Math 474 :
Day 23
.
1¥
Re U B
If
.
bounded
a
Area
Now
,
He
use
we
in
region
IECRD
:
=
He parameter
A real R)
regular surface given
a
of
by
I
:
UuR2
1R3
,
then
lxixxvldudv
very handy
identity
IT
+
Linens
VEGI
={{
Ahl RI
EFM
gets called
f
determinant
the element
dudv
area
of
]
[
the matrix
Help
has
=
to
write
where
§
=
Plane
Notice tht th
.
which
,
FEETµR2
To E
maps
( cosy
B the
quality
}
root of the
square
.
XTQ
III.
=
0
Using
Ot
,
stdd
the
he
tu rgin R
f
area
We already compiled
E
Sino
=
,
{(p
-
.
F
=
0
,
a)
,
:
0<
6=1
,
,
cent
)
cos
,
a
before
as
out
}
te
,
in th }
.
ah
sense
of the sphere
.
so
sina.s.nys.no
coordinates
spherical
eoffz
Aalst
Not
th
:
Sino
same
in the
.
is the
}
tht
=
2fj%ok
.
-
appears
When
,
integrating
writ
.
.
so
ftp.T.naao.ae
;{¥"jEdoae=f"j5o=Faoaa=
ktgmthfefr
Ps no drdody
ifno
=
spherical words
.
Sino d Odle
=
226%1
dll
=
4 'T
Which
,
is indeed
the
area
fte
sphere
The Gauss
Mp & the Secord Fundamental Form
.
Remnbr tht
noml
The
idea
we
Cwntwe of
defined the
dick
vector is dungy
far
Surtees
B
panmetized
the la
are
"
b)
which
1
measures
fastthe
too
,
.
less the
or
more
Her gk
on
same
Again ,
.
K€#D÷a
we
want to lode
how mah the unit homl
vector
B
:L
!
Iii:;
where
:imo
in
Suke
beds
distinguish
in
th
}
scenario
drum
opposite
day g.
"
"
The god B to
fp¥#§,g
-#
at
fan
one
mdiffat and
when the
lines
.
:
-
Df A safe
E
.
the
IT
np
Gm
a
is
}
poanetizth
NTP)
U E R
=
B
±
IN
-
,
diftadinbknp
a
the bays
mp
often called
Ri
if the
orientate
5TV tht
NTPLTPE
E
:
at
so
every
PEZ
.
man
write
are
orient He
D
explicitly
since
×¥
in
xnxxi
DANGEIT
Now hues
,
.
Not all
whe
Since D:
E
things
s
regular
stt
sorties
to
set
oksiy
.
Far exaple the Mob
,
:
us
strip (
see
µH
:
;
hands
-
on
demo
)
.
dpp
treahpeEuhaeffy@dNpiTpZ-TMp5.BtTmp5tNlp1TpZ.so1heBamtnlidenlifwKmTn.5Ed.Tp
Z
and
we
can
think
of
d Bp
as
a
mp
dttp
:
TPZ
TPZ
.
.
Of
course ,
you
first
qmstm shed
he
:
why
the hell
wild
we
do such
a
outing
thing
?