Math 479
:
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Now
,
to
pm
G
-
,
bass Bonnet,
.
§
Since
so :
,
Ezct
the gbhl
,
the RHS
Gas
K
B
=
2T E,
-
>
Barnet then ,
0
.
-
Bt
Ez
K
< 0
so
,
did
says
.
.
SSK
C
0
1
⇐
)
18
Global Guss Bonnet
th
-
If
E
is
carpet onenkbk Surface 40
a
§
kda
fray triplet
of
case
the
.
he
,
is
T
here to
Df A swfu
coordinate
f
E
sy
iht
hs
E
&
np
XT
3
.
#€€ # ¥
.
a
triangular
if
D in bj
T
Now
gin
,
a
body
of
triaynbth
E
V
define
we
#
=
ah
.
shzs
triage
tht
a
either
eapty
,
the
Di
vertex
common
a
is
imxf
the
or
,
a
cannon
*
Not ok
ok
intern B
to He
↳
left
define
of triangles
=
#
of edges
=
#
of
WHY
XHZ )
the Euler dvedvislr
Tetrahedron
D;
if oh
c-
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XH E)
Fg
{
Not
F
&
=
D
ok
Orient the
then
.
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,
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=
ES trim
He
to
:
⇐
body
,
V
:f÷¥
=
-
of
the
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tiamghtdsrtaee
Et
-
so
4
X 2
E
=
6
F
=
4
a
triangle in the plane
edge
we
a
heal
Et
V
Che
:
8
=
X= 2
E= 18
F
Exi Iusahidm
.
12
V=
E =3
F
Ej
=
X= 2
0
20
box
open
.
1 delete top
free
V= 8
fm
e=
ahs
GE
Picture
.
12
=
F
,
=
X=
p
I
10
V= 16
fame
X= 0
E= 48
F
of :
Tons
Helps
€(;@
÷
X=
I =3
better
=
32
the
seen in
squire , due
opposite
sits
are
identified
0
F =2
f.
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the Euler chodnsk
2
It
doesn't
ralf dyed
1kt
doesn't
much
not tearing
depend
an
)
the
on
you
geometry
done flviaghkn
of the surface
...
.
it
.
it
B
reafafdpebmlhewdrlyiyswfae
inmwt ( uxhenged)
under
body
.
& stetting
In
feet
,
flu
one
a
in
pres
first
a
-
semester
topology
case
-
QTIQ
to
In gmd
apt
for
.
oriented
,
sites
who
X=
bdy ,
2 -2g
,
uhw
the
is
g
f
genus
-2
-
site
the
uhih
,
handles
comb the #
of
(
Proof
66kt Guss Bonnet
-
of
:
3 F
local
By
Now
.
along
,
=
2
E
:§
Gass Bon
each
We hw
dji
edge
each triangle adds
,
at eh vote write
edge
skid
is
2
by
di
flat
holes
triangles ,
)
so
.
kda
the
=
!
¥
geodesic
,
Gi
bteah
,
kda
*
=
9
.
I
=
amative irtgd
[
2*
appears
SEE ;i
-
twin
tens net at
ante
##g
Now
3 edgy
or
w
/
I
-
§ ?sHds]
.
opposite
to
o
signs
,
so
the
geodesic
nrkx
mist
.
i
q¥€¥±giµ
,
he
So we
2
§
Bt
no
,
if
it
-
§= Eji
,
Kahn
we
=
think
§= Oji
-
it
so
,
the
alone
red
to
us
,
=
? I Eogi
aht
sun
mig
.
it
]
armies
ins ted
f tvs
,
the
angles
in
kt
to
a
single
sun
to
25
?
so
Oji
Es
§
2tV
=
Kd Am
In fat this pray anazyft
1
fit
,
beg
by
664
she (
an
me
Gass
.
Barnet
Some
me
consequences
D
2
B
E
hk
keep
F
-
2 E) =2# ( V
nlgusht
obviosf
s
to
(3
Ftt
it
-
conpt anwthle surface
a
&
f
+
§
Kd Are
the extnw asly at
E
+
Ei
corners
=
f
/
w
,
f
the choice
tetdttatf
F)
Et
-
flviayldh
2 HXH E)
=
#
.
.
relied edgy & bandy vrtuslogs )
interior
piecewise
smith
bondy
,
ZIXIE )
52
.
:
A caput safe wlo body
As a key stted
,
geodesics
sipf
-
,
,
Kolar
he
we
Shifrin )
Kg l Dds
{i
§
F
it
=
it
2*V
=
well (you he
mere
see
If
:
§
the the
bit
little
a
I.
IF
-
since
,
,
I
V
2A
=
Since
.
E be
Siple
a
in
psi the
of
and solves
ngakef
ornate
Gaussian
a
,
is
Hpolgrdf )
it meet twice
so
Hit
a
sphe
the resin
.
bw the ntxekb
is
.
came
:
tpdogid cylinder wlngatk ornate ( tr exapk, the Gtnoid)
Then
.
E by at nest
one
closed geodesic
¢
±
PT
:
son
them
Cmseewtne
2 called
,
intruding bad
*
€
=
&
C
a
snf
D
.
If
connected
c
nd
resin
,
=/
¢ fm
,
uhih Btpossihkhy
2
.
anew
9€
Bt
then
the rgin bw C
=
NOD
:
p1#
X
Smu
arte
idgneht flvignhth
s
.
just
we
he
.
bass Bonnet
By
ants
bah ±2+
&
,
an
or
¥
-
For the
Sok
agathe
an
annulus
:
.
,
the
in
In
.
st
we
and
to find
triangle th for
a
ants
san
&
ftht
X
K¥2
tightly
Pfuj
3
0
=
.
.
hY@*
÷
claim
D
&
Kd
plane
fit
O
,
the
.
,
2
c
ZTXHH
wehr
cylinder
Art faksds =2#X I A)
1<=0
,
"
& the HH amatwesfte
ids
Let
XHH
the rginblw
C
be gpsitef
=
muted
,
inner
so
& ok ads
me
the gods hamlets
and
Di
0
.
& D hang 11=0
.
so
,
by
Gas , Burnet
.
,
SS KdHr=o
hhih
4
Any
sink
two
p§
.
is ipassike
closed
K
since
gwdeks
Supp why didn't
on
a
< 0
#
.
surface
f
posit
karate
most mtrsut
.
TETLEY
:S
:X
:
.mn?I:Imk
€i*
,
'll start
As we
So
long
week
,
disunity next
a
awake
Next
week
hw
you
&
the
this
no
red need to
first fdanbl fun
geodesics
will talk abt the
,xn
can
assume
apt
svtns
are
contained
the Ch Dtoffd symbols
IN
in
,
& time the bassin
.
,
.
hyperbolic plane ,
nhih
is the
fdamhl satin wl oft nglnecnntwe
.