2006 Fall MATH 100 Lecture 22 1 MATH 100 Lecture 22 Introduction to surface integrals t , , , be density let the : lamina bent a of Mass z y x R y x dA f f y x f y x M M R xy y x f z z y x 1 , , , by defined is lamina the of mass the then , region the is plane on the lamina this of projection the if and ; , equation the has , , density with lamina curved a If : Def 2 2
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2006 Fall MATH 100 Lecture 221 MATH 100 Lecture 22 Introduction to surface integrals.
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2006 Fall MATH 100 Lecture 22 1
MATH 100 Lecture 22 Introduction to surface integrals
then,,, bedensity let the :laminabent a of Mass zyx
R yx dAffyxfyxM
M
Rxy
yxfzzyx
1,,,
by defined is lamina theof mass the
then ,region theis plane on the lamina thisof projection theif and
;,equation thehas ,,density with lamina curved a If :Def
22
2006 Fall MATH 100 Lecture 22 2
Definition of density function:
zyxSS
Mzyx
,, containing area theofsection samll theis where
lim,,
R yx dAffS 1 is area theand
area, surface the toequals mass the,1 when :Remark
22
2006 Fall MATH 100 Lecture 22 3
kkkykkxkkkk
kkkkk
Ayxfyxfyxfyx
SzyxM
1,,,,,
,,
22
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 4
MATH 100 Lecture 22 Introduction to surface integrals
dAffyxfyx
Ayxfyxfyxfyx
MM
yx
R
kkkykkx
n
kkkkk
n
kk
1,,,
1,,,,,
Thus
22
22
1
1
2006 Fall MATH 100 Lecture 22 5
mass its find ,,, ,10 , 1Ex 022 zyxzyxz
156
144
1
3
2
142
12
14
122
2,2 ,1: :Sol
2
30
1
0
2
3
0
1
00
2
0
1
0
20
220
22
u
duu
rdrdr
dAyxM
yfxfyxR
R
yx
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 6
upproduct thesum and
area surface with into Subdivide .on definedfunction
continuous a ,, and surface finite with surface a be Let :integral Surface
1
1
ni
n
ii
S
zyxg
n
kkkkk Szyxg
1
,,
k
n
kkkk
nSzyxgdSzyxg
g
1
,,lim,,
:over of integral surface thegiveslimit Its
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 7
dAyyzyzyfggdS
zyfx
dAyyzzxfxggdS
xzRzxfy
dAffyxfyxgdS
yxfzg
zx
R
zx
R
R
yx
1,,,
then,, If (c)
1,,,
thenplane, onto projection theis and ,by defined is If (b)
nimplicatio wider has expression The
1,,,
then ,, and If (a)
22
22
22
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 8
1 ofoctact first over the Evaluate 2Ex zyxxzdS
24
3
223
2
13
1111
10,10:, ,1 :Sol
1
0
22
1
0
1
0
22
1
0
1
0
22
dxx
xx
dxxyyxxy
dxdyyxxxzdS
xxyyxRyxz
x
x
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 9
24
3
12
3
111
10,10:, ,1
:2Ex osolution t eAlternativ
1
0
2
1
0
1
0
22
dxxx
dxdzxzxzdS
xxxRzxyx
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 10
20,:over Evaluate :Ex 222 zyxzσdSyz
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 11
2
21
2
cos1
2
21
sin2
21
6
sin2
sin2sin2
2
20,21:, 21 so
, :Sol
2
0
2
0
22
0
2
1
26
2
0
2
1
252
0
2
1
22
222222
22
2222
d
ddr
drdrrdrdrr
dAyxydSzy
rrRzz
yx
yz
yx
xz
R
yx
yx
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 12
Surface integral of vector functions, we have studied line (curve)integral with orientation, now we go to surface with orientation.In general, a surface is given by G(x,y,z) = 0
The particular cones are
zy
zx
yx
xxGzyxxG
yyGzxyyG
zzGyxzzG
,,1 ,0,
,1, ,0,
1,, ,0,
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 13
There are 2 unit normal vectors
A surface has 2 orientation, corresponding to the 2 normal direction.The orientation should be chosen in the way that there is no sudden change in the normal direction when we transverse along the surface.
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 14
The 2 possible orientation: inward normal and outward normal
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 15
Surface integral
.over of
component normal theof ginteractin surface or the
,over of integral surface or the
,over of integralflux thecalled is
thenn,orientatio theofr unit vecto theis ,,
if and , surface oriental on the components continuous
has ,,,,,,,, If :Def
F
F
F
dSnF
zyxnn
kzyxhjzyxgizyxfzyxF
2006 Fall MATH 100 Lecture 22 16
dSnFkzjyixzyxF
xyyxz
evaluate ,,,let and
normals, upwardby oriented be let
plane,- above 1portion theis Suppose :Ex 22
1
2 ,2 :normalunit upward :Sol
22
yx
yx
yx
zz
kjzizn
yzxz
MATH 100 Lecture 22 Introduction to surface integrals
(continuous next page)
2006 Fall MATH 100 Lecture 22 17
2
3
1
1
122
22
11
:Sol
2
0
1
0
2
22
2222
22
22
22
rdrdr
dAyx
dAyxyx
dAzyx
dAkjzizF
dAzzzz
kjzizFdSnF
R
R
R
R
yx
R
yx
yx
yx
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 18
downward oriented is if
b
upward oriented is if
a
thenplane,-on of projection theis ,,: If
:Theorem
R
yx
R
yx
dAkjzizFdSnF
dAkjzizFdSnF
xyRyxzz
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 19
dSnFkzFazyx
evaluate , normals, outward oriented : :Ex 2222
3
2
upward , ,on
:Sol
3
2
0 0
22
222222
2221
1
21
πa
rdrdra
zdA
dAkjyxa
yi
yxa
xkzdSnF
nyxaz
dSnFdSnFdSnF
a
R
R
MATH 100 Lecture 22 Introduction to surface integrals
2006 Fall MATH 100 Lecture 22 20
3
2
downward , ,on :Sol
3
2
0 0
22
222222
2222
2
πa
rdrdra
zdA
dAkjyxa
yi
yxa
xkzdSnF
nyxaz
a
R
R
MATH 100 Lecture 22 Introduction to surface integrals