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-N.S.PRIYAASSISTANT PROFESSOR
DEPARTMENT OF APPLIED MATHEMATICS
onAcceleratiVelocityExamplevectora
calledisdirectionandmagnitudebothhaswhichquantityAVector
TimeeTemperaturExamplescalaracalledismagnitudeonlyhaswhichquantityA
Scalar
,:
:,:
.:
potentialelectricalandspaceinpoanyateTemperaturExample
functionposcalaracalledisspaceinpoaofpositiontheoffunctionaisthatquantityphysicalscalarA
functionpoScalar
int:.intint
:int
.:
.intint
:int
forcenalgravitatioandparticlemovingaofVelocityExample
functionpovectoracalledisspaceinpoaofpositiontheoffunctionaisthatquantityphysicalvectorA
functionpoVector
axes. Zand YX, thealong rsunit vecto are k,j,i Where
z
ky
jx
i∂∂
+∂∂
+∂∂
=∇
as defined isoperator aldifferenti vector The
:Operator alDifferentiVector
zk
yj
xi
∂∂
+∂∂
+∂∂
=∇φφφφ
( )
bygivenfunctionpovectortheThenspaceofregioncertain
aindefinedfunctionposcalarabezyxLetint.
int,,φ
:)int( functionposcalaraofslopeGradient
φφ gradbygivenisvalueimumtheandgradofdirectiontheinimumisderivativeldirectionaThe
maxmax
:Note
ngradderivativelDirectiona ˆ.φ=
:derivativelDirectiona
xykxzjyzi
++=
( ) ( ) ( )xyzz
kxyzy
jxyzx
i∂∂
+∂∂
+∂∂
=
zk
yj
xi
∂∂
+∂∂
+∂∂
=∇=
φφGrad
:Solution
:Example
(1,1,1)at xyzif grad 1.Find =φφ
( )( ) kjigrad
++=1,1,1φ
( )( ) ( )( )
( )( )222
222222
log
loglog
zyxz
k
zyxy
jzyxx
i
++∂∂
+++∂∂
+++∂∂
=
zk
yj
xi
∂∂
+∂∂
+∂∂
=∇
φ
:solution
( ) φφ ∇++= findzyxIf 222log.2
[ ]2222
2 zyxrrrr
++===
[ ]kzjyixrrzyx
++=
++
= 2222
( )kzjyixzyx
++
++
= 2222
++
+
++
+
++
= 222222222222
zyxzk
zyxyj
zyxxi
kzjyixrwhererrrfrfthatove
++==∇ ,)(')(Pr.3
)]([)]([)]([)( rfz
krfy
jrfx
irf∂∂
+∂∂
+∂∂
=∇
zrrfk
yrrfj
xrrfi
∂∂
+∂∂
+∂∂
= )(')(')('
rzrfk
ryrfj
rxrfi )(')(')('
++=
rrrf
kzjyixrrf
)('
][)('
=
++=
rrwherernrriirrri
thatprovekzjyixrIf
nn
==∇=∇
++=
−2))
.4
kzjyixrGiven ++=
)1(2222
222
→++=
++==
zyxr
zyxrr
)2(→∂∂
+∂∂
+∂∂
=∇zrk
yrj
xrir
rz
zrz
zrr
ry
yry
yrr
rx
xrx
xrr
From
=∂∂
⇒=∂∂
=∂∂
⇒=∂∂
=∂∂
⇒=∂∂
22
22
22
)1(
)2(invaluesabovethengsubstituti
rkzjyix
rzk
ryj
rxir
++=
++=∇
rrr
rr
=∇∴
=
nn rz
ky
jx
ir
∂∂
+∂∂
+∂∂
=∇
)()()( nnn rz
kry
jrx
i∂∂
+∂∂
+∂∂
=
zrnrk
yrnrj
xrnri nnn
∂∂
+∂∂
+∂∂
= −−− 111
∂∂
+∂∂
+∂∂
= − kzrj
yri
xrnrn 1
rnrrrnr
rnr
n
n
n
2
1
1
−
−
−
=
=
∇=
rnrr nn 2−=∇∴
( ) ( )
( )22
2222
4
44
xzyzxz
k
xzyzxy
jxzyzxx
i
+∂∂
+
+∂∂
++∂∂
=∇
φ
3ˆ kjin
−+
=∴
kjinHere −+=
:Example
kjiofdirectiontheinatxzyzxofderivativeldirectionatheFind
−+
+=
)1,1,1(
4.1 22φ
:solution
3916 −+
=
3).96( kjikji
−+
++=
nderivativelDirectiona ˆ.φ∇=kji
96 ++=
( ) )81()1()42()1,1,1( ++++=∇ kji
φ
)8()()42( 222 xzyxkzxjzxyzi ++++=
32−
=
kzjyix
842 +−=
)42(
)42()42(
222
222222
zyxz
k
zyxy
jzyxx
i
+−∂∂
++−∂∂
++−∂∂
=∇
φ
:solution?max
?max42
)1,1,1(int.2222
derivativeldirectionaimumthisofmagnitudetheiswhatimumazyxofderivative
ldirectionatheispothefromdirectionwhatIn
+−=
−
φ
( ) kji
842)1,1,1( −−=∇∴ −φ
.intˆ
int
pogiventheatsurfacethetonormaldrawnoutwardtheofdirectiontheinnvector
unitthemeanspoaatcsurfaceatonormalUnit =φ
:normalUnit
( ) kji
22)1,1,1( +−=∇φ
kxyzjxziyzx
)3()3( 22 −+−−=∇φ
φφ tonormaldrawnoutwardofdirectiontheinacts∇
33 zxyzx +−=φ
:solution
)1,1,1(int1.1 33
potheatzxyzxsurfacethetonormalunittheFind =+−
21
21.cosφφφφθ
∇∇∇∇
=
bygivenisandsurfacestwobetweenangleThe 21 φφ
:surfacestwobetweenAngle
)22(31 kji
+−=
φφ
∇∇
=∴ n̂
244)6()8(12 2221 =−+−+=∇φ
( ) kji
6812)3,4,6(1 −−=∇φ
kzjyix
2221 −−=∇φzxyzxy −+=2φ
2221 zyx −−=φ
:solution)3,4,6(int18
11.1 222
potheatzxyzxyandzyxsurfacesthebetweenangletheFind
=−+=−−
( ) kji
29)3,4,6(2 −+=∇φ
kxyjxzizy
)()()(2 −+++−=∇φ
−
=∴ −
524624cos 1θ
524624
−=
86244)29).(6812( kjikji
−+−−=
21
21.cosφφφφθ
∇∇∇∇
=
86)2(91 2222 =−++=∇φ
( ) kbjaia
122)2,1,1(2 ++−=∇ −φ
( ) kji
24)2,1,1(1 +−=∇ −φ
kbzjaxiaxy 22
2 32 ++=∇φ
kyjzix
22)910(1 −−−=∇φ
322 bzyax +=φ
xyzx 925 21 −−=φ
:solution
)2,1,1(int40925
tan.2322
−=+=−−
potheatlyorthogonalcutmaybzyaxandxyzx
surfacesthethatsobandatsconstheFind
14 == bandagetweandsolving )2()1(
)2(48 →=+−∴ ba.4
,secintint)2,1,1(32 =+
−
bzyaxonliesit
surfacestwotheoftionerofpoaisSince)1(04 →=+−⇒ ba
0246 =+−⇒ ba
0)122).(24( =++−+−∴ kbjaiakji
0.,sin 21 =∇∇ φφlyorthogonalcutsurfacesthece
)3,9,1()2,2,2(int.3 2
−−−=
andspotheatzxysurfacethetonormalsthebetweenangletheFind
.linesnormalthealongactingvectorsthebetweenangletheasoutfoundbecanlinesnormaltwothebetweenAngle
2),,( zxyzyxHere −=φ
kzjxiy
zxykzxyjzxy
2
)(z
)(y
)(x
i 222
−+=
−∂∂
+−∂∂
+−∂∂
=∇φ
( )
( ) 2)3,9,1(
1)2,2,2(
69
422
nkji
nkji
=++=∇
=−−−=∇
−
−−
φ
φ
.int21
spogiventheatsurfacethetonormalsthealongactingvectorsarenandn
21
21nn
.ncos
n=θ
11824)69).(422( kjikji
++−−−=
−
=∴
−=
−11711cos
11711
1θ
)3()36( 223 →−=∂∂ yzxz
zφ
)2()23( 32 →−+=∂∂ zxxy
yφ
)1(2 32 →−=∂∂ xyzy
xφ
,equationstwoabovetheComparingz
ky
jx
i∂∂
+∂∂
+∂∂
=∇φφφφ
,definitionBy
kyzxzjzxxyixyzy
)36()23()2( 2233232 −+−++−=∇φ:solution
kyzxzjzxxyixyzygrad
iffunctiontheFind
)36()23()2(
.42233232 −+−++−=φ
φ
czyyzxxy +++−=∴ 4322
233φ
)6(),(23 324 →+−= yxfyzxzφ
,..)3( ztorwpartiallygIntegratin
)5(),(3 322 →+−+= zxfyzxxyyφ
ytorwpartiallygIntegratin ..)2(
)4(),(322 →+−= zyfyzxxyφ
,..)1( xtorwpartiallyofsidesbothgIntegratin
vectorsolenoidalisFthenFIf
,0. =∇
:vectorSolenoidal
zF
yF
xFFFdiv
asdefinediskFjFiFFfunctionvectoraofdivergenceThe
∂∂
+∂∂
+∂∂
=∇=
++=
321
321
.
:vectoraofDivergence
vectoralirrotationcalledisFthenFIf
0=×∇
:vectoralIrrotation
321 FFFzyx
kji
FFcurl∂∂
∂∂
∂∂
=×∇=
asdefinediskFjFiFFvectoraofCurl
321 ++=
:vectoraofCurl
.int.,
:.int
int.
:
volumeunitperpoconcernedthefromissuingisheatwhichatratetherepresentsFthenfluxheatrepresentsFIf
Examplepothatfromissuingis
quantityphysicalthewhichatvolumeunitperratethepoeachatgivesFthenquantityphysicalanyrepresentsFIf
divergenceofmeaningPhysical
∇
∇
.2int,tan
),,(int:
ωω
representspothatatFcurlthenvelocityangulartconswithaxisfixedaaboutrotatesthatbodyrigid
aofzyxpotheofvelocitylineartherepresentsFIfFcurlofmeaningPhysical
yzzxz 323 82 +−=
9821).( )1,1,1( −=−−=∇ −F
( ) ( ) ( )423 22 yzz
yzxy
xzx ∂
∂+−
∂∂
+∂∂
=
)22).((. 423 kyzjyzxixzz
ky
jx
iF
+−∂∂
+∂∂
+∂∂
=∇
:solution
)1,1,1(int22..1 423 −+−=∇ potheatkyzjyzxixzFforFFind
2−=⇒ a011 =++⇒ a
0)()2()3( =+∂∂
+−∂∂
++∂∂ azx
zzy
yyx
x
0))()2()3).((( =++−++∂∂
+∂∂
+∂∂ kazxjzyiyx
zk
yj
xi
0., =∇ FsolenoidalisF
:solution
.)()2()3(
''.2
solenoidaliskazxjzyiyxFvectortheifaofvaluetheFind
++−++=
( )kj
kjiF
43
43)0()1,1,1(
+=
++=×∇ −
)4()3()22( 224 xyzkxzjyxzi −+++=
)]()2([
)()2([)]2()2([
32
3424
xzy
yzxx
k
xzy
yzx
jyzxz
yzx
i
∂∂
−−∂∂
+
∂∂
−∂∂
−−∂∂
−∂∂
=
321 FFFzyx
kji
FFcurl∂∂
∂∂
∂∂
=×∇=
:solution
)1,1,1(,22.3 423 −+−= atFcurlfindkyzjyzxixzFIf
)]2()3([
)]2()24([
)]3()24([
azyxy
zybxx
k
azyxz
zcyxx
j
zybxz
zcyxy
i
++∂∂
−−−∂∂
+
++∂∂
−++∂∂
−
−−∂∂
−++∂∂
=
zcyxzybxazyxzyx
kji
FFcurl
2432 ++−−++∂∂
∂∂
∂∂
=×∇=
:solution
.)24()3()2(
,,tan.4
alirrotationiskzcyxjzybxiazyxFvector
thethatsocbatsconstheofvaluetheFind
+++−−+++=
0,
=×∇ FalirrotationisF
)2()4()1( −+−−+= bkajci
2,4,102;04;01
0)2()4()1(
==−=∴=−=−=+⇒
=−+−−+∴
bacbac
bkajci
0
=
kyyjxzxzi
)22()44()11( −+−−+−=
)]2()2([
)]2()22([
)]2()22([
22
222
2
xzyy
zxyx
k
xzyz
zyzxx
j
zxyz
zyzxy
i
+∂∂
−−∂∂
++∂∂
−+−∂∂
−
−∂∂
−+−∂∂
=
zyzxzxyxzy
zyx
kji
FFcurl
2222 222 +−−+∂∂
∂∂
∂∂
=×∇=
:solution.
)22()2()2(.5 222
potentialscalaritsfindhenceandalirrotationiskzyzxjzxyixzyFthatShow
+−+−++=
)1(),(222 →++= zyfzxxyφ
;.. xtorwpartiallygIntegratin
22 2xzyx
+=∂∂
∴φ
zk
yj
xi
F
∂∂
+∂∂
+∂∂
=
∇=φφφ
φ
φbeFofpotentialscalartheLet
.alirrotationisF
∴
czyzzxxyandFrom ++−+= 2222),3()2(),1( φ
)3(),(222 →++−= zyfzyzzxφ
;.. ztorwpartiallygIntegratin
zyzxz
22 2 +−=∂∂φ
)2(),(2 →+−= zxfyzxyφ
;.. ytorwpartiallygIntegratin
zxyy
−=∂∂ 2φ
1;1;6 ===∴ cba0)6(;0)1(3;01 2 =−=−=−⇒ axbzc
0)6()33()1( 22 =−+−−+−⇒ kaxxjbzzic
0
33
0,
223
=
−−+∂∂
∂∂
∂∂
∴
=×∇
yxzczxbzaxyzyx
kji
FalirrotationisF
..)3()3()(
,,tan.6223
FofpotentialscalarthefindAlsoalirrotationbemaykyxzjczxibzaxyF
thatsocbatsconstheofvaluestheFind
−+−++=
:solution
yxzz
−=∂∂ 23φ
zxy
−=∂∂ 23φ
36 zxyx
+=∂∂φ
zk
yj
xi
F
∂∂
+∂∂
+∂∂
=
∇=φφφ
φ
FofpotentialscalarthebeLet
φkyxzjzxizxyF
cbaofvaluesthesegU
)3()3()6(
,,,sin223 −+−++=
cxzyzyxgetweFrom ++−= 323),3(),2(),1( φ
)3(),(
)2(),(3
)1(),(3
3
2
32
→+−=
→+−=
→++=
yxfyzxz
zxfyzyx
zyfxzyx
φ
φ
φ
getweiablesconcernedthetorwpartiallygIntegratin ,var..
])2()[( 22 dyyxydxxyxc
+−+−= ∫
)].()2()[( 22 kdzjdyidxjyxyixyxc
+++−+−= ∫
∫=c
rdFFbyWorkdone .
xyiixyicurvethealongto
fromplanexytheinparticleadisplacesjyxyixyxFforceawhendoneworktheFind
==
−+
−+−=
2
22
)(,)()1,1(
)0,0()2()(.1
FUNCTIONSPOINTVECTOROFINTEGRALLINE
:solution
])2()[( 221 ∫
==
+−+−=dxdy
xydyyxydxxyxW
xylinethealongiCase =:)(
dxx∫ −=1
0
22
32
−=
xycurvethealongiicase =2:)(
])2()[( 2
)2(2
22 dyyxydxxyxW
ydydxyx
+−+−= ∫
==
32
−=
dyyyy )22(1
0
35 −−= ∫
10var,,
.,.2
32
2
tofromiestandtztytxcurve
theisCwhererdFevaluatekzjyzixzFGivenC
===
++= ∫
)(
)).((.
2
2
dzzyzdyxzdx
kdzjdyidxkzjyzixzrdF
C
CC
∫
∫∫
++=
++++=
10var3
223
2
tofromiestdttdztz
tdtdyty
dtdxtx
=⇒=
=⇒=
=⇒=
∫∫ ++=CC
dzzyzdyxzdxrdF )(. 2
dtttt
dttttdttdtt
)32(
)3()2(
8641
0
2651
0
4
++=
++=
∫
∫
10586
31
72
51
372
5
1
0
975
=
++=
++=
ttt
kzejxixyzeFwhereCpaththe
oftindependenisrdFegraltheofvaluethethatShow
xxC
)()1()2(
.int.3
2 ++−+−=
∫
0,. =∫ FcurlifCpaththeoftindependenisrdFC
zexxyzezyx
kji
Fcurl
xx +−−∂∂
∂∂
∂∂
=
212
0
)2()1(
)2()()1()(
2
2
=
−∂∂
−−∂∂
+
−
∂∂
−+∂∂
−
−∂∂
−+∂∂
=
xyzey
xx
k
xyzez
zex
jxz
zey
i
x
xxx
:' theoremsGreen
.
)(
,),(),(sin
directioniseanticlockwtheindescribedisCwhere
dxdyyP
xQQdyPdx
thenRincontinuousaresderivativepartialorderfirstitsandyxQandyxPandplaneXYtheinRregionagenclocurveclosedsimpleaisCIf
C R∫ ∫∫
∂∂
−∂∂
=+
yxQ 6−=∂∂
yyP 16−=∂∂
xyyQ 64 −=
22 83 yxP −=
dxdyyP
xQQdyPdx
C R∫ ∫∫
∂∂
−∂∂
=+ )(
,' theoremsGreenBy
10,0
])64()83[(
'.122
=+==
−+−∫
yxandyxlinesthebydefinedregionthe
ofboundarytheisCwheredyxyydxyx
forplaneintheoremsGreenVerify
C
:solution
35
=
1
0
32
3105
−=
yy
∫ −=1
0)1(10 dyyy
∫ ∫−
=1
0
1
010
yydxdy
∫∫=R
ydxdyofSHR 10)1(..
)1(10])64()83[( 22 →=−+−∴ ∫∫∫RC
ydxdydyxyydxyx
∫
∫∫
+
−−+−−−+=
1
0
1
0
221
0
2
4
])}1(64{)}(8)1(3[{3
ydy
dyyyydyyydxx
])64()83[(
])64()83[(
])64()83[()1(..
22
11
22
00
22
dyxyydxyx
dyxyydxyx
dyxyydxyxofSHL
BO
dydxyx
yxAB
dyy
OA
−+−
+−+−
+−+−=
∫
∫
∫
−=−==+
==
35
=
2323
111 −
−++=
∫∫ ∫ −−++=1
0
1
0
1
0
22 4)3411(3 ydydyyydxx
.'sin1,1,1,1
})(){(.2 222
theoremsGreenguyyxxlinesthebyformed
squaretheisCwheredyyxdxxxyEvaluateC
=−==−=
+++∫
22
2
yxQ
xxyP
+=
+=
:solution
1
1
1
1
2
2−−
∫
=
x
∫ ∫− −
=1
1
1
1xdxdy
∫∫∫ =+++RC
xdxdydyyxdxxxy ])()[( 222
dxdyyP
xQQdyPdx
C R∫ ∫∫
∂∂
−∂∂
=+ )(
,' theoremsGreenBy
xxQ
xyP
2=∂∂
=∂∂
021
211
1
=
−= ∫
−
getwexQandyPTaking ,22
=−
=
dxdyyP
xQQdyPdx
C R∫ ∫∫
∂∂
−∂∂
=+ )(
,' theoremsGreenBy
1
)(21
Pr.3
2
2
2
2=+
−∫
by
axellipsethe
byboundedareathefindhenceandydxxdybygiven
isCcurveclosedsimpleabyboundedareathethatove
C
∫ −−=C
dabdbaA )]sin(sin)cos(cos[21 θθθθθθ
θθθθθθ
dadydadxayax
cos;sinsin;cos=−=
==
∫ −=C
ydxxdyellipsebyboundedArea )(21
CbyenclosedRregiontheofArea
dxdyR
=
= ∫∫
dxdyydxxdyC R∫ ∫∫
−−=−
21
21)(
21
abπ=
πθ 20)(
2ab
=
∫
∫
=
+=
π
π
θ
θθθ
2
0
2
0
22
21
)sin(cos21
abd
dab
∫∫∫∫∫ ∇=VS
dvFSdFthenVinsderivativepartialorderfirst
continuouswithfunctionpovectoraisFifandVvolumewithspaceofregionagenclosurfaceclosedaisSIf
theoremdivergenceGauss
).(.
intsin
:
∫∫∫ ++=V
dxdydzzyx )222(
( )dvkzjyixz
ky
jx
idvFV V
222.).( ++
∂∂
+∂∂
+∂∂
=∇∫∫∫ ∫∫∫
∫∫∫∫∫ ∇=VS
dvFSdF ).(.
theoremdivergenceGaussBy
.0,,0,,0
,.1 222
czandzbyyaxxplanesthebyformedcuboidtheofsurfacetheisSwhere
kzjyixFfortheoremdivergenceGaussVerify
======
++=
:solution
)(
222
cbaabcabccabbca
++=++=
dzabzabba
dzazyayya
c
bc
)2(
)2(
2
0
2
00
22
++=
++=
∫
∫
dydzzxyxx
dxdydzzyx
c b a
c b a
∫ ∫
∫ ∫ ∫
++=
++=
0 00
2
0 0 0
)22(
)222(
dxdynkzjyix
dxdynkzjyix
dxdznkzjyix
dxdznkzjyix
dydznkzjyix
dydznkzjyixSdF
kncz
knz
jnby
jny
inax
inxS
ˆ).(
ˆ).(
ˆ).(
ˆ).(
ˆ).(
ˆ).(.
22
ˆ
2
22
ˆ0
2
22
ˆ
2
222
ˆ0
22
ˆ
2
22
ˆ0
2
++
+++
+++
+++
+++
+++=
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫∫∫
==
−==
==
−==
==
−==
)(
2220
2
0
2
0
2
0 00
2
0
2
0 0
2
cbaabcabccabbca
adyccdxbbdza
dxdycdzdxbdydza
bac
b aa cc b
++=++=
++=
++=
∫∫∫
∫ ∫∫ ∫∫ ∫
dxdyzdxdyz
dxdzydxdzydydzxdydzx
czz
byax yx
∫∫∫∫
∫∫∫∫ ∫∫∫∫
==
== ==
+−+
+−++−=
2
0
2
2
0
22
0
2
∫∫∫∫∫ ∇=VS
dvFSdF ).(.
theoremdivergenceGaussBy
.1,1,1.2 2
±=±=±=++=
zyxbyformedcubetheoverkyzjzixFfortheoremdivergenceVerify
:solution
dxdynkyzjzix
dxdynkyzjzix
dxdznkyzjzix
dxdznkyzjzix
dydznkyzjzix
dydznkyzjzixSdF
knz
knz
jny
jny
inx
inxS
ˆ).(
ˆ).(
ˆ).(
ˆ).(
ˆ).(
ˆ).(.
ˆ1
2
ˆ1
2
ˆ1
2
2
ˆ1
ˆ1
2
ˆ1
2
++
+++
+++
+++
+++
+++=
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫∫∫
==
−=−=
==
−=−=
==
−=−=
.0
2
)2(
1
1
1
1
1
1
1
1
1
1
verifiedistheoremdivergenceHence
dydzy
dxdydzyx
=
=
+=
∫ ∫
∫ ∫ ∫
− −
− − −
∫∫∫ +=V
dxdydzyx )2(
( )dvkyzjzixz
ky
jx
idvFV V
++
∂∂
+∂∂
+∂∂
=∇∫∫∫ ∫∫∫2.).(
011
11 1
2
1
2
=
+−+
+−++−=
∫∫∫∫
∫∫∫∫ ∫∫∫∫
=−=
== −=−=
dxdyyzdxdyyz
dxdzzdxdzzdydzxdydzx
zz
yx yx
.int
..,
int
:'
SsurfacetheofpoanyatnormaldrawnoutwardtheoftippositivethefromseenasdirectioniseanticlockwtheindescribedisCwhere
SdFcurlrdFthenSonsderivativepartialorderfirstcontinuous
withfunctionpovectoraisFifandCcurveclosedsimpleabyboundedsurfacesidedtwoopenanisSIf
theoremsStoke
SC∫∫∫ =
kxjziy
−+= 2
xzyzxyzyx
kji
FFcurl
−−∂∂
∂∂
∂∂
=×∇=
2
∫∫∫ =SC
SdFcurlrdFistheoremsStoke
..'
.32,0,1,0
tan2'.1
planeXOYtheabovezandyyxxplanestheby
formedipedparallelopgularrectheofsurfaceopentheisSwherekzxjyzixyFfortheoremsStokeVerify
=====
−−=
:solution
1
20
1
−=
= ∫ xdx
∫∫∫∫
==
==
==
==
+++=
00
02
'
01'
00
dxxBO
dyy
BC
dxxAC
dyyOA
xydxxydxxydxxydx
( )0
)2()1(..
'
'
==
−−=
∫
∫
zplanetheonliesCboundaryxydx
zxdzyzdyxydxofSHL
BOAC
BOAC
)1().2()2( →−+=−−∫ ∫∫C S
Sdkxjziyzxdzyzdyxydx
dxdynkxjziy
dxdznkxjziy
dxdznkxjziy
dydznkxjziy
dydznkxjziySdF
knz
jny
jny
inx
inxS
ˆ).2(
ˆ).2(
ˆ).2(
ˆ).2(
ˆ).2(.
ˆ3
ˆ2
ˆ0
ˆ1
ˆ0
−+
+−+
+−+
+−+
+−+=
∫∫
∫∫
∫∫
∫∫
∫∫∫∫
==
==
−==
==
−==
.'1
2
1
0
2
0
2
2
0
1
0
verifiedistheoremsStoke
x
xdxdy
∴−=
−=
−=
∫
∫ ∫
∫ ∫∫ ∫∫ ∫∫ ∫∫ ∫ −+−+−=2
0
1
0
1
0
3
0
1
0
3
0
3
0
2
0
3
0
2
022 xdxdyzdzdxzdzdxydydzydydz
dxdyx
dxdzzdxdzzydydzdydzy
z
yx yx
∫∫
∫∫∫∫ ∫∫∫∫
=
== ==
−+
+−++−=
3
21 0022
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