Tutorial
.
① Cyli under
M {
lx ,
g. z) E 1123 :
xztyz = R
?
} R > 0
f- : U → IR V.= IR?
IILO,
Oiz) : ZEIR } .
f- (x
, yiz) =
xzty
?
-
R
?
f-
'
(o) = M .
Day! =
(2x , Zy ,
O ) is surjediettkiy.DE U .
=) MEIR
'
2 div
.
subuefd .
•
Graphen
around point s (x ,
yiz) wiky # 0
glx , z) ±
Lx , z) z V ERR)
× IR
Mn ! =
grlg) Xx
, glxcz) , z ) EIR
}
} .
ER?
4 :
-
LR
, R ) x IR> ox
IR
< o
Simi
Leary ,
bar pocht, hey ,
7) EM wie X ⇐ 0
glyiz ) - ±
FREI ( az ) e V
Ut : = IR
;
× LR
.
R) × IR
Parameterzahl .
Y : La , z ) → ( Rosa ,
Rsina ,
z )
£ E [ 0,2T) × IR
Taz ,
4 =
III § ) is injedie MIR
?
Since Cosa and Sona
hat zero of Kil Soul
Trivialen Cylimaroad ••
aiade,
time .
(¥,
}
=
[ Ii!
) r
> 0 aeto , zit)
d : KIR'
Kos → IR
}
011M) { (¥ ) : r -
R}.
LE) - Ez)
② Double neue
<
M =
{ Key , z ) EIR
?
: Lxztyz) tanze =
z
?
}
flxiy , z) =
Katy2) train - zz
Day! = (2 taia ×
,
Ztaix
y ,
Zz
)
vannes at love) ; f mal
aeguar
teure .
=) MIKO, 0,0) } is a
submfd .
of IR?
Grey
: z = ± ten a
Fly
z
÷:: s ..
.
.in#f¥4
!
(y , s) m (s cosa
cosy ,
s cosa
suq ,
s sind
""
=p:O:: :
÷:): -
"
it is in
g- eure IR ?
- 1123 .
Trivialitaeten : Sp her ad Coordinator
③ Homr UR
"
, IR
"
) E Har ( IR"
.IR
"
) - IR
" "
To e Hon
,
LIR"
,
IR
"
)
IR
"
= E ⑦ Et Et = Kerl! ) deine (F) = r - duale) .
&)
1km = F ⑦ Ft F = Im LT. ) a
Win respect ho k) we war
ideehty Te Hau KR"
,
LR"
)
wir a
marie r n -
r
TI §)
At Han LEE)
m) BE Han LEI , F)
( t Han CE , Ft )
DE Hamlet ,
Ft)
-
„ U :
=
{ T =
(E3) : A iuuerwble} open
Sunset ,
heute
C- IR
" '
"
ie
G- Kr) E Hahn)
To EU open .
„
! ) Ao inuowble .
Fix TEU .
TE Hau .HR "
,
1km ) ← dein LKERLT) ) n
-
r
TK) - o ⇐ (± :B:) =/ :)
4 , w) EE ⑦ Et
=
IRN f) V = -
A-
'
Bw
( A-
'
Bw =
DW
=) dem Werk) ) = nr
←
tweet (A-
^
Bw = Dw
⇐ CA
-
'
B =D .
g :{ (E !) E Hau LIRYIR
"
) : A is inuehble }
aOpen
Sunset
of here vedarsubspeocee
9. SIE ! ) e- Heult! Ru)}
E Hau LIR! )
→ ( A- '
B E Hen LE ,
Ft) .
Smooth and
grlg) = Us Haar LIRYIR"
) .
ÜEB) KEI) ,
g E3 )) = (t
In particulier ,
it r = min
µ ,
m } ,
then dim ( Hay HR"
.IR"
)
= h .
Uh .
Haar LIRYIR "
) E Hear HR ! IR "
)
gebe
Sabah .
① Grass ueouinäeu
variety / maifeld .
Er (r
, n) =
{ W : WEIR"
is a r die
.
seebspoce
GR"
}
= Set of all k r -
dem
.su/sspocesefR
"
.
G- (1 , n ) = IRP
""
( projedive) .
Er ( r
, n ) is a sabueld .
of How
.
LIRYIR
"
) of dem .
r ( nr
) .
Idee ,
We
identity r-dim.arbspcesof.IR
"
wie tue er
koag .
=
projeoieuseukotuau.hn In Kers
wog tue
noealize Er lr , n ) as a Sunset Han
,
UR
"
,
IR"
) .
Fo E IR
"
sub tpoce ab dem .
r
,
IR
"
E.
⑦ EI .
# )
Po :
IR
"
→ IR
"
arkeog .
groj .
ho Eo
g
www.kswihaesreahoE) just .
es P
.
Ilie ) .
Open heighbh .
of Po E Homr ( IR "
,
IR
"
) is
gar bg
U : III
{* sß) : A sinnende } .
Recall a dinner map P : IR
"
→ IR "
is on
okeog .
projediae
to some sub Space of R "
⇐ P? P and PEP .
When EU an
erluogoa .
protease ?
⇐ *
III.e.) = LEE µ)- →
A - At , B = Ct ( KA
-
'
B)E- (ARB .
)'
\
TET
µ
TBC AB t B CARD
CATCH Bc CB + ((* ,
B) 2)=
(I c
!)
=) AZ t BL = A
( AA tccz A )
} lfthi' is 'disfreah all den
•
icdeaeitres al :
AB t ( A - A) A-
'
B =
AB t B AB
=
B .
-
=
BC
f- :/ 3) e- Haut IR! Ru) :
A EGLI r ) }
→ (AEA TEC - A ,
B- Ct,
D- CAYB) .
PY.EE
"
I
"" " ) " da"
? ÄH
!!
;)
A- = ÄA x etc (A)t
= Id + LCÄITKÄ)
"
A-
'
=) A =
(It ZTZ)
-
1
E- (A- ?
Pa raudi zoten ↳
4
: Hau In: nur) an> z ↳
(
KEE!!
B
:)Z A CA
-
'
B
In+ ztz )
= : C
- dim = r
'
t r (n r
) + (mr
) r
= r ( htm Zr
+ r ) = r (ntm -
r