2006 fall math 100 lecture 221 math 100 lecture 22 introduction to surface integrals

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


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


kkkykkxkkkk

kkkkk

Ayxfyxfyxfyx

SzyxM

1,,,,,

,,

22




dAffyxfyx

Ayxfyxfyxfyx

MM

yx

R

kkkykkx

n

kkkkk

n

kk

1,,,

1,,,,,

Thus

22

22

1

1


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



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



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



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



24

3

12

3

111

10,10:, ,1

:2Ex osolution t eAlternativ

1

0

2

1

0

1

0

22

dxxx

dxdzxzxzdS

xxxRzxyx



20,:over Evaluate :Ex 222 zyxzσdSyz



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



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,



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.



The 2 possible orientation: inward normal and outward normal



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


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


(continuous next page)


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



downward oriented is if

b

upward oriented is if

a

thenplane,-on of projection theis ,,: If

:Theorem

R

yx

R

yx

dAkjzizFdSnF

dAkjzizFdSnF

xyRyxzz



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



3

2

downward , ,on :Sol

3

2

0 0

22

222222

2222

2

πa

rdrdra

zdA

dAkjyxa

yi

yxa

xkzdSnF

nyxaz

a

R

R


2006 fall math 100 lecture 221 math 100 lecture 22 introduction to surface integrals

Documents

fall math

surface integral slide

outward normal math

surface integrals continuous

page slide

normal direction

possible orientation

unit normal vectors