Physics 131: Survey of the Universe Mathematical Review As many

!"$#%&
' (*,+.-/0(*,1324(65879-:;13-=<
)
> @?BA#=C 4A#EDF$GAHI#?9AI&J
#ALK@
M%NO8!4P4=@%Q?B%P4=SRTUAVP4WJ
WAV&V&X%K
&?BY!Y?BAI!Z&4M%[!AIUK\ID]M%MRTV
#P
#Z&
^4&_V`4a&M
WabMcb&V&dV!%
&^K@#ZK\&H
#PJ#e44f4?gK84&H_ih$K\X4N.%[K@%M
M.#fRTVAjIA%MkARM%U4#l!m?B%W!&?
4Q&^n6#6AMo_
pJqQrJs^t 7 pCuvxwaygzQpUzu,ygw
> M%P4R!A%B|{}&&%4#gA&^~L&#=&#NK@&&!#
#44?B}AiKQAj_aOLK@%M
MM%}04
#W&GA#W
#Pa4#&%W&ARM0S44&?B?LRTm!AIjNA#WENRTXb4&C&
#Pd!Z&M%4C&?N&AW
!?A@#=!#_
z +.- s^ (65@;1b
S&%?B}M%@AM
PR&A
f6AI!%4#$A8~A%!?BX!
I
J J
b48X{A?g}M
I_S
Q%\jAM%M%BAm&#=!#NA#WJ%QjAWSfIAkARM
N N%\AM!Jj_]Z6A
!V6AMc
P4#e?BA#f%[!6A^&
#P4#l&VM
Xf
WLI&LAI!%4#Z
f|{A&M%d6AM&
&
#P#e!m&
P_[#a4Z6Aj43A[K8VWC&jfAe ¡|¢3 =£¤
¥$4#e!LM
XjN[&M
M%$4
K@6AI^&VIAM
m8&6AIfIAkARM
m%_V¦$4&?BAM%M%4N§^&Hl4^¨@4?BA#ZM
X&&&gb4UAM%}6ARf
f!
¨@4?9AI#AM%}6ARGA9Wb4UIA&
ARM%N©Ri44ª4M
WJ~LAmA&%M%«CA#=?LR4M4
K\A#=j_
p ¬ ¬©,®¯ 1o/0(*-5b° sfE (5 z\®
C
`4JK@
M%ME&#&mA#X{}&
#dM%
HI
± ² j
Y
S%jAWNSSIA&
ARM
IN ± N=
X³=³¡!´µ4¢¶g ·¸¥X¤º¹86AM6!Y_]aS%RA&%jAM%M%i?BA#!6A ± ?9Aj
R6AM;!B»¼¾½Y4î4$¿¼¾ÀN$4?g!
#PYM
%H^!AIj_O$#Wl#$RUX{A&M%J6AM;!N6R
&4M
WRAYIAM%^&4?g&
#PVM
HI^%_
s[¯ (/ ¬ 5b-ÂÁ
O?B%P4=QM%4HgAISA#J#W&P4&AW6A!^&W#$&Â@Ãf&|{MÄN6AI#WENb\![?B4S}6AS!AjN
ÅV²)Æ I&¼
ÎA&
ARM%N Å NGAI#WSb4\&UAP4^.&U!W#=jNÂAI#WAM
M!^#=!#?BA#S%\!6A
S
!9&W#L
iA}}{%?9AI&M Æ jAm4M%WE_Ç64?9A#=a!W#=!Nc&
UK@
M%MRT9AdP4W
P4&_
¦I!
L!6AU
#Z!%^X{A?g}M
IN;O[W%W#EDF\~&A Åe²ÈÆ _VOfAIM
&J&WÊÉË6¢·Ì@ÍejA_gOÎ#
&QK84&WN6O#WW!V}T%9#S#M%BKÏ4M%W9!^!W#=@KQAIN6RSK@!S!6A
#?LRmK\AVKÈ?9AI#W6AjN?B#&N4#WN84m
#0&
LA&IN4jAI&V4M%W0!6AL!6A
}&4#eK\A_ > }T&&#eK@C% Æ g?B#&$4M
WEN6b4$X{A?g}M%N6K\M
WlA&WMRTiAB4M
M%P4
&W#Ð
E 1o/135b( gz\®EÑyVÒ^yVIÓ -
4?g&
?gNK@#dK8mA^RT
#PYX4#dM
&@}&%&N6K8^K@
M
M&A#X{}&
4#M
%HI
ÔLÕ Lb
ÖeAI&?BAI!%jAM¨$X
K\Íl
!"$#%&
%jAWN8SiIAkARM
N Ô NT%L´L´¡¥X¡[gbj_
#=&
&NTA#WdA&&4#4?B&$
#}6A!
MkAIN
S
b&#=!M9WmjAIM
MkAI&
4#%#gK@%G9![#?LR8?B%P4=8R4N9?9AM%M©o_]44N&Aj|N4%
?g
P4=SR^44N64M
A&P4VoAYR
M%M
#EN=b4SX{A?g}M
jX_?BX!
?gNAM%MTK8K\A#=@!gH#KÏ
@AIRT4
KÏR
PY&4?g&
#PV
_
Of%b4$!%[jA4#l!6Af&
#&
!fd´¡ ¥X¡g´ i¶g Ë6¢·oÉ ¥@!%?9A!
_ QA
AM
M4NA#l4W
?BAP4#!W^%@AL3A&4@\j_O4?BX!%#PY
gPWJ!gA#4&W@?BAP4#%!WY%Q?BA#S!6A
%L
LK@%&
# & &4&XYAIM
I_ªSN8&a!Ajª!6AgAa!W#=9%#ª!M
A&V
Õ LbX@GAM%MÄN%@AV4&X$GA!?g#=N6&
#^!XJK@
M%M}4R6ARMCRi½ mbjN6&AI!S!6AI#EN64#A
}M
A#@IP4
A#=!S4#K@%G&J?g
PQRTY4Vb@GAIM
MÄ_
S%$%S4R=
4M%9#@!^&&?ÈK\ID]WKQA#=@!Yi%#}&
UjAM%M
AI!%4#_
®Â¬®Â *,1 ® .(65 z\®
Ç%#6AM
M4NK\@?BAmH#K !AIÎK\U!
#P4AI&S&MkAI&WENAI#WEN4b4|{AI?B}M%NK8$?9AjLH#KÊ!6AIK@#
4#^#?mRTm!UAIP4N&Aj4N6AL&?BAM
MTG
M%WÂ8P4X!SM
A&P4N!#eA#!S#?LRN !fK8%P4=@
!AI@G
M%WÂ\P4X!@MkAP4QASK\M
MÄ_S
SK\M
WRUK@%!#AM%P4R!A%jAM%M%BA
ÏÅ
O&AWNfSVIA&
ARM%N N.%}&4}4&
4#AM©&&gIAkARM
N Å _ 6AIU?BA#^!6Ai84
b4#WÎK8YG
M%W&#ENAI#WJ!fn6SK\ASW4RM
[&UAP4Î &f#WEN!#![n&SK84M%WAM
RW4RM%f!^K8
P$ !4#WE_
s[¯ (/ ¬ 5b-ÂÁ
> #4M%WZ
Æ jAU4M
WENA#WZK8
P
=C}4#W
_ 8&Îe
^9jA^4M
W_ih$K ?LGZW
&K8%P4 D M
MÂWm&
#B?B4WGAI
MMkAI&4#C%#B![&NRQ4#K\AB.4M%
#Pi|{}&&%4#
M
%H!%Â&i!J#&m!AI
\X&Îl%f Æ Af4M%WZA > &#4M%WE_USN;V?LfK\
P4= Æ
A@?LGdA@%?d_ Æ =V}T#W 4V}4#WN6&!
\X&ÎCK84M%WK8%P4"
Y}4#W_
!AI!?B#&@}&4}4&
4#AM
Î9AI&#D AMKQAj@&g&
?g}M%_Ç64S|{AI?B}M%N
±#
A#W
Ô $
AiAIM
&LIAM%
WJ|{}&&%4#_
SMkA$X{}&&
4#EN #
Ô $ NÂ?gjA#S&6AI$% 44dW4RM
N6!# Ô P4}dR=AV3A&4@% _O
4&&
}M
{TN Ô PS}dR=AL3AX!4S»NÂA#WB#E_
s[¯ (/ ¬ 5b-ÂÁ
S6AI&
4#Jb4Q&A&AVcAY%&M
f%
('*)$
&
OÎS!&AVKQAj K@&!
#PL!%@A$AY}&4}4&
4#AM
Î+
&4}4!
#;AX{}&&
4#.K@%GiP4X.&%WIAIM
MI&4#GA#=&N&A@
?g
M
AT|{}&&&%4#
K84M
WR
&
,)$
ÖeAI&?BAI!%jAM¨$X
K\Í Æ
!"$#%&
fjA#9GGH9&
_cO;4J}M
Pm
#CAi!AW%\If
#=!i!$n6&SX{}&
4#N44Cn#WC!6A
!^
M%f6A@A#AjAL_ _cO 4}M
PV%# Æ %#=!L!fn6&@X{}&&
4#EN4n6#WJ!6A
!S
M%Q6AA#YAI&jA^ Æ _ ½ _c¦I!\&6AI!
c%% &@A&A^TAf
M%\K@%& )Ï _cOÎ#
&[K84WNER=eW4RM%
#Pg!i!AW
NT4«6AIW&}M
m&LAjANTK@
GZ%[X{AX!MdK@6A
& ,) $ ?gjAI#_
s[¯;¬® .-T* > #dAM%P4R&A
[&#=&#iK@
M%MT&#W!Y6Aj4&S!&?g@
#J%_Ç64SX{A?g}M
I

S%Q&AWNÎ{9&L!^# Ä!J}K\_ > #WQ?gjA#\!6AQ44?LM%!%}M% R=C!&M%l&
?g_S
IA&
ARM%N N6%SjAM%M
WC!G´¥
X¢Ä¥XË6·oNK@
M%«%SH#K@#lAQ&UX{}T##=j_
Ç64SX{A?g}M%N+ " _
E ( -e7 ®E® * S@!&?gN $ A#W jN6Aj4[}T
AM6#6A?g_cSn&8
jAWBAm¸{ Î&6AI&WNK@
M
S&4#W
L¸{ ÎRTWE_ ª
#=&
&Q&#J&6AI&f!
#P4NA#WENAQ4CK@
M%MTIN&4?g&
?gK\^#WJ!in6#W
AV#?LRN N6Gd&6AI $ ± _S
@X{}&
#e?BACAM
YRT^K@&&&#A
±V
K@%GdjAWNSUIA&
ARM
IN ± N AM
S!VÌÉ ¡!¥¡!´´·;©&^IA&
ARM
IN _
ÖeAI&?BAI!%jAM¨$X
K\ÍJ
!"$#%&
vYy ' ' ygweq s r 7 sfs qs^zmzs 7d
> @O8?B#&
4#WAR,N§^&HJM
X&&&@AI&^b&#=!MJ&WlASIAkARM%Q%#AM
P4R!AI
fAI!%4#_
4?g^ !U?g4@4?g?B4#M%B&WZ§^&HCM
X&&&$A
N > > M%}6A3Sn6$M
X&&S !i§^&HJAM%}6ARG
N \XGAd3SU#WeM%&&!
N \§UA?g?9AJbSU!%&WM
X&!G
N 8ÃfMGA3SUb&M%!!
N ,S!Ae!%SK@
M
Mb&#=!MRT^Wd
#A#P4M%!
N AI?LRTWA
N 8
' N ^?gP4AdbS^n6#6AM;M%!!
ÖeAI&?BAI!%jAM¨$X
K\Í
!"$#%&
ygquwZr pJqSrCs^t 7 pCuvÈs pUziuygw
N4#KÏH#K)K &BjAWAM%P4R!A%f6AI&
4#_Ç64SX{A?g}M
IN^}M%D]S!
&WM
AK bK@
G
K@
M%MM%jA#dAIRT4@
#M
A&!S%
$ Å `4YK\M
Wg&AWg&
AN©S$IAkARM%N YN=&6AWB
6AM&&SIA&
ARM
IN Å NRWE_ «Ãf4#EDF
K84&ARTSK@6AI Å A#
W }&#=[b4S#KU_8OOQP4A^44eAYIAM%^ Å NK84M%W4dRiARM
!L&M%MT?g[K@6AI\AIM
fI ?BAHQ!%Q6AI&
4#C&& dO.44A#EN4JH#KÏK!9Ì´¤º¥!
AM%P4R&A
[6AI&
4#E_
SU!
GHB&Y&4M%#PV!UAI!%4#
Q!YP4X@!U6A!
#%#=!gALb4&?xM%
HI

SY&M
L%[!6A^K@6AI!X44aWJ!C!VM
f&
WV8&V6A!
#l44«#W«&JWJ!V&A?g
!
#PY!Y!&%P4=$&
W_@N©4e!AHU!4
P4%#6AM6AI!%4#EN
$ ÏÅ N,KÏWg4lP4Xc~N
4#J&UM
"8C!AH
#PV&U&6AiÐ
;[%W
$ ¨@
P4=%W
Å N
Å Ä4?Bm44a?BAjejAIM
M.!6AI Å jA#aAM%&CRLK@&&&#aA Å $ NER^W4#D ^K84&ZAR4!6A
#KU_¾
N6 ÅV NbSX{A?g}M
INK@6AI$
¿_
d3ASK\!AjKÏRb&|_M%P4P4%#PL! Y
#=!Y4SjAM%M
AI!4NK\P4
;D]@WBAVR#Gd|{A?B}M
_
s[¯ (/ ¬ 5b
- 6Á
4M4^b4 ± _
$ ±Y R!!AI $ b4?xR!
W_
;[%W
$ ± 0 $ ±
¨@
P4=%W
$
±Y $
ÖeAI&?BAI!%jAM¨$X
K\Í½
!"$#%&
s[¯ (/ ¬ 5b-EÁ
4M4^b4 ± _
Ãf%%WfR= _
;[%W
±
¨@
P4=%W
Ãf%%WfR= _
;[%W
¨@
P4=%W
±
±Y À
s[¯ (/ ¬ 5b-EÁ
4M4^b4 ± _
¨@
P4=%W
Æ ¿
À
R!!AI _
;[%W
±V Æ ¿
± ± 0 ±

± ±
o
¦I!
Q!6AW%
W
#PUA^#P=AI!4@R=YAU#P=A!%SP4%A^}T&%&%NA#WgW%W
#PUAU#P=AI&%
R=AY}T4%&%4^P%4$AY#P=AI&%_
N
±V s[¯ (/ ¬ 5b-.Á
4M4^b4 ± _
±Y
%f
^A9&&%GHÐY

_i ± _¨@?g?LRNE%4a6Aj4YA9b!AI!%4#eK@!%#ZA
S
b!AI!%4#ENE&#«!WK@#GA%&bW#?B%#6AI!G[!W,K@#GAI
&ªb
#Z!%^jAN
?B$}&Y&d}GA%&eb#?B!AI&4G|_
ÖeAI&?BAI!%jAM¨$X
K\Í
!"$#%&
wduz
OÎ#eP#!AMoNÂK8UK@
M
MEK\A#=[&B&m#%&$%#&
$M
A&_QSXeAI&J¥µ4·o¡!¥X¶9¥X¤º¹^
?g}4!A#=jNÂA$O8&&
W
!B
M%M
!!A!^AR4I_SOÎ#lP4#!AIMÄN!&mA&U!&UÎ}Tf#%&$K@
GdK@%M
M;4?BU}eAP=A%#eAI#W
AP=AI
#E
q -.6*,+Á\?g!&$b?C|N=#=&
?g!&3?C|N4bXfj_¾XN=M%
P4= Ä4A&$bM%XN=}6AI&&[3}|NAI!4#4?g
AM
#! > "fXN6%#GLb
#Â|N&_
> @!%S}T
#=jNO84#MCX{}X4&YH#,K !U?g!
f!?d
Æ ¼¾½ ©¼
4 4 4 z 13/0- Á8&4#WL3!XN4A&i!XN§^
P=AjjAmo§^!XNX!_
S&MkAI&
4#QA&I
4 ?g
#_
4V?g
#E_ ^_
Æ Y UW6Aj
=½LW6Aj U
4N 44N¾4L ² ^_
')( Á@H
M%4P4&A?Bf3HP|N6P4!A?gUbPXNMkAÖeA&.|N&_
4YP UHP
4?gV8!Y#!AR4VK@
M%Mc&? #3AI?B%M
kAI_mO$K@
M%MX{}M
A
#a4&g4&IN b4^|{A?B}M
N
K@6AIAY}6A&
_hK\X4N
#dA#=J#=jNÂ44d#Wd&gH#KÏK@AI&YWgK@#d4dn#Wd4?g
#!S%#dAI#eAM%P4R!A%fX{}&
4#_
Ç64[|{A?B}M
N
?BAP4%#^44ZAYA#lAI!%4#ZM
%H a Æ À Z_fhK\X4N.
?BAP4%#^!6AI[O@A&HIW
b4Q&A#K8$
# Z_6AI@A#44W `4?BAjaAMKQAji?mM%&
}Me4#B
WYQA#0AI!%4#R=«AJb!AX!%4#EN.&dM
4#PA^&B#?g&AI!
A#WW#4?g
#6AI&4Q !Ub&A&
4#AI&U6AMo_
6AI$O?BA#
S!
Sj4 Z_N!Ub&A&
4#
j4
©~$M
%H^?LM!%}M%
#PmR=a4_ > S4dAM
MoN% 4d?LM%!%}M%gR=a4N#!%#PYG6A#P$&
Æ À J

M
Aj4@!6AI!%4#m~AS&&ASRb4_
O 44d6Aj4U!U!AI?B^#%!S%#J!U#?g!A!4@A#WdW#4?g
#6A!4N!jAI#dA#M;4_
ÖeAI&?BAI!%jAM¨$X
K\ÍÀ
!"$#%&

s[¯ (/ ¬ 5b-6Á
Æ À Æ À
¼ À

4 4 Æ
± V
$
;D]&4M4
##%&I Z_S¦$,KUN#&
&6AI@!&mA& Æ 3AI!&$ 4#!RTI&!?dN
&
4 4 04 4!44
±Y
$
¦I!^!6ASK\6AWJ!g?LM!%}M%gR=C!i4#=4&%4#3AI!QÎK@
I_ > M
N6#&
f!6AIS&
AiÎK8
&?BjA#M%WZ4#ZR!d!m&4}ZA#WlRTI&!?dNTA#WeÎK8 &?BjA#M%W
4#J!U&4}dA#WRT!4?_
s[¯ (/ ¬ 5b-EÁ
4#
W
O J iA#W Ôi
M
MoNn6N6#!^&6AIj

±
Y
Ô$
4N,K WYK\U&M%4^&

ÔS
$
, $\
$
$
$
j4
$
$
¨@?g?LRN% 44&6AI&&4?g&
#PLK@!#!N446Aj4U!Y&6AI&iAM
MEI©&U#%&_
N
±Y 4
h K8N&?B?LRT9!AI9&
#PBK@#46Aj4aAb&A&
4#K@%&
# A«b!AX!
#ENQ!6A

!
#P4[
#l!«WK@#!A
&J&«WK@#GA%&&P4}!A
&_US%^
[&VjAI&LK@%&
!
$_
±V 4 ¼% $
$
ÖeAI&?BAI!%jAM¨$X
K\ÍJ¿
!"$#%&
s[¯ (/ ¬ 5b-EÁ
4#
WS!U6A!
#
Æ 46 Æ ¿ $
±
Y
Æ $
hK WgK8U&4M4^b4Q&
N6
#M
W%#PL#%! &}«4 §U4}AM%M !U B
#&[AI#Wd#%&Q!4P4X!
j
Æ 4# Æ ¿ 0j
Æ
&} Æ ÖdM!
}M%9AM
MT©&Ug%#=!
$ $
±L
±Y Æ ¿G44
$
$
&}d ?g}M%%V!#%&_
¦ I!
$&6AI84#B&&4}9A8&&[#!;M%#P&ENK@
M%@&[R!4? 6AI\4#MLÎK8_^#

;&[#=!%?BX!\b&? &fR&&4? jA#9RTf&WC&LjAI#MTK@%&JAm#=!%?BX!8b&4? !
!}E_

$ $
$ Æ ¿G44
±

Æ ¿G44

Æ ¿G44 $ Æ ¿G44j4
Æ !¿44G4

4 ÖeAI&?BAI!%jAM¨$X
K\ÍJ»
!"$#%&
vmuXs^wdziuQu,vxwaygz\pzu,yYw
¦I!
$&6AI\
#B&fAIRT4X{A?g}M
IN&[#?mRT&QAI&[GA!%#Pi&mP4XQ}&ÎgR%P_cS6AQ6A}}T#
AmM%\%#JA!&#4?mN&%#$K\^WjAMK@%!B4CR%PJ3A#WCA&%4#6AM%M%C&?BAM%MÄN=#?LR_OQ%
RjAI& !%Q!6AI@K8^K@
M
MT&AM
MCWAMEK@%&ZÌ X¢Ä¥XË6·3¢ UË´·Ä ·o¢Ä´Ë6_
t ( 1327 5b- ;D]@RP4%#K@!&?BUX{A?g}M%
½¼]¿ 0 ¼¾» ¼¾½ Æ 0
6AIWiAM%M6T&[#?LR8A%#B4?g?B4# JSBAM
M6Aj4fAI#B4W
#Am#?LRTfbRTXÎK\#
YA#W0=$&
?gfAdBK@%&aA#ZX{}4##_USV#
m!
#PJAIRT4LfA&m&6AI[%4Z?LM%&
}M
}TK8N64dP$AUK@! &AI!S%j_cÇÂSX{A?g}M
IN
j44
O44aAI&L4#bWaR=d&
NE&?B?LRT&LWXn6#%&
4#eX{}4##=&_i¨$?B?LRTN© 0_M%PV!&#?LRT&@
#=&Y4@jAIM
MkAI&4SA#WJ&C!AI@44P44_
;
HIK@
N %!Y|{}4##=L
^#P=AI&%N.!YX{}4##i!M
M
44K ?9AI#lW
P4![40Ag&
!g¡|¢ ·; !W%?9AIM}4
#=j
$ ¼¾
¦P=AI&%4$X{}4##&QA[AR%4#b
#P_c¨$?B?LRTb&4? P4g&G4MÂ!6Aj _N
$ $ _Sj $ 4_SN. $ j4 ¼]4_§^$% S&
V4#J?B4JjA_6AgARTJ ¦&?BAM
M4NK8dW4#EDFY4# R!LK@&!%#Pe!
N
RjAI&
3OD]$AJUK@%!
&
&_
`4jAI#d|{}&fA#=J#?mRT$%#&%#=!n6U#!AI!%4#E_
Ç64SX{A?g}M%NÂ½»N¾4
&}«4
4#=gKÈ?9A#=0}MkAL&W
?BAM\}4
#=LK84M%W 6Aj4J!a?g,J&Z!AIY4#M04##?mRT
K84M
W«RY&&YM
XUQj_YOÎ#«&
UA&IN©^K\4M
W«6Aj4Y!?B4Y4 }M
A_gS
^K@
M%McR
4@X{}T##=j_
&} Æ > &@?g,%#PL!UW%?9AM}T4%#=jNM%4HBAIS&U#?LRTS446Aj4M
XjN&
S%Q44$ g
#=_
½»44
½¼]»
&}d
4?LR
#f&^!?B_
½»G44 ½¼¾#
»
`4jAI#&C!
SK@%&4@jAIM
MkAI&4_Qj 44NA#Wl½¼]» 0!44 ½»G44_
ÖeAI!?BAI!%jAM¨$X%K\Íl
!"$#%&
s ¯ (/ ¬ 5b-6Á
[
6AI$% ½L
#&
#=&%n6^#IGAI&
4#
Öd
#PJ&YW
?BAM}4
#=YY}M
AL&&VM
XUM
A^4«K@! _¾½NE!Y|{}4##=i%V
A#W! g
#=% _¾½_
½ ¼ ½ s ¯ (/ ¬ 5b-EÁ
[
6AI$%@_¾4¿4»L
#
#=!%n#GAI&
4# OÎ#L!
cA&IN444V6Aj@&f?g,\!SW
?BAM}MkAQf}M
AN·Î´U· 6¥@¡|¢ ·oN4f!S|{}T4##=
Î_SU g
#=
@¿_]»N
¼]44¿4» ¿¼¾» 0
j ' 5*,1 ¬ 531324(*,1 ® ª(6 Ó 1b:;1 1 ®
> #!@#
}&4}ÎJ©}TK8&$8V
Q&6AIj
j %Nb4SX{A?g}M%N&6AI $ 0 j
j_
SU3AS&6AIS4eA#AWWX{}T##=!?9AIH@?LM%!%}M%%#PmR%PY#?LRT&@&%?B}M
_
s[¯ (/ ¬ 5b-ÂÁ
o½¼¾»

o# $

½¼]» Z#j
,À¼¾À#0 4¼¾À4#
À 0 $ ¦ I!
i&6AIfM
A&}E_^
#9,À_ ÀB%R
P4P4$&6A#jNOS6AIWe!C?g4!LW%?9AIM }MkA

4#U?g4&f&
?gNA#WO\AWWW?g4f!Y?mJ|{}4##=j_
Ãf%%&
#JAM
VK\4H#%M%

s[¯ (/ ¬ 5b-ÂÁ
j Æ
Æ 0
j

® <L- Ç%#6AM
M4N&6A
#PgAY}TK8@8NbSX{A?g}M
IN6
Q4
?g}M
I
j $\ $ ÖeAI!?BAI!%jAM¨$X%K\Íl4
!"$#%&
NbSX{A?g}M
IN. | $ _
&9&6AIU4 $ 4G4G44_
¬ - 2=1b(65 w (6/0- > CA0n6#6AM$!4}%NS4 &4M
W RZAjKQAl!6AC&?Bl}K\&Jg6Aj4Z&}kAM#6A?g_ Ç64
X{A?g}M%N6.K\iA&f!AM
H%#PYAR4@#!@©M
#P!
U#6AI#4?g!i3#?9
U?g
&4?g&U E?C
U?g
M%M
?g&3?g?9
$ U#=!%?BX!ib?9
?g!3?9
H
M%4?BX!^3H?9
mÖdP=A?g!ÄÖd?9
m§^
P=A?g&fÄ§^?C
S&S}n{cK84HK@!L#%!©Â?BA&NI!%?BIN4A#=!c#j_©Ç64|{AI?B}M%NA4i4?g?B#
#@ !
?gf%Q!i§^
P=Aj4AN6§^NK@
G
^ jA_
ÖeAI!?BAI!%jAM¨$X%K\Íl Æ

Download Report

Physics 131: Survey of the Universe Mathematical Review As many

Paperzz.com

Your Paperzz