adding up all divf
oriented outward
is
gives you
,
F : R-IR3
Vector field
E
=
: closedcura
↑
boundaries
3
E(X z))x2
· boundary
of S1770
S=
closed
,
y2 + z
+
y,
=
13
·
11
& (X Y 8)(x * y2
,
,
=
um
13
xy
I
adding
not closed
a closed surface S
the inside
To
·
to
points
normal wester
says
I
S
=
<Y 2z x2)
,
Compute
,
-
sides
---
offor
+
,
=
(x y 6 2x 2x)
-
OR
(r 0
=
E(x y)(x2 y2 =163
+
,
[UcoSv usinv 6-2(rcosv)- 2(rsinv)] (UV) (0 13 [0 2n]
,
=
,
,
outside (unbounded
243
2x
-
-
(xy)ED
-
,
r(k v)
=
+
z= 6
,
,
,
-
part of 2x 2y + z 6 such that x y216
parameterize in I ways
is
can
3(x y z))x2 y =16
,
+
,
,
,
(xo((iv))fldA
orientation
through S
go
,
oriented outwards
S
·
go
from inside to outside need
L
zo
F
;
3
f(x y)
inside (bounded
an
z = 4-X2 y2
plane in
S=
enclosed surface EFIR3
the
is
has
eX
ex
top bottomsheets a
13
in
surface with boundary C2
"positive" orientation
C has
Both depend on orientation
------
-
Compute (dr
make it closed
&
·
a
w
Sis oriented
~
()(dirfor (s
thei
not closed
not
THM
ke
F : E-IR3 vector field
closed means no boundary
e)
sure t
:
flow through C
let S be a closed surface enclosing solid EEIR3
THEORM :
Divergence
Assume S
D
over
f = [0, 0 4)
,
= [LOSV Sinv -Losu-Isinv)
to outside
,
2
,
xin zis
= Eusinv Kcosu Zusinv-Ikcosv]
e
E
=
EXy9
)))2ztmov
,
,
Zu
Y
-
,
Compute
(
% os
=
0
:
f
=
change
by divegerTHM :
+
E
=
Els 0 z)) 02113
,
2π
so
0
( + 2z = 2z + 2
E to cylindrical :
,
, zrdzord (e Pr
2π
e
Compute
3
+
g
S
,
bottom
side
parametrize by
=>
~C =
01z15
3
(2π)(z)(35)
=
s
cosures on
.
F(r(k v)) =
,
F (X y z2)
=
Evh xi)
.
,
,
=
,
250
:
=Ucosv
:
,
,
usinv
[Isudad
,
xin
25]
0
I
=
1 0 0 [2v ?,? )
·
=
))0x
4
O
g
10r1
,
.
·
ds
,
(f) Lixy e
-
-
=
*
(70s +3, e
I
E
=0, 0 , roseusin
=
100 7
,
Vos
---
----
/
dr
() Lududus (zudus
L
=sinv kosu o)
,
.
=
16(24)
=
32π
-----------
+
=
,
(f
:
Zit 4
·
(Fds )Firuvixid
= Kost sinv o
:
Zu
=
&
(Eds ()
oxfliur) rixi
so
--------
⑭
I
,
top
+,
,
(2z 2dz
"
=
,
integral
flux
S + Sz + Sz
=
(1 70s
0207 2
----------
as
positive : points up
curl of F
Compute flux throughs)
,
divF
I
=
vosivusit
Do rest of the S. S , and add
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