

V

U = 0e

r

V

1 1 +

1/2+1 e

V

r

(1 2)1/2

k= 0 e

V

r

12

k =1e

V

r

12

U k UU = 1

kk= 0


22/82

m

V =e2 [r 2 m(u, r)]

m

m

+P 2

1 2 += 1

4 r (r 2 m)m,u e2 +(r 2 m)

r m,r

1

1 +( P ) = m,r

4r2

1 1 +

( +P) =r 2m

2r2 ,r

P = ,ure2

4 + 1

81 2m

r2,rr + 42,r ,rr

+3,r(1 2m,r) m,rr

8r

P

m

= P

1 +

P =P

1 +


23/82

P

= 0

(1 )2 ( +P) +=

m,u e24r(r 2m)

= m,r4r2

+ P =r

2m

2r2 ,r

P = ,ure2

4 +

1

8

1 2m

r

2,rr + 4

2,r

,rr

+3,r(1 2m,r) m,rr

8r

m P

e=m,r4r2

Pe=2,r(r 2m) m,r4r2

e Pe


24/82

= e+P Pe

= ee+P

= m,u e2

4r(r 2m) (e+Pe)

(1 )2

m

P

dr

du=

V

r

1

m u

dm

du = m,u+ m,r

dr

du

dm

du =

4 r2 e2 1 2 mr P 1 +


25/82

T1 ; = 0

e2

P+

1 2m/r

,u

+P

r +

P+

1 2m/r

4rP+m

r2

=

2

r(P P)

H(u, r)

H(u, r)

2

r

(P

P) +e2

P+

1 2m/r,u H(u, r)

H= 0


26/82

P

r

+P+

1 2m/r 4rP+m

r2= H(u, r)

=

P =P


27/82

= (u, r)

= ( +P)

1 + = ( + P)

1

> P > P

< P < P

e2

P+

1 2m/r

,u

+P

r +

P+

1 2m/r

4rP+m

r2

=

2

r +

= ,r+ 2m

r(r 2m) 2,ue2

1 2m/r +

4r

1 2m/r

( + 3P) 22 + 41 + 4

2e2(PtP)/r Pt

2e2

r (Pt P) =

Pt=r

2e2 +P


28/82

= 0 = 0 = 0 Pt

Pt =P

Pt

Pt = PD+PR+P

PD

PR

PD+PR = r

2e2

= 0

r ,,,H

= P = P

,P, m,

m,r = 4 r2

,r = 2 r2

r 2 m( +P)


29/82

P,r = P+1 2m/r

4rP+

m

r2

=

= 3 M

4 A3

m=

4

3 r

3

=1

2ln

3

1 2 M/A2

1 2 m/r 1

2

P =

1 2 m/r

1 2 M/A

3

1 2 M/A

1 2 m/r

M A


30/82

r =

a(u) = A(u) a A


31/82

r= A

(0 r A) ds2() (r

A)

ds2(+) u,r,,

ds2(+)=

1 2M

r

du2 + 2 dudr r2 (d2 + sen2 d2)

ds2() ds2(+)

r= A

V

(u, A) = 0

m(u, A) =M(u)

(u, r)

(u, r) =A r


32/82

= e (2 A +V /r)1/2 (A 0 1)

= e (2 A +V /r)1/20 + ( A +V /r) 1

A dA/du u=

l= u= 0 l =e21

u=

L= l =e (2 A +V /r)1/2

r= A

() ; |a =(+) ; |a

L

L = L

L() |a =L(+) |a

,ue2 +

1 2 m

r

,r m,r

2 r

a

= 0


33/82

r r = A

,u |a= A ,r|a

Pa= aa

a

Pa= 0

r = A A(u) F(u)

(u) E(u)

F = 1 2 MA


34/82

= 1

1 a

E= 4 A2 a

A a = (u, a)

(1/2, ) M = m(u, a) E

A=dA

du =F( 1)

m

u r= A

M= F E

E

F >0

F

F =

2 E+ (1 F) ( 1)A


35/82

(a)

F

F+

aa

+2F Ha

a+ (1)

4Aa

(3 1)

3 +F2 A

+F

a,ra

= 0

A M E


36/82

r r = 0

R

R=RR

R= 64 2

2 + 3 P2 + 2 ( +P)

1 1 +

R= 64 2

2 + 3 P2 + 2 ( + P)


37/82

P

r = 0

m

m, r(u, 0) = m, r r(u, 0) =, r(u, 0) = 0

C =C

C

C

C = 48

m

r3 4

3

2

m r3

C|r=0= 0


38/82

m= m3(u) r3 + O(r5)

= 0(u) +2(u) r2 + O(r4)

m r

r

P(u, r) =P0(u) +P1(u) r+ O(r2)

P0 =4 2 3 m3

4

P1=

e20 2,u

2

(u, r) =0(u) +1(u) r+ O (r2)


39/82

0=3 m34

1= e20 2,u

2

(u, r)

(u, r) =1(u) r+

O(r2)

1 = e2 0 2,u

2 2

(u, r)

(u, r) =1(u) r+ O (r2

)

1 =e2 0 (2 2,u m3,u)

4

P =P()


40/82

(0+ 3 P0), u 61 e2 0 = 0

0+ 3 P0 6

1 e20du=

0+ 3 P0=

P1= 1

P <

P1 = 1(0+P0)


41/82

P1

1


42/82

u

= f(u)

f u

A(u)

P =f(u)

(1 3 a)(1 83f r2)1/2 (1 a)(1 83f A2)1/23(1 a)(1 83f A2)1/2 (1 3a)(1 83f r2)1/2

m

m=4

3f r3

= 1

2ln

(1 a)

3 (1 83f A2)1/22 (1 8

3f r2)1/2

12

+a


43/82

= f(u) =3(1 F)

8A2

P =3(1 F)

8A2(3 2) F

3

F (3 2)

(u, r) = 1 2m/r = 1 (1 F)r2/A2

=1

2ln

1

3

F

2

1

2

+

1

r A F E


44/82

P = 3 r16 Y2A2

(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r24

F 3A22Y

+

F

3/2

F+

3 r

F(1 F)

2

Y + 3 (

F)

32 A2 2 Y3 3

+ 3 r (1 F)8 A3 Y2 3/2

3 r2 (1 F) 2

F Y

F4 3/2 A2 Y

+ F A

3(1 F)8 A2

3

F(1 F)[3(1 F) r2 A2 (3 +)]32 Y A4 3/2

Y =e2 = 3

2

F 1+ 1


45/82

= 3 r

16 Y2A2(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r2

4

F 3A22Y

+

F

3/2

F+

3 r

F(1 F)

2

Y + 3 (

F)

32 A2 2 Y3 3

+ 3 r (1 F)8 A3 Y2 3/2

3 r2 (1 F) 2 F Y F4 3/2 A2 Y

+ F A

+3(1 F)

8A2

F 3 + 2

3

F 3 + 2

+ 3

F r2 + 3 F3/2r2 +2

F A2 + 3

F A245/2A2Y

= ee+P


46/82

e= 3 r16 Y2A2

(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r24

F 3A22Y

+

F

3/2

F+

3 r

F(1 F)

2

Y + 3 (

F)

32 A2 2 Y3 3

+ 3 r (1 F)8 A3 Y2 3/2

3 r2 (1 F) 2

F Y

F4 3/2 A2 Y

+ F A

+3(1 F)

8A2

F 3 + 2

3

F 3 + 2 1

+ 3

F r2 + 3 F3/2r2 +2

F A2 + 3

F A245/2A2Y


47/82

e+P = 3 r

16 Y2A2(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r2

4

F 3A22Y

+

F

3/2

F+

3 r

F(1 F)

2

Y + 3 (

F)

32 A2 2 Y3 3

+ 3 r (1 F)8 A3 Y2 3/2

3 r2 (1 F) 2

F Y

F4 3/2 A2 Y

+ F A

+3(1 F)

8A2

3

F r2 + 3 F3/2r2 +2

F A2 + 3

F A2

45/2A2Y

= e2 r

8 A3

A F 2(1 F)A

34A2

(1 F)

F3

F 3 + 2

(1 )2

= 0


48/82

A F

r= a

F

F + (1 F)

+

3 (1 F)2 ( 1)(2 1)2 A

= 0

m

m(u, r) = m3 r3

(u, r) = 0+ 2 r2 + 4 r

4 +O r6


49/82

m3 = MA3

0 =1

2 ln

(1 a)

3

F

2 1

2

+a

2 = 3 (1 a)

F M

2 A3 (1 a) 3

F

1+ 2 a

4 =9 (1 a)

2 F(1 a)

F(1 3 a)

M2

4 A6

(1 3 a) 3

F(1 a)

m r

P(u, r) =P0(u) +P1(u) r +P2(u) r2 +P3(u) r

3 + O r4

P0=

3 M

4 A3 (1 3 a) (1 a)

F

3 F(1 a) (1 3 a)


50/82

P1 =9 (1 a)

a+ 2 aF

3/2 3 aF+F 2 F3/2 + 1

F E

4 A3 (1 3 a) 3

F(1

a)

3

3 a(1 F)

18 F3/2 (1 a) F+ 27 aF 9 9 a

F

8 A3

(1 3 a) 3

F(1 a)3

P2 = 3 M2

2 A6(1 a) (3 a 1)

F

3

F(1 a) (1 3 a)

2

P3 = 9

F (1 F) (1 a) T1 E

8 A5

(1 3 a) 3

F(1 a)4

+ 9

F(1 F)2 a[3 T1+ 8 F(3 a 1)]

16 A5

(1 3 a) 3

F(1 a)4

T1 = 18 (1 a)2 F2 + (3 a 1)

18 (1 a) F3/2

+ 5 (3 a 1) F 3 (1 +a)]

(u, r) =0+P1 r+P3 r3 + O(r4)

0= 3 M

4 A3 = f(u)


51/82

(u, r) =1 r+2 r2 +3 r3 + O (r4)

1 = P1

0+P0

2 = P10+P0

2

3 =P3 (0+P0)

2 P1 P2 (0+P0) +P31(0+P0)

3

(u, r) = e20 03 +P1 r P210+P0 r2

P3+2

9 (0 3 P0) e20 0

r3 + O r4

0 = 3 F4 A3

E+3 MA

a1 a

E


52/82

a

A F a

C = 2 MA

=RS

A

RS= 2 M

(P0 )

A 9 RS8

(1 a)2(1

3 a/2)


53/82

W = (1 a)2

(1 3 a/2)

A 9 Rs8

W

RS= 2 M

RB = (9 RS/8) W

RB RS= RS8

(1 3 a)2(1

3 a/2)

a = 2/3 a

1< a < 2/3

W

(1, 0) (1/2, 3/5] (0, 1/2)

(RB = RS) a = 1/3

Rs RB < A


54/82

0.8

1

1.2

1.4

1.6

1 0.6 0.2 0.2 0.6

W

!a

a 1 0 0,5 0,6


55/82

0.85

0.9

0.95

1

0 0.1 0.2 0.3 0.4 0.5

W

!a

a


56/82

u

m= C(u) r3

1 +B(u) r2

=1

2ln

(1 K a)

3

F

2

1

2

+K a

= 1 2 C r2

/(1 +Br2

)

B = B(u) C = C(u)

r= A ma = M P a= aa

B=3 a(1 K)A2 (1 a)

C=MA3 (1 + 2 a 3 K a)(1 a)

m

K = 1 B = 0 C = M/A3


57/82

= C4

(3 +B r2

)(1 +B r2)2

P =

3 +B r2

(3 +B r2)

(1 3 K a) 3

F(1 K a) (1 +B r2)

3

F(1 K a) (1 3 K a)

P

r a

a> 1

3 K 2 K= 1

K

r= A

A F


58/82

= (2 3 + 3 K 3 K) ( K +K)(1 F) F 3 ( K +K)2 2K F

+ ( 1)

2 A

3 ( K +K)2 2K

= 3 ( 1)(42 F+ F 1 +F) K2 4 2 (6 5) F K

+(12 F 3 5 F)2

K= 0

= (2 3)3F(1 F) F+

(12F 5 F 3)6 A F

A

u

=A2(1 F)

F

(5 F+ 3)

12A2(1 F)+ b

a = 7 F 3 12 b A2 (1 F)5 F 3 12 b A2 (1 F)

b

b=12 F00 5 F0 3

12 A20(1 F0)

u = 0


59/82

12 F 5 F 3

12 A2

(1 F) =b

P =3 r

(3 F

F

) (3 r2 C++B r2) 3 C F r2

2 3/2 (1 +B r2)3 (3

F )3 C

+3 C r3

(3 F

F

) (3 r2 C+ 2 + 2 B r2) 3 C F r2

2 3/2 (1 +B r2)4 (3 F )3 B

+ 3 C r

4

F(1 +B r2)2 (3

F )3 F

C

(

+ 3

F) B r2 + 3

F 3

4 (3

F ) (1 +B r2)3


60/82

= 3 r (3 F F) (3 r2 C++B r2) 3 C F r22 3/2 (1 +B r2)3 (3

F )3 C

+3 C r3

(3 F

F

) (3 r2 C+ 2 + 2 B r2) 3 C F r2

2 3/2 (1 +B r2)4 (3

F )3 B

+ 3 C r

4 F(1 + B r2

)2

(3 F )3

F

+C

(2 B2 r4 + 3) (3

F ) + 3 B r2 (5

F 3 )

4 (3

F ) (1 + B r2)3

= +P


61/82

= 3 r (3 F F) (3 r2 C++B r2) 3 C F r22 3/2 (1 +B r2)3 (3

F )3 C

+3 C r3

(3 F

F

) (3 r2 C+ 2 + 2 B r2) 3 C F r2

2 3/2 (1 +B r2)4 (3

F )3 B

+ 3 C r

4 F(1 + B r2

)2

(3 F )3

F

+B C r2

4

(3

F )Br2 5 + 3

F

(3

F ) (1 +B r2)3

+P =3 r (3 F

F

) (3 r2 C++B r2)

3 C F r2

2 3/2 (1 +B r2)3 (3

F )3 C

+3 C r3

(3 F

F

) (3 r2 C+ 2 + 2 B r2) 3 C F r2

2 3/2 (1 +B r2)4 (3

F )3 B

+ 3 C r

4

F(1 + B r2)2 (3

F

)3

F

+ C

4

(3

F )B2r4 + (5 + 9

F)B r2 + 6

F

(3

F ) (1 +B r2)3


62/82

= C r2 B (1 +B r2) C2

(1 +B r2)2 (3

F )

3 C

F

2 (1 +B r2)2 (3

F )

(1 )2

B=4 (2 3 ) E+ ( 1)(1 F) (7 F 6 + 1)

2 A3 (1 F)

C=4 (1 +F) (2 3) E+ (1 )(1 F) (7 F+ 6 F+ 6 1)

4 A3

K= 0

Pt a

= PDa

+PRa

K = 0

Pta = 0


63/82

PRa =

PDa

PDa = (1 F) a Z1

32 A3 F2 (1 + 2 a)3 (1 a)

PRa= Z2

2 A3 F(1 + 2 a)2 E

Z1= 192 F A2 (1 + 2 a)

2 2 F(1 a) (5 2a 46 a 31)

+ 9 (1 +a) (1 a)2 +F2 (109 3a+ 159 2 + 3 a 55)

Z2= 8 A2 (1 + 2 a)

2 + (1 + 6 F) 2a+ (1 + 3 F) a

2

E

E= a(1 F) Z116 F(1 + 2 a)(1 a) Z2

a 1/2

a 1

a M A

M= 1 A= 5


64/82

1

0.5

0

0.5

1

0.4 0.1 0.2 0.5 0.8

E

!a

a M = 1 A = 5


65/82

A M

A= (12 b A2 + 7) M

6 A1

3

M= F E

m

K= 0

m(u, r) = C r3 C B r5 + O r7

(u, r) =1

2 ln

3

2

F 1

2

+

3

F C

2 (3

F 1) r2

+3 CF4

3 2

F

1 C 2 3

F

1 B3

F 12 r4 + O r6

m

r r


66/82

P(u, r) = P0(u) +P1(u) r+P2(u) r2 + O r3

P0= 34

C

F 1

3

F 1

P1= 3C

4

F

3

F 13 F 3 F

2

3

F 12 C

P2= 32

F C2

3

F 12 + 12

9 F 18

F+ 5

BC

3

F 12

P3= 3 CF

(3

F 1)2 B 3 C4

2 B (3 F 1) + 3 CF(3

F 1)4 F

+3

F

2

6

F B 9 F C 2 B+ 6 C

(3

F 1)3 C

(u, r) =0(u) +1(u) r+2(u) r2 +3(u) r3 + O r3


67/82

0 =3 C4

1 = P1

2 = 3 C B

F3

F 1

3 = P3

P2n+1= 2n+1, n 0

(u, r) =1(u) r+2(u) r2 +3(u) r

3 + O r3


68/82

1 = P10+P0

2 = 21 3 C2

F(18

F 9 F 5) B

8 2 (3

F 1)3 (0+P0)2

3 = 31

P2+20+P0

1+ P3

0+P0

(u, r) =1(u) r+2(u) r2 +3(u) r

3 + O r4

1=1

3 0 e2 0

P1

2= 1 P1+(18

F 9 F 5) C B4 (3

F 1)2

3=1

30 e

20 B+e20

12 [3 B 2 (0+ 3 P0)] C(0 P0) P3

(0+P0)

13

[2 (0+P0) (0+ 3 P0) + 3 (P2 1)] 1


69/82

(P0 )

A 9 RS8

(1 K a)2(1 3 K a/2)

W = (1 K a)2(1 3 K a/2)

A 9 Rs8

W

K= 0

RB = (9 RS/8) W

RS= 2 M

RB

RS=

RS

8

(1 3 K a)2

(1 3 K a/2)

a K= 2/3

1< a K


70/82

W K= 0

K a (K, 0) (K/2, 3K/5] (0, K/2)

(RB =RS) K a= 1/3

RS RB < A K


71/82

L

u=


72/82

K= 0

r

r


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alvarado j

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