Tutorial 3
-
f.
ge
CLN) (ft g) In)
Heinen
F-flqtf.HN① MIN topdog .
Spaces ,
F : M → N Continuous
map .
=
F-
*
: [ (N ) → COLM ) F (f) =
f. F : M → IR
④ F
"
linear ,
f. ge
COLN ) ,
be IR FHtJg)=f7+JF§f
Flftsg) ( m ) =
# tsg ) . F) ( m ) ¥ fltlm)) TIGHT)
Htt g) ( Fln))
← =
#after Fi) .
er
③ Assmue M ,
N (Smooth) mfds .
F- is Smooth Fiel (N) ) E (M) .
"
'
g
E ( N )
F-
g
=
got : M → IR is scuoodu Os Composites
of smooth fcls .
F-
"
Fix + EM and let ( V. n) te a and of N with
FK) EU . We war extaud u to usuooh tot
ü : Na IR
"
ü _
III
)
s.t.lw.l.org. ) ülu - u .
By assumption ,
Für'
= üio F : M → IR is scuooh Ki Support
(V.v ) is a work ea N wir × EV .
)
Ü .
For
-
^
\
: v ( Un F Yu)
) → ü ( O )
-
„
µ vlvnt
-
'
Lu ) ) UIU )
is sueooh vs
Loup .
of saudi µ weops .
Heure ,
F i ) Icuodu
by del .
of scuooheness for
mops M → N .
⑤ Assuue F is a www.eau.hekweeesmookuf Mad N .
F- ^
:
N → M wartinnen s inverse .
F is a diffeau .
Es
F
'
: CNN ) - OLM ) is
on
isomorphen .
F. F "
=
idar F-! F -
1dm
( Fo F
-
i
)
'
=
Idi =
Idc.
µ,
# F)
'
g
=
g.
F. F- 1
„
E)IE =,
¥µ
-
KIF}
So the statement fellows for ⑥ e-
id
② s Key ,
z ) =
2¥ §
+ 3
¥ vf .
an IR
>
-
-
In
cyliudricd koordiniertes : v
-
^
(r
, y , z ) =
(rcosy , rsiuy , z -7 )
Jacobi en of now -1
= idov -
^
i. ÷::
:)
"
÷:
:)
:L:
÷: :) ! ) .
=L:: : . .
) .
*
+ 3 d-
Oz
.
In
Spheiwd wo dindes :
v
- ^
lr .
y ,
0
) =
( r sin 0
cosy ,
r sie -0
siuy , rcos G)
Jacobi on
of idov.li v . id
-÷
÷::*:
::
::::::
::)
=) 9 =
(2 6)
y sin 6 sie
E
singt
3 los -0 ) ¥
-
2 sie
y t los
4 d
-
-
suo
Oy
+
2 los 0
Gsg wsfs.my
-
3 sie 0
-
¥
Jacobi an idov-1
:
÷: : :
:
"
: :
÷:)
③ M =
{ (x ,
y ,
z) EIR
'
: xzty
?
= 1 }
M =
f-'
( o) f (x.
y ,
z ) =
xzty
?
- 1
Tun .
M = Kerl !!) = { lxiyl.tt/tIR3ixx'xyyzo}
4. f Key , >
) =
If (¥! ) = 2x
kzty?
- 1) =
O
f-er
(x.g)EM-
Heule ,
9 is
taugen to M ( i - e .
9)
µ
E HIM) ).
q.tk ,
y ,
z ) = If %) =
2×2+25 * 0
for Key A) EM .
=
) h is net
taugen? to M .
④ M = IR ?
Sky ) =
y # ↳ y
) 4¥
"
ftyky ) .
[
sind .
Ey ¥ ,
f) =
9×95 ¥
.
dt) =
) is an
integre
curve her Is , n
]
⇐
eilt) = -
¥)
2
↳ lt ) =
↳ It) GH) .
=)
↳ ( t ) =
2a
haushoch
+ ta
(ZH ) =
Ltte )? b b lastet .
Inka conditions do) =
Key) a =
¥ ad b-
;]
9!) FEHLT
, Kag ) ) -
(III.
HEY:3 )
Net deficient fort = ¥
; Lance Es , n
] is not
uaeplehe .
Some
applications of the Frobenius Then :
Cousine the followiug linear system of PDES for f : IR! IR
(*)
-
Zz
?
0¥ + 2x
9¥ =
O
-
3. z
?
0¥ + 2g ¥ = 0
Does 4) los
any non weiterer
Solutions !
X = -
Zz?
! t
2x
¥
|
Spon
o renk 2 diskutier
E on the Sunset
Y = -
3 z
?
% t
2g % ✓ =3 µ , y, a)ER?
: ZEO}
( X f
= 0 4. f--
o is (*) ) .
ER ?
More over
,
IXM ] = -
12×7
% t 8 yz ¥ =
¥4 -
4¥ X
Heule ,
E is
integrale by Froh .
Then .
7 loudly arend ay Ko . go.z.IE V v Chart ( Ku )
s . t . E is
spannend by # and
# -
* in mordendes
VI.¥!!! ) is
eguw.
ho
¥ =
!-0
.
Hanel , f =P is a sanken and any
Idee hier in
usuftic .
Sind
heiß bh .
of k.
. y.io ) is of the force fk,
g. z ) =
gleite,
g. z)
for Sword for .
g
in one neide .
Ein System of PDES for f : IR
?
→ IR .
.
0¥ kiy) = a (x. y ,
f- Ing ) ) (** )
•
¥ ( x.
g) =
ß
V. 9 ,
f Hg) ) a
, es
smookfcts
del .
an
grenzend VEIR?
I When does (**
this a sdutreu ?
Necessery condition for a and ß :
%a4.y.ftyD-dpk.gl/ny-o&suceIIg--ItTox
Chain rule =)
%'
M¥0 =
0¥ t d
dfg (** ).
µ * * ) is ne coss .
Condition for Ex) to love a Solution
in o
neigen
bh .
of my point
↳ , yo
) wir Obi
tag
vduefk.igd-to.BYFroh .
Thun .
H# * ) is also sutticieul : it in
pcie,
that for any
K. .
yoizo ) EV 7 an
gar heiß .
U
of Kay.
) EIR ?
and a
unique
Solution f ' UTIR of Ex) weh
fko, yo )Zo
.
Wwf ( * * ) pnescnbes the
tagen plane
holen gnepn
of f. Collection of the
hagel pures defies
renk 2 deiner hutie of V
and ¥) is eqvüv .
to
iuuoeuliuihy .
Suppoge f : U → IR were a Solution ( UEIR?
Open Subset ) of # ×
)
Then 4 : U → IR }
4kg) =
(x. y ,
fk , y ) ) ( Y Aarau .
afsubuf.gr/f) .
)
is a
diffeou . aeho grlf) -
T
#yßrlf) is spannend by
„
dkiy,
ft#y) )
Trunk ( ff ) =
¥ +8¥ ¥
Tunis =
¥ +
¥+79444911
× =
¥ + <
¥
vf.eu Vi
Y =
Ey t ß #
Spon a rank 2 dir .
E au
V .
E is in wanne ← (* **) hdds .
A- In Wis lose
through my point Koiyoito)EV
7 on
Integra schuf . NEVER
?
while lowdlg Los
tue form
grlf ) for a Sund fd . f. UHR ,
U open
# Kongo)zo
. neigt.
of Kaya