math methods physics - astronomy.nmsu.edu
TRANSCRIPT
Math Methods in Physics 1
Prof Wladiwir by a
Sep 8th.
616
Class # 6 .
logarithms of complex numbers
-= ln t = lure .
= la r + io
. . lni = kits= IIz
An - lit net '=
it.
ei "+1=0 → ei#= -1 → en H = it
Complex
powerswi€lnw = i.lu i
÷i hfei " " )
= - Iz
6
.ii=w=e€
I'
is a real number
-
.
hint 4h thatthere are
several solutions
i-eit.tn
.
Heit"Yi= e
-#2
= or tl" it
,?ggI÷m¥?ea±¥÷¥e;!eek÷aiI¥¥#.
iii if i)'
= ( iMYE e
-
t.eu#tIe'' " " n ' '
=it
,
is"e'
i2
:
= ( i )' ' =p" " )
' 2
.
-
e
T
( est,
is?,
. ).
what is
this? des it
mean
thatit : e
M?
.
Nope .
This is like when m will VT = ± 2,
and obviously.
- 2 ± 2.
But this illustrates the s: waken with imaginary numbers
is not unlike that with negative numbers .
why is it that ( l ) ( - A =L ? One can saythat t multiply
by . 1 is equal to'
'
revert ''
the magnitude in the real axis.
In this case,
:- the complex plane .
to multiply by i is
1- rotck forward by so ?
tell a bit more of the history of complex numbers.
Mph.int.
on .
.
md¥e= -KX harmonic oscillator
X= Aeiwt lansdt )
ii 's Aiw eiwt
x
' '
- . awzeint
: . - Aw2e#t.
.- K4PiW÷Uwig
-ok asks them to memorize ,do problem in your need .
I need to tell
themHat it s not needed ¢ memorize anything or do anythingn one 's heed .
-
cos Z = 2 ; arc cos 2
Gt:e¥e' it
= 2
2
n = eit; e-
it= Yu
.
.
. n the = 4 → n'
- 4 u +1 =D
"4¥11 : 2
:±JJei In = z±vJ
Z
.
.
Take los on both sides
.
it = leg n = log ( ztvz ) = Ln ( 2 ± JI ) t Znti
z = Zut - i Lu ( ZIJJ )
=me
notationlnz : lu ( re¥±#"
) : lur + it± 27N
Ln 2 = principal value of but, for which OEOEZT
ln isanoperator,
ineatttgebrahmb: nation of < lgebra and geometry .
Give a brief refresher on matrices
.
Some useful matricesThe rotation Mattix
:
Example:4→Series ofmatrices .
words of warning ( At B) ? A 't ADTBA +132( Don't write ZAB )
EATB =/ EA. e
'
if A,
and B don'tcommute .
Dopit 29 and tbnl ( 5 .]l=
RecapitulationsZx - z = z
6×1 5y+37=7X=(§) b = ( }y)Zx - y
= 4
A =( 2g E,
so ) Ax=bor
: Aijnj = b;
j=i
Einstein notation
3
E
Ajxj = 5 : ⇒ Aj x ;
T bijet
Omit
summations.
Sum over repeated indices apply .
.
Applications .
write equations in component form
JY = 2 it
J . A = 2 : A ;
Jxa = @; Ak - 2kApi
FA =2jjA÷=11 -
Row reduction ( Gauss elimination )
,A
. 6×1 + 3-2 = . 6- a ,
5y +6 z = 1
⇐÷¥kH¥
2×-7=2( A )
5y+ 6z = 1 ( B
'
)
Zx- y =4 ( c ) C
'= C - G- A
an.
dj =D}
. dietZx - y = 4 q- Zx tz :
- 2
÷- 7=2
Erchenge B and C
A Zx - z =z
131 5y + 6z = 1
1 - y + 7=2 a"= C
'
- bi B
4
%+⇐,
d"
'd ''aegis
- 5y i 52:#AtlZx - z = 2
- ytz= 2
z = 1
eiiuinaetzre,.÷3
, ⇒Q###,
)
Honea .Fosler :
" Matrix augmentation"
HE!:D " '¥,
!DI"¥.IM .
Est.I*K .it?h..HYs "
¥;D "EeiH&:c:D:c ::g#÷a
,
Kramer rule
a ,x + b
, y= C
.
GZX tbzyozX = c
, bz . czbl
9¥ ,
Yi 914cga
, bz .
azb ,
Kiwi
'kn'/ 9
× = C zlaTy51€. Ki"bd
Ax =b
× ; = detail A ; is the Mattix formed by replacing the
¥1 i. th column of A by the column vector b.
3×3 systems
qxtb , ytc ,-2 =
¥⇒↳3Fs⇐Ft
in b, c ,
=-D,
Kiss.DE#kt*kxyx.tk#;HHkI;HeiskYbsHs.Elx'=y1x.oHi=rlt.rEyo=uHutr.TEjb;:..rn-rrt=n
'
y{x+rt=t'
FEYAHIYin .H#:*
'
:L
n . l¥' YI + .
. Hanil←MIYI KmtYI1 ], ;yr / =P ;
sir 'k2 = yyn - Me ) = i
#.
ryy- n 's trrt
'
. ylnivt' )
HE.nti / = ritsirki:
r ( t'
+ nig )in=rkitrt' )t -r
Hintz)
.