M7R0
I,Integrator3=6"%&falseGE\[l_o%2traadModelRumGrayGf*6%%"rG%"BG%$netG6'%&DeltaG%
#kuG%#kvG%#kwG%$sysG6#%HRev.~29.12.03~(S.Markvorsen@mat.dtu.dk)GF$C,>&8$6#"""-%
&evalfG6#*&,&&9%6#""#F:&FAF9!""F:&9&F9FE>&F8FB-F<6#*&,&&FA6#""%F:&FA6#""$FEF:&F
GFBFE>&F8FR-F<6#*&,&&FA6#""'F:&FA6#""&FEF:&FGFRFE>8%f*6$%"vG%"wGF$6$%)operatorG
%&arrowGF$-%+spacecurveG6%-T$6%%"uG9$FA/F[p;&T&F9&F`pFB/%&colorG%%grayGF$F$6&F*
F\pF+FA>8&f*6$F[pFaoF$FboF$-Ffo6%-Fio6%F\pF`oFA/F`o;&F`pFR&F`pFOFbpF$F$Fep>8'f*
6$F[pF`oF$FboF$-Ffo6%-Fio6%F\pFAFao/Fao;&F`pFin&F`pFfnFbpF$F$Fep>&8(F9-%(displa
yG6#-%$seqG6$-Fer6$-F]o6$,&FQF:*&%"qGF:FIF:F:,&FhnF:*&%"tGF:FVF:F:/F]s;""!FT/F`
s;FcsF[o>&F`rFB-Fbr6#-Fer6$-Fer6$-Fgp6$,&FDF:*&%"pGF:F7F:F:F^s/Fbt;FcsFFFds>&F`
rFR-&Fbr6#%&plotsG6#-Fer6$-Fer6$-Fcq6$F`tF[sFctFas-Fbr6#7%F_rFgsFft6#QWViser~tr
|`ydmodel~af~rumlige~omr|`yde~~r(B),~hvor~B~er~3d.F$F$F$%.StokesRandIntGf*6&F*F
+%"VG%'netrawG6en%#rPG%'JacobiG%*integrandG%%tangG%#rrG%&kurveG%+kurveTotalG%#A
fG%%maxiG%%miniG%*extensionGF.F/F0F2%'deformG%&fladeG%%feltGF,%&normaG%%VmaxG%"
mG%&uTangG%&vTangG%&KrydsG%(divisorG%%spanG%&spanxG%&spanyG%&spanzG%'questrG%'q
uestBG%'questnG%'questVG%+feltGlobalG%%minxG%%maxxG%%minyG%%maxyG%%minzG%%maxzG
%%maxVG%)maxVglobG%$rrPG%*feltTotalG%-tangIntegralG%%tilfG%%fluxG%.fluxIntegran
dG%%pindG%$hatG%$pilG%$NenG%(pindNenG%'hatNenG%'pilNenG%(feltNenG6#%HRev.~21.01
.04~(S.Markvorsen@mat.dtu.dk)GF$CJ6-8/8B8C%'questfG8R%"MG%'MkoordG8)%*kurveWpos
G%*kurveWnegG%(maksvalG@%/9#FPC$>869'@%3-%%typeG6$&FezF9.%(numericG-Fjz6$&FezFB
F][l>8D%%trueGC$>Fc[lF%>Fez7$FjnFjnC$>Fc[lF%>FezFh[l>&FeyF9-F<6#*&F?F:F\[lFE>&F
eyFB-F<6#*&FMF:Fa[lFE>8+7#-Fer6$-Fer6$-F<6#-F\p6$,&FDF:*&FbtF:F]\lF:F:,&FQF:*&F
]sF:Fb\lF:F:/Fbt;F:F\[l/F]s;F:Fa[l>8,f*6#%"kGF$FboF$-%$maxG6#-Fer6$-F<6#&&T#6#%
"iG6#F\p/Fi^l;F:*&&T%F9F:&F__lFBF:F$F$6&FdvFg\lF,Fez>8-f*F\^lF$FboF$-%$minGF`^l
F$F$Fa_l>8.-F_^l6#-Fer6$,&-Fj]lF\^lF:-Fc_lF\^lFE/F]^l;F:FS>8>*&$""(FEF:Fh_lF:>F
fy-Fjz6$-%$sumG6$-F[al6$-F<6#&&Fg\lFh^lF\^l/.Fi^l;F:*&F\[lF:Fa[lF:/.F]^lFa`lF][
l>Fgy3-Fjz6$-F<6#FDF][l-Fjz6$-F<6#F@F][l>87f*FeqF$FboF$-F<6#-_%'linalgG%(dotpro
dG6%-F`p6#-%#opG6#-Fio6$F\pFAF]cl.%+orthogonalGF$F$6&F*F\pFjuFG>F_rf*6#F[pF$Fbo
F$-F`p6$F\p&FioFRF$F$6&F+FAF*F\p>Fgsf*6#F`oF$FboF$-F`p6$&FioFBF\pF$F$F]dl>Fftf*
FiclF$FboF$-F`p6$,&FcdlF:F\pFE&FioFOF$F$F]dl>&F`rFOf*F`dlF$FboF$-F`p6$&FioF9,&F
idlF:F\pFEF$F$F]dl>8O-%(unapplyG6%-%%diffG6$-&F`rFh^l6#F`sF`sFi^lF`s>F]of*6$Fi^
lF[pF$FboF$-%%sqrtG6#-Fibl6%-Fg^l6$F\pF`sFdflFdclF$F$6$FcxFbel>Fgpf*6$Fi^lF`sF$
FboF$-%)simplifyG6#-Fibl6%-F`p6#-F`cl6#-&Fg^lFj^l6#FA-T'FcclFdclF$F$6(FavF`rFju
FGFcxFbel>&FcqF9-F[gl6#-%$intG6$-Fgp6$F:F[p/F[p;FDF@>&FcqFB-F[gl6#-F^hl6$-Fgp6$
FCF`o/F`o;FQFN>&FcqFR-F[gl6#-F^hl6$-Fgp6$FSF[p/F[p;FcsF?>&FcqFO-F[gl6#-F^hl6$-F
gp6$FPF`o/F`o;FcsFM>8Q,*FjglF:FehlF:F_ilF:FiilF:@%3FfyFgy>8E-Fjz6$-F[al6$-F[al6
$-F<6#-FdblF`]l/.FbtFf]l/.F]sFh]lF][l>FhjlF%@%33FfjlFhjlFc[lC@>80f*F`dlF$FboF$-
Ffo6%-Fio6$F[pF\pF]p/Fcp%$redGF$F$Fep>81f*FiclF$FboF$-Ffo6%-Fio6$F\pF`oF^q/Fcp%
%blueGF$F$Fep>&82F9-Fbr6#-Fer6$-F\\m6#Fc]l/F]s;FcsFa[l>&F_]mFB-Fbr6#-Fer6$-Fe\m
6#Fa]l/Fbt;FcsF\[l>83-Fbr6%7$F^]mFi]m/%(scalingG%,constrainedG/%+projectionG$""
)FE>84-%'plot3dG6'-F\pFeqFbhlF\il/%%gridG7$,$F\[lFS,$Fa[lFSFg^m>88-F`fl6#-F_^l6
#-Fer6$-Fer6$Fa[mFe]lFg]l>8M-F_^l6$$"#5FEFj_m>&F\zF9-%)tubeplotG6(-F_rFiclFbhl/
Fe_m7$"#]Fjn/%'radiusG,$Fh_l#F:"#qFg^mFj^m>&F\zFB-F\am6(-FgsF`dlF\ilF_amFbamFg^
mFj^m>&F\zFR-F\am6(-FftFiclFfilF_amFbamFg^mFj^m>&F\zFO-F\am6(-F[elF`dlF`jlF_amF
bamFg^mFj^m>8*-Fbr6#-Fer6$&F\zFh^l/Fi^l;F:FP>8Uf*F^flF$FboF$-F\am6)-%&evalmG6#,
&FcglF:*(F`sF:F_glF:FgglFEF:/F`s;Fcs*&$FjnFEF:T)F:/Fcam*&$Fjn!"$F:F]dmF:/Fe_m7$
FCFjn/%&styleG%,patchnogridGFg^m/Fcp%&blackGF$F$6*FavF`rFjuFGFaxFd`mFbwFc`l>8Vf
*F^flF$FboF$-F\am6'Fdcm/F`s;*&F\dmF:F]dmF:*&$FgnFEF:F]dmF:/Fcam*&$FCFEF:,&*&F]d
mF:FcemF:F:F`sFEF:Fbdm/Fedm%&patchGF$F$Fidm>8Wf*F^flF$FboF$-Fbr6#7$-Fg^lFccl-F_
_lFcclF$F$6&FixF`cmFjxF[em>&85F9-Fbr6$7#-Fer6$-F\fm6$F:,&FDF:*(F?F:FbtF:F\[lFE#
F:FS/Fbt;FcsFg_mFg^m>&FffmFB-Fbr6$7#-Fer6$-F\fm6$FC,&FQF:*(FMF:FbtF:Fa[lFEF`gm/
Fbt;FcsFh_mFg^m>&FffmFR-Fbr6$7#-Fer6$-F\fm6$FS,$F_gmF`gmFagmFg^m>&FffmFO-Fbr6$7
#-Fer6$-F\fm6$FP,$F]hmF`gmF^hmFg^m>8P-Fbr6#-Fer6$&FffmFh^lF]cm>8F-%,fieldplot3d
G6(-FG6%%"xG%"yG%"zG/Fbjm;-Ff_l6$FE-Fc_lF9-F_^l6$F:-Fj]lF9/Fcjm;-Ff_l6$FE-Fc_lF
B-F_^l6$F:-Fj]lFB/Fdjm;-Ff_l6$FE-Fc_lFR-F_^l6$F:-Fj]lFR/Fe_m7%FjnFjnFjn/%'arrow
sG%&THICKG-%&printG6#Q0Fladestykket~F:F$-Fc\n6#-Fbr6'7$F__m-%'base3dGFajmFg^mFj
^m/%%axesG%%noneG/%+lightmodelG%'light3G-Fc\n6#Q0Vektorfeltet~V:F$-Fc\n6#-Fbr6'
7$F\jmF[]nFg^mFj^mF]]nF`]n-Fc\n6#Q=Randkurven~for~fladestykket:F$-Fc\n6#-Fbr6'7
%Fc^mFgbmF[]nFg^mFj^mF]]nF`]n-Fc\n6#QAVektorfeltet~V~langs~randkurven:F$-Fc\n6#
-Fbr6'7&Fc^mFgbmFeimF[]nFg^mFj^mF]]nF`]nF$-%'printfG6$QEParameterfremstilling:~
r(u,v)~=~%a~|+F$Fc_m-F\_n6$Q8Definitionsomr|`yde:~%a~|+F$FA-F\_n6$Q<Vektorfelt:
~V(x,y,z)~=~%a~|+F$F`jm-F\_n6#Q"|+F$-F\_n6#QioDe~tangentielle~kurveintegraler~l
angs~hver~enkelt~rand-koordinat-kurve-stykke:F$Fe_n-Fc\n6#/-F[gl6#-%(combineG6#
Fjgl-F<Fb`n-Fc\n6#/-F[gl6#-Fa`n6#Fehl-F<Fj`n-Fc\n6#/-F[gl6#-Fa`n6#F_il-F<Fban-F
c\n6#/-F[gl6#-Fa`n6#Fiil-F<FjanFe_n-F\_n6#Q\oDet~totale~tangentielle~kurveinteg
ral~langs~randkurven~er~derfor:F$Fe_n-Fc\n6#/-F[gl6#-Fa`n6#Fcjl-F<Febn6#Q[sFind
er~og~viser~(hvis~muligt)~det~tangentielle~kurveintegral~af~givet~vektorfelt~la
ngs~~|+totale~randkurve~for~given~fladestykke.F$F$F$%)planCmGoGf*6&F*F+%#ffGF[v
6CFfxFfwFgwFhyFhwFbjmFcjmF^wF_wF.F2F/F0%%sysTG%$kuTGF]w%$kvTG%'parallG%&MdiscG%
$radG%$cenGF,Fdv%%polyGFhv%%fyldG%"AG%+JacobiSignGF^v%)arealAppG%,arealEksaktGF
jyF[zF3F$C)>Fg\l-Fdel6%-F[gl6#-Fgel6$Fc_mF[pF[pF`o>Fj]l-Fdel6%-F[gl6#-Fgel6$Fc_
mF`oF[pF`o>8?f*FeqF$FboF$-F[gl6#,&*&&FafmF9F:&FbfmFBF:F:*&&FafmFBF:&FbfmF9F:FEF
$F$6&F^wFg\lF_wFj]l>8@f*FeqF$FboF$*&-%'signumG6#FafmF:FafmF:F$F$6$FhcnF[en>Fc^m
f*F`dlF$FboF$-F[gl6#-F^hl6$-F[gl6#*&-F`p6#-F`cl6#F`\mF:-FgglFa\mF:/F[p;&T*F9&F`
gnFBF$F$6*F*F\pF\cnFGF^vFhenF+FA>Fgy-F[gl6#-F^hl6$-Fc^mF`dlF\il>Fc[l7$-F[gl6#*&
-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(&Fc_mF9F:-FG6#-F`cl6#Fc_mF:-FhenFeqF:FbhlF\ilF:Fg
yFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(&Fc_mFBF:FhhnF:F\inF:FbhlF\ilF:FgyFE6
#QcrFinder~og~viser~(hvor~muligt)~planintegralet~af~'masset|aytheden'~ff(x,y)~o
ver~givet~plant~omr|`yde,~eksakt~og~approksimeretF$F$F$%*fluxIntGoGf*6%F*F+Fju6
)F^wF_wF`wFawF^vFgxFhx6#%HRev.~01.02.03~(S.Markvorsen@mat.dtu.dk)GF$C(>F8F]dn>F
]oFddn>Fgpf*FeqF$FboF$-_Fjbl%*crossprodG6$FafmFbfmF$F$6&F^wF8F_wF]o>F`rf*FeqF$F
boF$-F`fl6#-Fa`n6#-Fibl6%FafmFafmFdclF$F$6$F`wFgp>Fgbmf*FeqF$FboF$-F[gl6#-Fibl6
%Fafm-T(6#-F`cl6#-F`pFcclFdclF$F$6(F`wFgpF*F\pFjuFG-F[gl6#-F^hl6$-F[gl6#-F^hl6$
-FgbmFeqFbhlF\il6#Q]qFinder~fluxen~af~vektorfeltet~V(x,y,z)~gennem~givet~flades
tykke,~orienteret~ved~parametriseringen.F$F$F$%(rumImGoGf*6(F*F+F\cn%#ptG%"eGF[
v6WFgwFfwFhyFhwFfxFdvFevFfvFgvFdcnFccn%%MSphGF.%#kfGF/F0F1F_cnF`cn%$kwTGF2F^cn%
*alleSiderG%#mmGF]wF^wF_w%&wTangG%)standardGFhvFgcnFecn%+parallepipG%*volEksakt
GFjyF[z%'volAppGF,F_v%'questpG%'questeG%$conG%$enhGFacnFfcnF`wF^vFicn%#IMG%%aks
eG%&akselG%(nettingG%'afstSQGF3F$C->8=-Fdel6&-F[gl6#-Fgel6$-F\p6%F[pF`oFaoF[pF[
pF`oFao>Fc`l-Fdel6&-F[gl6#-Fgel6$F__oF`oF[pF`oFao>F[en-Fdel6&-Fgel6$F__oFaoF[pF
`oFao>Ffyf*F`_oF$FboF$-_Fjbl%(augmentG6%-Fg^l6%F\pFAFG-F__lFd`o-FgglFd`oF$F$6(F
^wFh^oF_wFc`lFh]oF[en>Fiyf*F`_oF$FboF$-F[gl6#*&-F\fn6#-_Fjbl%$detG6#Fc`oF:-F[gl
F^aoF:F$F$6$FgcnFfy>8N-Fecm6#*&9(F:-F`fl6#-Fibl6%FjaoFjaoFdclFE>Fd`mf*F`_oF$Fbo
F$-Fecm6#,&-FioFd`oF:F`pFEF$F$6&F*F\pF`]oFfz>8Xf*F`_oF$FboF$,&-Fibl6%Fc`oFc`oFd
clF:*$-Fibl6%Fc`oF__lFdclFCFEF$F$6&F_^oFd`mF`^oFfao>8;f*F_oF$FboF$-F[gl6#-F^hl6
$-F[gl6#*(-F`p6#-F`cl6#FhoF:-FgglFjoF:-F]dmFjoF:/F[p;&T,F9&FcdoFBF$F$6,F*F\pF\c
nFGFe^oFgboF^vFiyF+FA>8<f*6#FaoF$FboF$-F[gl6#-F^hl6$-Fg^l6$F`oF\pF^qF$F$6&Fg]oF
acoF+FA>8T-F[gl6#-F^hl6$-FgdoFido/Fao;FhnFen6#Q_oFinder~inertimomentet~af~givet
~v|aygtet~omr|`yde~i~rummet~om~given~akse.F$F$F$%(kurveKmGf*6(F*F+F\cnFjuF`]oF[
v6VFfxFfwFgwFiw%'kurve1GFd^oFgvF.F[wF\wFevFfvFbvFixFjxF[yFjvF]vF^vF_vF,FhwFdv%*
torqueVecGFccnFdcnFc]oFexFjwFcwFdwFewF[xF\xF]xF^xF_xF`xFgxFaxFbw%(torqueAG%(tor
queBG%(torqueCGFhyF]^o%'torqueG%0torqueIntegrandG%(torqueLG%+integrandAG%+integ
randBG%+integrandCG6#%HRev.~12.03.04~(S.Markvorsen@mat.dtu.dk)GF$CH@%/FbzFgnC$>
Fj_m9)@%-Fjz6$&Fj_mF9F][l>89Fd[lC$>FhgoF%>Fj_m7#FjnC$>FhgoF%>Fj_mF\ho>FgpFjal>&
Fg\lF9-F<6#*&F?F:FfgoFE>8:7#-Fer6$-F<6#-F\p6#,&FDF:*&FbtF:FbhoF:F:/Fbt;F:Ffgo>F
]o-Fjz6$-F[al6$-F[al6$.-F<6#&&FghoFh^lF\^l/FdalFbioFgalF][l>Fj]lf*FiclF$FboF$-F
<6#-Fibl6%-F`p6#-F`cl6#-FioFj^lFfjoFdclF$F$6&F*F\pFjuFfz@%33-Fjz6$-F<6#&FjaoF9F
][l-Fjz6$-F<6#&FjaoFBF][l-Fjz6$-F<6#&FjaoFRF][l>FcjlFd[l>FcjlF%>Feim-Fjz6$-F[al
6$.-F<6#-FG6#-F`cl6#&Fgho6#%"jG/.F^]pFbioF][l>Fcq-Fjz6$-F[al6$-F<6#-Fj]lF^io/Fc
[mFbioF][l>Fffm-Fdel6$-Fgel6$-F\pF[flF`sF`s>Fezf*F[flF$FboF$-F[gl6#-F`fl6#-Fibl
6%-Fg^lFj^lFh^pFdclF$F$6$F]vFffm>8Sf*FiclF$FboF$-F[gl6#-Fhjn6$-Fecm6#,&*&%$aaaG
F:FjjoF:F:*&%$dddGF:F`pF:FE-F\\oFgjoF$F$6(F*F\pF`]oFjaoFjuFfz>F`cmf*FiclF$FboF$
*(FfjoF:&-FgglFj^lF9F:-F]dmFj^lF:F$F$6*F*F\pF\cnFGFffoF[_pF^vFez>F[emf*FiclF$Fb
oF$*(FfjoF:&F^`pFBF:F_`pF:F$F$F``p>F\fmf*FiclF$FboF$*(FfjoF:&F^`pFRF:F_`pF:F$F$
F``p>Fd`m-F[gl6#-F^hl6$-F`cmFeqFbhl>Ffao-F[gl6#-F^hl6$-F[emFeqFbhl>Fbel-F[gl6#-
F^hl6$-F\fmFeqFbhl>Fiy-%%subsG6%/Fe_pF:/Fg_pF:7%Fd`mFfaoFbel>Fbeo-F`fl6#,(*$&Fi
yF9FCF:*$&FiyFBFCF:*$&FiyFRFCF:@%33333F]oFgpFcqFhgoFeimFcjlC;>F`r-F\am6(F_^p/F`
sFchl/Fe_m7$,$FfgoFjnFjn/Fcam$Fjn!"#Fg^mFj^m>F\z-F`cl6$F:-F`cl6$F:F`r>Fh_lf*F\^
lF$FboF$-F_^l6#-Fer6$-Fer6$&Fg^l6%Fi^lF^]pF\p/Fi^l;F:,$F^_lFjn/F^]p;F:FjnF$F$6&
Fd^oF\zF,Fj_m>Feyf*F\^lF$FboF$-Ff_lFedpF$F$Faep>Fgbm-F_^l6#-Fer6$,&-Fh_lF\^lF:-
FeyF\^lFEF``l>8L*&F\dmF:FgbmF:>F\\m-F\am6(F_^pFfcpFgcp/Fcam,$Fgbm#F:"#gFg^mFj^m
>F^fpFgbm>8K-F_^l6$Fg`mFc_l>Faco-Fdo6&Fjao-Fecm6#Fiy/%'lengthG-Ff_l6$F^fpFbeo/F
cp%'yellowG>Fgdo,$Fgbm#F:"#:>Fh^o-F<6#Fjao>Fc`l-Fa_m6)7%,&&Fh^oF9F:*(FgdoF:-%$s
inGFiclF:-%$cosGF`dlF:F:,&&Fh^oFBF:*(FgdoF:FehpF:-FfhpF`dlF:F:,&&Fh^oFRF:*&Fgdo
F:-FhhpFiclF:F:/F[p;Fcs%#PiG/F`o;,$FcipFEFcip/Fe_m7$Ff`l"#9FddmFbpFg^m>Fc_l-F`f
l6#-F_^l6#-Fer6$Fh]p/Fbt;FcsFfgo>Fe\mf*F[flF$FboF$-F\am6)-Fecm6#,&FjjoF:*(F[pF:
FfjoF:FgglFEF:/F[p;Fcs*&F\dmF:F]dmF:/Fcam*&$F:F\dpF:F]dmF:FbdmFddmFg^m/Fcp%&gre
enGF$F$6*F*F\pFjuFfzF\wFc_lFbwF^fp>F_]mf*F[flF$FboF$-F\am6'Fgjp/F[p;*&F\dmF:F]d
mF:*&FcemF:F]dmF:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F[pFEF:FbdmFiemF$F$Fc[q>Fc^mf*F
[flF$FboF$-Fbr6#7$Fh^p-F__lFj^lF$F$6&FixFe\mFjxF_]m>F__m-Fbr6$7#-Fer6$-Fc^m6#,&
FDF:*(FbtF:F?F:FfgoFEF`gm/Fbt;Fcs,$FfgoFSFg^m>Fhen-F^jm6(-FfzFajm/Fbjm;-Ff_l6$F
E-FeyF9-F_^l6$F:-Fh_lF9/Fcjm;-Ff_l6$FE-FeyFB-F_^l6$F:-Fh_lFB/Fdjm;-Ff_l6$FE-Fey
FR-F_^l6$F:-Fh_lFRF]\nF_\n-Fc\n6#Q(Kurven:F$-Fc\n6#-Fbr6'7$F\\mF[]nFg^mF`]nFj^m
F]]n-Fc\n6#Q.Vektorfeltet:F$-Fc\n6#-Fbr6'7$FhenF[]nFg^mF`]nFj^mF]]n-Fc\n6#QenVe
ktorfeltet~(langs~kurven)~og~kraftmoment~om~givet~punkt:F$-Fc\n6#-Fbr6'7'F\\mF_
_mF[]nFc`lFacoFg^mF`]nF]]nFj^mF$-F\_n6$QCParameterfremstilling:~r(u)~=~%a~|+F$-
F\pFiclF__n@%Fhgo-F\_n6$Q1Net-finhed:~%a~|+F$Fj_mF$-F\_n6#Q)Punktet:F$-Fc\n6#/F
c[mFjao-F\_n6#Q.V|aygtfunktion:F$-Fc\n6#F`jm-F\_n6#Q;V|aygtfunktion~langs~flade
n:F$-Fc\n6#/.%"fG-F[gl6#-FG6#-F`cl6#F[aq-F\_n6#Q,Vektorfelt:F$-Fc\n6#/.FfzFg]q-
F\_n6#Q9Vektorfelt~langs~kurven:F$-Fc\n6#/F_cq-FfzFfbq-F\_n6#Q(Jacobi:F$-Fc\n6#
/.-FezFiclF^dq-F\_n6#Q7Kraftmoment~af~V~om~p:F$-Fc\n6#/.%#KmGFiy6#Q_pFinder~og~
viser~kraftmomentet~(eng.:~torque)~om~punkt~af~vektorfelt~paa|+v|aygtet~kurveF$
F$F$%/fladeTotalVridGf*Fiu6^oFdvFevFfvFgvF.F/F0F2FhvFivFjvF,%&fluxAG%&fluxBG%&f
luxCG%#mCG%#mAG%#mBGF[wF\wF]wF^wF_wF`wFawF^vFbwFcwFdwFew%"WG%'normaWG%%WmaxG%%m
axWG%'pindVrG%&hatVrG%&pilVrG%'feltVrG%&WkorrGF_v%+integrand2GFfwFgwFhwFiwFjwF[
xF\xF]xF^xF_xF`xFaxFbxFfxFgxFhxFixFjxF[y%)feltGlobGF\yF]yF^yF_yF`y6#%HRev.~17.0
1.05~(S.Markvorsen@mat.dtu.dk)GF$CH@%FazC$>FeyFfz@%3-Fjz6$F]\lF][l-Fjz6$Fb\lF][
l>FbelFd[lC$>FbelF%>FeyFh[lC$>FbelF%>FeyFh[l>F_r-F<6#*&F?F:F]\lFE>Fgs-F<6#*&FMF
:Fb\lFE>F87#-Fer6$-Fer6$-F<6#-F\p6$,&FDF:*&FbtF:F_rF:F:,&FQF:*&F]sF:FgsF:F:/Fbt
;F:F]\l/F]s;F:Fb\l@%33Fjal-Fjz6$-F<6#FQF][l-Fjz6$-F<6#FNF][lC$>FfaoFd[l>Fd`m-Fj
z6$-F[al6$-F[al6$-F<6#&&F8Fh^lF\^l/Fdal;F:*&F]\lF:Fb\lF:FgalF][lC$>FfaoF%>Fd`mF
%>FhgoF]dn>FghoFddn>Facof*FeqF$FboF$FgjnF$F$6&F^wFhgoF_wFgho>Fh^of*FeqF$FboF$-F
`fl6#-F[glFa[oF$F$6$F`wFaco>Fezf*FeqF$FboF$FfblF$F$Ffcl@%3Fd`mFfao>Feim-Fjz6$-F
[al6$-F[al6$-F<6#-FezFahq/Fc[mFghq/Fe[mFihqF][l>FeimF%>8fnf*FeqF$FboF$-F[gl6#-F
hjn6$F]clFfglF$F$6(F*F\pFjuFGF`wFaco>8Jf*FeqF$FboF$-F[gl6#-Fecm6#*&Fi\rF:-F]dmF
cclFEF$F$6*F*F\pFjuFGF`wFacoF^vFh^o>Ffyf*FeqF$FboF$-F[gl6#-Fecm6#*&Fi\rF:,&Fd]r
F:$F:FEF:FEF$F$Fe]r>Fgyf*FeqF$FboF$-F<Fa[oF$F$6$FbeqFfy>F__mf*F`dlF$FboF$-F[gl6
#-F^hl6$&-Fg^lFa\mF9F]pF$F$6&FhxFe\rF+FA>Fffmf*F`dlF$FboF$-F[gl6#-F^hl6$&Fj^rFB
F]pF$F$F[_r>Fc^mf*F`dlF$FboF$-F[gl6#-F^hl6$&Fj^rFRF]pF$F$F[_r>F\\m-F[gl6#-F^hl6
$-F__mF`dlF\il>Fe\m-F[gl6#-F^hl6$-FffmF`dlF\il>F_]mFdgn@%33Ff[rFbelFeimCC>F]of*
F\^lF$FboF$F^^lF$F$6&FdvF8F,Fey>Fgpf*F\^lF$FboF$Fe_lF$F$F]ar>Fcq-F_^l6#-Fer6$,&
-F]oF\^lF:-FgpF\^lFEF``l>Fdbl-F`fl6#-F_^l6#-Fer6$-Fer6$F`\rFfhqFhhq>Fc[l-F`fl6#
-F_^l6#-Fer6$-Fer6$-FgyFahqFfhqFhhq>F\zf*F`dlF$FboF$F^\mF$F$Fep>Fgbmf*FiclF$Fbo
F$Fg\mF$F$Fep>Fbho-Fbr6#-Fer6$-F\z6#Fdhq/F]s;FcsFb\l>&Fg\lFB-Fbr6#-Fer6$-Fgbm6#
Fbhq/Fbt;FcsF]\l>Fc`lFcq>Fgbo-F_^l6$Fg`mFdbl>Fhjl-F_^l6$Fg`mFc[l>Fj]l-Fbr6%7$Fb
hoFicrFg^mFj^m>Fc_l-Fa_m6'Fc_mFbhlF\il/Fe_m7$,$F]\lFjn,$Fb\lFjnFg^m>8gnf*FeqF$F
boF$-F\am6)-Fecm6#,&FbclF:*(F`sF:F]clF:FgglFEF:/F`s;Fcs*&F\dmF:F]dmF:/Fcam*&F`d
mF:F]dmF:FbdmFddmFg^mFgdmF$F$6*F*F\pFjuFGFaxFgboFbwFc`l>8hnf*FeqF$FboF$-F\am6'F
ier/F`s;*&F\dmF:F]dmF:*&FcemF:F]dmF:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F`sFEF:FbdmF
iemF$F$Fbfr>8inf*FeqF$FboF$F^fmF$F$6&FixFeerFjxFdfr>Fh_l-Fbr6$7#-Fer6$-Fer6$-Fa
gr6%FbhqFdhq$FSFEF`drFfcrFg^m>F\jmf*FeqF$FboF$-F\am6)-Fecm6#,&FbclF:*(F`sF:Fbfm
F:FgglFEF:/F`s;Fcs*&F\dmF:F]dmF:/Fcam*&F`dmF:F]dmF:FbdmFddmFg^mFgdmF$F$6*F*F\pF
beqFfyFeeqFhjlFbwFc`l>8Gf*FeqF$FboF$-F\am6'Fchr/F`s;*&F\dmF:F]dmF:*&FcemF:F]dmF
:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F`sFEF:FbdmFiemF$F$F\ir>8Hf*FeqF$FboF$F^fmF$F$6
&FfeqF\jmFgeqF^ir>8I-Fbr6$7#-Fer6$-Fer6$-F[jrF]hrF`drFfcrFg^m>Fcjl-F^jm6(F`jm/F
bjm;-Ff_l6$FE-FgpF9-F_^l6$F:-F]oF9/Fcjm;-Ff_l6$FE-FgpFB-F_^l6$F:-F]oFB/Fdjm;-Ff
_l6$FE-FgpFR-F_^l6$F:-F]oFRF]\nF_\n-Fc\n6#Q.Fladestykket:F$-Fc\n6#-Fbr6'7$Fc_lF
[]nFg^mFj^mF`]nF]]nFh_q-Fc\n6#-Fbr6'7$FcjlF[]nFg^mFj^mF`]nF]]n-Fc\n6#Q;Vektorfe
ltet~(ved~fladen):F$-Fc\n6#-Fbr6'7%Fj]lFh_lF[]nFg^mF`]nF]]nFj^m-Fc\n6#-Fbr6'7%F
c_lFh_lF[]nFg^mF]]nF`]nFj^m-Fc\n6#QLVridningsvektorfeltet,~V~x~n,~(ved~fladen):
F$-Fc\n6#-Fbr6'7%Fj]lF_jrF[]nFg^mF`]nF]]nFj^m-Fc\n6#-Fbr6'7%Fc_lF_jrF[]nFg^mF]]
nF`]nFj^mF$F[_nF__n@%Fbel-F\_n6$F_aqFeyF$-F\_n6#Q5Vektorfelt~V(x,y,z):F$Fiaq-F\
_n6#Q9Vektorfelt~langs~fladen:F$-Fc\n6#Fhhn-F\_n6#Q-Jacobi(u,v):F$-Fc\n6#/.-Fh^
oFeqF\`s-F\_n6#QHNormalvektorfeltet~genereret~af~r(u,v):F$-Fc\n6#/.-%"NGFeq-F[g
l6#-FacoFeq-F\_n6#QLVridningsvektorfeltet,~V~x~n,~langs~fladen:F$-Fc\n6#/.%$Vxn
G-F[gl6#-F]]rFeq-F\_n6#Q\oDen~totale~vridningsvektor~af~givne~fladestykke~med~v
ektorfeltet:F$-Fc\n6#/.-%$IntG6$F`as/%#muG;&%"FGFj^l%!G-F[gl6#7%F\\mFe\mF_]m-Fc
\n6#/Fjas-F<Febs6#QdoFinder~og~viser~det~totale~vrid~for~givet~vektorfelt~af~gi
vet~fladestykkeF$F$F$%(kurveCmGf*F[cn6<F]vF^vF_v%&CMposGFc]oFccnFdcnF,FhwFdvF.F
fwFgwFhyFfxFjyF[zFbvF]zF^zF_zFd^oFgvF`foFevFfvFayF$C8@%FazC$>Fg\lFfz@%-Fjz6$Fbh
oF][l>Fj]lFd[lC$>Fj]lF%>Fg\lF\hoC$>Fj]lF%>Fg\lF\ho>F\\mFjal>&Fh_lF9-F<6#*&F?F:F
bhoFE>Fc_l7#-Fer6$-F<6#-F\p6#,&FDF:*&FbtF:FadsF:F:/Fbt;F:Fbho>Fey-Fjz6$-F[al6$-
F[al6$.-F<6#&&Fc_lFh^lF\^l/FdalF`esFgalF][l>Fe\m-Fjz6$-F[al6$.-F<6#-FG6#-F`cl6#
&Fc_lF]]p/F`]pF`esF][l>F8F[^p>F]of*F[flF$FboF$Fb^pF$F$6$F]vF8>Fgpf*F[flF$FboF$-
F[gl6#*&FfjoF:F^`pF:F$F$6(F*F\pF\cnFGF^vF]o>Fc^m-F[gl6#-F^hl6$-FgpF[flFfcp@%5-F
jz6$-F<6#Fc^m.%)positiveG4-Fjz6$F`hsF][l>F__m7%-F[gl6#*&-F^hl6$-F[gl6#*&&F_^pF9
F:F[hsF:FfcpF:Fc^mFE-F[gl6#*&-F^hl6$-F[gl6#*&&F_^pFBF:F[hsF:FfcpF:Fc^mFE-F[gl6#
*&-F^hl6$-F[gl6#*&&F_^pFRF:F[hsF:FfcpF:Fc^mFEF$@%33FeyF\\mFj]lC+>Faco-F\am6(F_^
pFfcp/Fe_m7$,$FbhoFjnFjnFjcpFg^mFj^m>Fhgo-F`cl6$F:-F`cl6$F:Faco>Fgdof*F\^lF$Fbo
F$FddpF$F$6&Fd^oFhgoF,Fg\l>Fh^of*F\^lF$FboF$FdepF$F$Fe[t>Fgho-F_^l6#-Fer6$,&-Fg
doF\^lF:-Fh^oF\^lFEF``l>Fffm-F\am6(F_^pFfcpF[[t/Fcam,$FghoFefpFg^mFj^m@%F^hsC&>
FcqFd[l>F\z,$Fgho#F:"#D>Fgbm-F<6#-Fecm6#F__m>F`r-Fa_m6'7%,&&FgbmF9F:*(F\zF:Fehp
F:FghpF:F:,&&FgbmFBF:*(F\zF:FehpF:F\ipF:F:,&&FgbmFRF:*&F\zF:F`ipF:F:FaipFdipFgi
pFg^mC%>FcqF%>F\z$FcsFcs>Fgbm7%FcsFcsFcsF`_q-Fc\n6#-Fbr6%7$FffmF[]nFg^mFj^mF$@%
33FfjsFe\mFj]lC'>Fj_m-F_^l6#-Fer6$-%$absG6#-F<6#-FG6#-F`cl6#-F\p6#,&FDF:F^es#F:
Fjn/Fbt;FcsF][t>Fez-F\am6*F_^pFfcpF[[t/Fcam,$*(FghoF:-F_^l6$Fcs-FG6#-F`cl6#F_^p
F:Fj_mFEF^`t/Fedm%*wireframeGFgdmFg^mFj^m>Fdbl-F\am6*F_^pFfcpF[[t/Fcam,$*(FghoF
:-Ff_lFh`tF:Fj_mFE#FEFjnF]atFegpFg^mFj^m-Fc\n6#Q`oKurven~med~v|aygtfunktion~og~
massemidtpunkt~(hvis~det~er~veldefineret):F$@%Fcq-Fc\n6#-Fbr6&7'FffmFdblFezF`rF
[]nFg^mF`]nFj^m-Fc\n6#-Fbr6&7&FffmFdblFezF[]nFg^mF`]nFj^mF$Fh`qF__n@%Faz-F\_n6$
F_aqFg\lF$-F\_n6$Q>V|aygtfunktion:~f(x,y,z)~=~%a~|+F$F`jm-F\_n6#Q*Jacobi(u)F$-F
c\n6#/.-F]oFiclFbct-F\_n6#Q2Totale~masse,~M~:F$-Fc\n6#/Fc^mF`hs@%F^hsC$-F\_n6#Q
2Masse-midtpunkt:~F$-Fc\n6#/F__m-F<F`]t@%FdhsC$-F\_n6#Q3Masse-midtpunkt~:~F$-Fc
\nF`]t-F\_n6#QHMasse-midtpunktet~er~ikke~veldefineret.F$6#QeqFinder~og~viser~(h
vis~muligt)~|+massemidtpunktet~af~en~v|aygtet~kurve;~funktionen~f(x,y,z)~|+er~v
|aygtfunktionenF$F$F$%0traadModelFladeGf*F)6&F.F/F0F2FayF$C)>F7F>>FIFL>F]of*F`d
lF$FboF$F^\mF$F$Fep>Fgpf*FiclF$FboF$Fg\mF$F$Fep>Fjgl-Fbr6#-Fer6$-F]o6#F[sFas>Fe
hl-Fbr6#-Fer6$-Fgp6#F`tFct-Fbr6%7$FjglFehlFg^mFj^m6#Q:Viser~tr|`ydmodel~af~flad
e.F$F$F$%(fladeKmGf*F^fo6coFdvFevFfvFgvF.F/F0F2FhvFivFjvF,F[wF\wF]wF^wF_wF`wFaw
F^vFbwFcwFdwFewF_vF[fqFfwFgwFhwFiwFjwF[xF\xF]xF^xF_xF`xFaxFbxF`eqFaeqF_eqFbfoFc
foFdfoFhyF]^oFfxFgxFhxFixFjxF[yF\fqFafoFefoFffoFgfoFhfoFifoFjfoF\yF]yF^yF_yF`yF
e_pFg_pFccnFdcnFc]oF[goF$CM@%F_goC$>FeyFbgo@%Fdfq>FhenFd[lC$>FhenF%>FeyFh[lC$>F
henF%>FeyFh[l>F_rFagq>FgsFegq@%F][p>FiyFd[l>FiyF%>F8Figq>Fcjl-Fjz6$-F[al6$.-F<6
#-FG6#-F`cl6#&F8F]]p/F`]pFcjqF][l@%F[iqC$>F[enFd[l>Fc`lFhiqC$>F[enF%>Fc`lF%>Fc^
mF]dn>F__mFddn>Fffmf*FeqF$FboF$FgjnF$F$6&F^wFc^mF_wF__m>Fdblf*FeqF$FboF$-F[gl6#
-F`flFa[oF$F$6$F`wFffm>F\\mf*FeqF$FboF$FfblF$F$F[[p@%3Fc`lF[en>8A-Fjz6$-F[al6$-
F[al6$-F<6#-F\\mFahqFa\rFb\rF][l>FdjtF%>Fe\rf*FeqF$FboF$-F[gl6#-Fhjn6$-Fecm6#,&
*&Fg^lF:F`\oF:F:*&FgglF:F`gnF:FE-FcdoF]\oF$F$6,Fe_p8`oF*F\pFg_p8aoF`]oFjaoFjuFf
z>Fdfrf*FeqF$FboF$*(F]clF:&FfglF9F:Fd]rF:F$F$6*F*F\pF\cnFGFffoFe\rF^vFdbl>Fagrf
*FeqF$FboF$*(F]clF:&FfglFBF:Fd]rF:F$F$Fb\u>8jnf*FeqF$FboF$*(F]clF:&FfglFRF:Fd]r
F:F$F$Fb\u>Fifp-Fdel6#-F[gl6$-F^hl6$-FdfrFeqFbhlF`o>F^fp-Fdel6#-F[gl6$-F^hl6$-F
agrFeqFbhlF`o>Fd`m-Fdel6#-F[gl6$-F^hl6$-Fh\uFeqFbhlF`o>Ffao-F[gl6#-F^hl6$-FifpF
`dlF\il>Fbel-F[gl6#-F^hl6$-F^fpF`dlF\il>Feim-F[gl6#-F^hl6$-Fd`mF`dlF\il>8en-F]b
p6%/F\\uF:/F]\uF:7%FfaoFbelFeim>Feer-F`fl6#,(*$&Fg_uF9FCF:*$&Fg_uFBFCF:*$&Fg_uF
RFCF:@%3333FbjtFhenFiyFdjtFcjlC>>F]of*F\^lF$FboF$F^^lF$F$F]ar>Fgpf*F\^lF$FboF$F
e_lF$F$F]ar>FcqFaar>Fe\m-F`fl6#-F_^l6#-Fer6$-Fer6$F][uFfhqFhhq>F\zf*F`dlF$FboF$
F^\mF$F$Fep>Fgbmf*FiclF$FboF$Fg\mF$F$Fep>FbhoF`cr>FicrFjcr>Fj_mFcq>F_jr-F_^l6$F
g`mFe\m>8Z-Fdo6&Fjao-Fecm6#Fg_u/Fbgp-Ff_l6$Fj_mFeerFegp>8bo,$Fcq#F:"#?>8coF\hp>
8do-Fa_m6)7%,&&FdcuF9F:*(F_cuF:FehpF:FghpF:F:,&&FdcuFBF:*(F_cuF:FehpF:F\ipF:F:,
&&FdcuFRF:*&F_cuF:F`ipF:F:FaipFdipFgipFddmFbpFg^m>Fj]lFjdr>Fc_lF^er>F[emf*FeqF$
FboF$-F\am6)Fier/F`s;Fcs*&F\dmF:F]dmF:/Fcam*&F`dmF:F]dmF:FbdmFddmFg^mFgdmF$F$6*
F*F\pFjuFfzFaxF_jrFbwFj_m>F\fmf*FeqF$FboF$-F\am6'Fier/F`s;*&F\dmF:F]dmF:*&FcemF
:F]dmF:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F`sFEF:FbdmFiemF$F$F^eu>Fgbof*FeqF$FboF$F
^fmF$F$6&FixF[emFjxF\fm>Fh_l-Fbr6$7#-Fer6$-Fer6$-FgboF]hrF`drFfcrFg^m>Ffy-F^jm6
(Fg]qF[[sFc[sF[\sF]\nF_\nFc\sFf\sFh_q-Fc\n6#-Fbr6'7$FfyF[]nFg^mFj^mF`]nF]]n-Fc\
n6#QYVektorfeltet~(ved~fladen)~og~kraftmoment~om~givet~punkt:F$-Fc\n6#-Fbr6'7'F
j]lFh_lFfcuF[]nFfbuFg^mF`]nF]]nFj^m-Fc\n6#-Fbr6'7'Fc_lFh_lFfcuF[]nFfbuFg^mF]]nF
`]nFj^mF$F[_nF__n@%FhenF[_sF$F`aqFcaqFfaqFiaqF[bq-Fc\n6#/Fabq-F[glFd_sFibqF\cqF
`_s-Fc\n6#/F_cq-FfzFihnFgcq-Fc\n6#/.-FdblFeqFihuF_dq-Fc\n6#/FedqFg_u6#Q_pFinder
~og~viser~kraftmomentet~(eng.:~torque)~om~punkt~af~vektorfelt~paa|+v|aygtet~fla
deF$F$F$%&rumImGf*F_]o6fnFgwFfwFhyFhwFfxFdvFevFfvFgvFdcn%"aG%%maxfG%%minfGFccnF
c]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFg]oF]wF^wF_wFh]oFi]oFhvFgcnFecnFj]oF[^oFj
yF[zF\^oF,F[fq%+integrand3GF_vF]^oF^^oF_^oF`^oFacnFfcnF`wF^vFicnFa^oFb^oFc^oFd^
oFe^oF3F$CX@%F_goC$>F^fpFbgo@%33-Fjz6$&F^fpF9F][l-Fjz6$&F^fpFBF][l-Fjz6$&F^fpFR
F][l>FcqFd[lC$>FcqF%>F^fpF^\nC$>FcqF%>F^fpF^\n>&Fc^mF9-F<6#*&F?F:F_juFE>&Fc^mFB
-F<6#*&FMF:FbjuFE>&Fc^mFR-F<6#*&FZF:FejuFE@%333333FcqF[blF_blF]iqFaiq-Fjz6$-F<6
#FhnF][l-Fjz6$-F<6#FenF][lC%>F8Fd[l>F\z7#-Fer6$-Fer6$-Fer6$-F<6#-F\p6%,&FDF:*&F
btF:F^[vF:F:,&FQF:*&F]sF:Fc[vF:F:,&FhnF:*&FaoF:Fh[vF:F:/Fbt;F:F_ju/F]s;F:Fbju/F
ao;F:Feju>F]o-Fjz6$-F[al6$-F[al6$.-F<6#&F\cmF\^l/Fdal;F:*(F_juF:FbjuF:FejuF:Fga
lF][l>F8F%@%33-Fjz6$-F<6#&FfzF9F][l-Fjz6$-F<6#&FfzFBF][l-Fjz6$-F<6#&FfzFRF][l>F
eimFd[l>FeimF%@%F][p>FcjlFd[l>FcjlF%@%3F]oF8>Fgp-Fjz6$-F[al6$.-F<6#-FG6#-F`cl6#
&F\zF]]p/F`]pFa_vF][l>FgpF%@%FgpC$>Fey-F_^l6#-Fer6$Fcav/F^]pFa_v>F\\m-Ff_lF`bvF
$>FhenFi^o>FdjtFb_o>FfyFi_o>Fhjlf*F`_oF$FboF$F_`oF$F$6(F^wFhenF_wFdjtFh]oFfy>F\
fmf*F`_oF$FboF$Fj`oF$F$6$FgcnFhjl>F[_pFgao>Fg_uf*F[flF$FboF$-Fecm6#,&FioF:*&F\p
F:F__lF:F:F$F$6&F`]oFfzF`^oF[_p>Fiyf*F`_oF$FboF$FaboF$F$Febo>Feerf*F`_oF$FboF$F
iboF$F$6&F_^oFiyF`^oF[_p>Fbelf*F`_oF$FboF$-F[gl6#*(-F`p6#-F`cl6#FdboF:Ff`oF:-F]
dmFd`oF:F$F$6*F*F\pF\cnFGFe^oFeerF^vF\fm>Fc`lf*F_oF$FboF$FccoF$F$6,F*F\pF\cnFGF
e^oFeerF^vF\fmF+FA>Fd`mf*F_oF$FboF$FafmF$F$6$Fg]oFc`l>F[enf*FidoF$FboF$FjdoF$F$
6&Fg]oFc`lF+FA>Ffaof*FidoF$FboF$Fh^pF$F$6$F]wF[en>8Y-F[gl6#-F^hl6$-F[enFidoFheo
@%3333F\avFgpFcqFeimFcjlCL>Fgbmf*F\^lF$FboF$-F_^l6#-Fer6$Fe^l/Fi^l;F:*(F^_lF:F`
_lF:&F__lFRF:F$F$6&FdvF\zF,F^fp>Fg\lf*F\^lF$FboF$-Ff_lFcfvF$F$Fjfv>Fj]l-F_^l6#-
Fer6$,&-FgbmF\^lF:-Fg\lF\^lFEF``l>Ffbu-F\am6*-Fg_uF[fl/F`s;,$-F_^l6$$FjgpFEFj]l
FEF]hv/Fe_m7$FCFf`l/Fcam-F_^l6$$FSF\dp,$Fj]l#F:FaamFiemFb\mFg^mFj^m>Fj_mf*F_oF$
FboF$-Ffo6%7%F[pF\pFA/F[p;F_elFcdlFb\mF$F$6$F+FA>Fhgof*FipF$FboF$-Ffo6%7%F\pF`o
FA/F`o;F\dlFidlF[]mF$F$F_iv>Fghof*FeqF$FboF$-Ffo6%7%F\pFAFao/Fao;&FioFin&FioFfn
Fa[qF$F$F_iv>&FgdoF9-Fbr6#-Fer6$-Fer6$-Fj_m6$F[^v,&FhnF:*&F`sF:Fh[vF:F:/F]s;Fcs
Fbju/F`s;FcsFeju>&FgdoFB-Fbr6#-Fer6$-Fer6$-Fhgo6$Fi]vFjjv/Fbt;FcsF_juF^[w>&Fgdo
FR-Fht6#-Fer6$-Fer6$-Fgho6$Fi]vF[^vFj[wF\[w>Fgy-Fbr6%7%FajvFa[wF]\wFg^mFj^m>Fff
mf*F_oF$FboF$-Ffo6%FhoF]pFb\mF$F$Fep>Fezf*FipF$FboF$-Ffo6%F\qF^qF[]mF$F$Fep>Fdb
lf*FeqF$FboF$-Ffo6%FhqFjqFa[qF$F$Fep>&FacoF9-Fbr6#-Fer6$-Fer6$-FffmFijvF\[wF^[w
>&FacoFB-Fbr6#-Fer6$-Fer6$-FezFi[wFj[wF^[w>&FacoFR-Fht6#-Fer6$-Fer6$-FdblFe\wFj
[wF\[w>Fc[l-Fbr6#7%Fg]wF`^wFi^w>&F__m6$F:F:-Fa_m6(-F\p6%FDF`oFaoF\ilFheoFb\m/Fe
_m7$,$FbjuFS,$FejuFSFg^m>&F__m6$F:FC-Fa_m6(-F\p6%F@F`oFaoF\ilFheoFb\mF\`wFg^m>&
F__m6$FCF:-Fa_m6(-F\p6%F[pFQFaoFbhlFheoF[]m/Fe_m7$,$F_juFSF_`wFg^m>&F__m6$FCFC-
Fa_m6(-F\p6%F[pFNFaoFbhlFheoF[]mF^awFg^m>&F__m6$FSF:-Fa_m6(-F\p6%F[pF`oFhnFbhlF
\ilFegp/Fe_m7$F`awF^`wFg^m>&F__m6$FSFC-Fa_m6(-F\p6%F[pF`oFenFbhlF\ilFegpF_bwFg^
m>Fh^o-Fbr6%-Fer6$-Fer6$&F__m6$Fi^lF^]p/Fi^lFa`l/F^]p;F:FCFj^m/F^]n%&boxedG>Fh_
lFC>&F^irFg_wf*F`_oF$FboF$7&Fdbo,&FdboF:-F<6#*(Fe`oF:&FgglFBF:F]dmFEF:,(FdboF:F
\dwF:-F<6#*(-T+Fd`oF:&FgglFRF:F]dmFEF:,&FdboF:FadwF:F$F$6,F*F\pF_wFdjtF.Fc^mFbi
uFh_lFh]oFfy>&F^irFb`wf*F`_oF$FboF$-%$mapG6$f*6#FbjmF$FboF$,&F\pF:-F<6#*(-F]dm6
%FioF`pF\\oF:&FedwF9F:T-FEF:F$F$6.F[pF\pF`oFAFaoFGF^wF__lF.FgglFbiuF]dm-&Fg^lFg
_wFd`oF$F$6*Fj]oF^irF^wFhenF.Fc^mFbiuFh_l>&F^irFjawf*F`_oF$FboF$7&Fdbo,&FdboF:-
F<6#*(Fe`oF:&FgglF9F:F]dmFEF:,(FdboF:FbfwF:-F<6#*(FddwF:F_dwF:F]dmFEF:,&FdboF:F
gfwF:F$F$6,F*F\pF^wFhenF.Fc^mFbiuFh_lF_wFdjt>&F^irFcbwf*F`_oF$FboF$-F]ew6$f*F`e
wF$FboF$,&F\pF:-F<6#*(FeewF:&FedwFRF:FhewFEF:F$F$6.F[pF\pF`oFAFaoFGFh]oF__lF.Fg
glFbiuF]dm-&Fg^lFjawFd`oF$F$6*Fj]oF^irFh]oFfyF.Fc^mFbiuFh_l>&F^irFi`wf*F`_oF$Fb
oF$7&FdboFafw,(FdboF:FbfwF:FadwF:FgdwF$F$6,F*F\pF^wFhenF.Fc^mFbiuFh_lFh]oFfy>&F
^irFcawf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$,&F\pF:-F<6#*(FeewF:&FedwFBF:FhewFEF:F$
F$6.F[pF\pF`oFAFaoFGF_wF__lF.FgglFbiuF]dm-&Fg^lFi`wFd`oF$F$6*Fj]oF^irF_wFdjtF.F
c^mFbiuFh_l@%2,&FeyF:F\\mFE$F:Fadm>F\jmf*F`_oF$FboF$-%.polygonplot3dG6%7#-Fer6$
-Fer6$-&Fg^lF`cwFd`oFacwFbcw/Fcp-%&COLORG6$%$HUEGF\dmFg^mF$F$6$Fj]oF^ir>F\jmf*F
`_oF$FboF$-Fgiw6%Fiiw/Fcp-Fbjw6$Fdjw-F<6#*&,&-F\\o6#-F`cl6#-F`pFd`oF:F]dmFEF:,&
FedwF:F]dmFEFEFg^mF$F$6,Fj]oF^irF*F\pF\cnFGFdiuF\\mFciuFey>F`cm-Fbr6%7#-Fer6$-F
er6$-Fer6$-F\jm6%Fi]vF[^vFjjv/Fbt;Fcs,&F_juF:FEF:/F]s;Fcs,&FbjuF:FEF:/F`s;Fcs,&
FejuF:FEF:Fg^mFj^m-Fc\n6#QHParameter-omr|`ydet~i~definitionsm|ayngden:F$-Fc\n6#
-Fbr6&7&FajvFa[wF]\w-F\]nF`_oFg^mF]]nFj^m-Fc\n6#QVDe~afgr|aynsende~flader~for~d
et~parametriserede~omr|`yde:F$-Fc\n6#-Fbr6&7$Fh^oF[]nFg^mF]]nFj^m-Fc\n6#QWDet~p
arametriserede~billed-omr|`yde~med~omdrejningsakse:F$-Fc\n6#-Fbr6&7%Fc[lFfbuF[]
nFj^mFg^mF]]n-Fc\n6#Q_pApproksimerede~billed-omr|`yde,~omdrejningsakse~og~antyd
et~v|aygtfunktion~(masset|aythed):F$-Fc\n6#-Fbr6'7%F`cmFfbuF[]nFj^mFg^mF]]nF`]n
F$-F\_n6$QGParameterfremstilling:~r(u,v,w)~=~%a~|+F$F__oF__n-F\_n6$F_aqF^fp-F\_
n6$QFAngivet~masse-t|aythed:~m(x,y,z)~=~%a~|+F$F`jm-F\_n6#Q/Jacobi(u,v,w):F$-Fc
\n6#/.-F\fmF`_oF]`x-F\_n6$QEPunkt~p|`y~omdrejnings-aksen:~p~=~%a~|+F$Ffz-F\_n6$
QHAkse-retningen~er~angivet~ved:~e~=~%a~|+F$Fjao-F\_n6#QZAfstanden~fra~rumpunkt
et~r(u,v,w)~til~aksen~er~givet~ved:F$-Fc\n6#/.-*$&%%distG6#%"lGFCF`_o-FeerF`_o-
F\_n6#Q=Inerti-moment~om~givne~akse:F$-Fc\n6#/.-F\bs6$-%".G6$FbbqF\ax/F_bs;&%&O
megaGFj^lFcbs-F[gl6#Fdev-Fc\n6#/Fhax-F<Fcbx-F\_n6#Q8Inertimoment-beregning:F$-F
c\n6#/Fhax-F\bs6$-F\bs6$-F\bs6$*(.-FbbqFajmF:FaaxF:F\`xF:FbhlF\ilFheo-Fc\n6#/Fh
ax-F\bs6$-F\bs6$-F\bs6$*(-FG6#-F`cl6#F__oF:FaaxF:F\`xF:FbhlF\ilFheo-Fc\n6#/Fhax
-F\bs6$-F\bs6$-F\bs6$*(FadxF:FaaxF:F]`xF:FbhlF\ilFheo-F\_n6#Q9-------INTERMEZZO
:::::::F$-Fc\n6#/-F\bs6$-FbelF`_oFbhl&7#-F^hl6$FgexF[p6#Fbhl-Fc\n6#/7"-F^hlFfex
-Fc\n6#/-F\bs6$FeexF\il-F\bs6$-Fd`mF_oF\il-Fc\n6#/Fgfx&7#-F^hl6$FifxF`o6#F\il-F
c\n6#/F`fx-F^hlFhfx-Fc\n6#/-F\bs6$FefxFheo-F\bs6$-FfaoFidoFheo-Fc\n6#/F[hx&7#-F
^hl6$F]hxFao6#Fheo-Fc\n6#/F`fx-F^hlF\hx-F\_n6#Q9-------KONKLUSION:::::::F$Feax-
F\_n6#Q<===========================F$6#QhoFinder~og~viser~inertimomentet~af~giv
et~v|aygtet~omr|`yde~i~rummet~om~given~akse.F$F$F$%*GaussFluxGf*Fiu6H%#rfG%#BfG
Fd]oFdvFf]oFjv%#fLG%&totalGFcwFdwFewFevFfvFgvFjwF,FfwFhwFgwFiwF[wF.F/F0F1F2%#rv
GFgxFhvF[xF\xF]xF^xF_xF`xFaxFbw%+outsigningG6#%HRev.~13.01.05~(S.Markvorsen@mat
.dtu.dk)GF$CQ@%FazC$>Fc^mFfz@%33-Fjz6$F^[vF][l-Fjz6$Fc[vF][l-Fjz6$Fh[vF][l>Fffm
Fd[lC$>FffmF%>Fc^mF^\nC$>FffmF%>Fc^mF^\n>&FhgoF9-F<6#*&F?F:F^[vFE>&FhgoFB-F<6#*
&FMF:Fc[vFE>&FhgoFR-F<6#*&FZF:Fh[vFE@%333333FffmF[blF_blF]iqFaiqFc\vFg\v>FezFd[
l>FezF%>Fcq7#-Fer6$-Fer6$-Fer6$-F<6#-F\p6%,&FDF:*&%#ppGF:Fb[yF:F:,&FQF:*&%#qqGF
:Fg[yF:F:,&FhnF:*&%#hhGF:F\\yF:F:/Fg]y;F:F^[v/Fj]y;F:Fc[v/F]^y;F:Fh[v>F__m-Fjz6
$-F[al6$-F[al6$-F<6#&&FcqFh^lF\^l/Fdal;F:*(F^[vF:Fc[vF:Fh[vF:FgalF][l>Fj_mf*F`_
oF$FboF$-F<6#-Fibl6%FadvFadvFdclF$F$Ffcl>Fdbl-Fjz6$-F[al6$-F[al6$-F[al6$-Fj_m6%
,&FDF:*&FbtF:Fb[yF:F:,&FQF:*&F]sF:Fg[yF:F:,&FhnF:*&%"hGF:F\\yF:F:/Fc[mF_^y/Fe[m
Fa^y/.Fi`yFc^yF][l>&F]oFg_w7&FQFNFhnFen>&F]oFb`wF`ay>&F]oFi`w7&FDF@FhnFen>&F]oF
cawFeay>&F]oFjaw7&FDF@FQFN>&F]oFcbwFjay>&F8Fg_wf*F_oF$FboF$-F`p6%F_elF\pFAF$F$F
]dl>&Fc`lFg_w-Fdel6%-F]bp6$/F[pFDF]_oF`oFao>&F8Fb`wf*F_oF$FboF$-F`p6%FcdlF\pFAF
$F$F]dl>&Fc`lFb`w-Fdel6%-F]bp6$/F[pF@F]_oF`oFao>&F8Fi`wf*FipF$FboF$-F`p6%F\pF\d
lFAF$F$F]dl>&Fc`lFi`w-Fdel6%-F]bp6$/F`oFQFf_oF[pFao>&F8Fcawf*FipF$FboF$-F`p6%F\
pFidlFAF$F$F]dl>&Fc`lFcaw-Fdel6%-F]bp6$/F`oFNFf_oF[pFao>&F8Fjawf*FeqF$FboF$-F`p
6%F\pFAF^jvF$F$F]dl>&Fc`lFjaw-Fdel6%-F]bp6$/FaoFhnF[`oF[pF`o>&F8Fcbwf*FeqF$FboF
$-F`p6%F\pFAF_jvF$F$F]dl>&Fc`lFcbw-Fdel6%-F]bp6$/FaoFenF[`oF[pF`o>F[enf*6&Fav%$
rrvG%#BBG%#VVG6'F^wF_wF`wFhxF^v6#%HRev.~12.01.05~(S.Markvorsen@mat.dtu.dk)GF$C'
>F8F]dn>F]oFddn>Fgp-Fdel6%-Fhjn6$-F8Feq-F]oFeqF[pF`o>Fcqf*FeqF$FboF$-F[gl6#*&-F
\fn6#-Fibl6%FafmF`\oFdclF:-Fibl6%Fafm-F`gn6#-F`cl6#-F\\oFcclFdclF:F$F$6*F`wFgpF
hfyFAFavF\pFjfyFfz-F[gl6#-F^hl6$-F[gl6#-F^hl6$-FcqFeq/F[p;FFFT/F`o;F[o&FGFO6#QY
Finder~fortegnsmodificerede~UD-fluxer~p|`y~de~6~sidefladerF$F$F$>Fgbmf*F`cwF$Fb
oF$-Fedw6&&Fg^lFccl&F__lFccl&FgglFcclF`gnF$F$6,FeixF8FiixFc`lFfixF]oFjuFGFgxF[e
n>Fg\l,.-FgbmFg_wFE-FgbmFb`wF:-FgbmFi`wFE-FgbmFcawF:-FgbmFjawFE-FgbmFcbwF:@%333
F__mFezFdblFffmC:>Fghof*F_oF$FboF$F\]wF$F$Fep>Facof*FipF$FboF$F`]wF$F$Fep>Fgdof
*FeqF$FboF$Fd]wF$F$Fep>Fchp-Fbr6#-Fer6$-Fer6$-Fgho6$Fe`y,&FhnF:*&F`sF:F\\yF:F:/
F]s;FcsFc[v/F`s;FcsFh[v>Fjhp-Fbr6#-Fer6$-Fer6$-Faco6$Fc`yF]\z/Fbt;FcsF^[vFa\z>F
^ip-Fht6#-Fer6$-Fer6$-Fgdo6$Fc`yFe`yF\]zF_\z>Fhen-Fbr6#7%FchpFjhpF^ip>&FgpFg_w-
Fa_m6(Fj_wF\ilFheoFb\m/Fe_m7$,$Fc[vFS,$Fh[vFSFg^m>&FgpFb`w-Fa_m6(Fe`wF\ilFheoFb
\mF_^zFg^m>&FgpFi`w-Fa_m6(F\awFbhlFheoF[]m/Fe_m7$,$F^[vFSFb^zFg^m>&FgpFcaw-Fa_m
6(FfawFbhlFheoF[]mF[_zFg^m>&FgpFjaw-Fa_m6(F]bwFbhlF\ilFegp/Fe_m7$F]_zFa^zFg^m>&
FgpFcbw-Fa_m6(FfbwFbhlF\ilFegpFf_zFg^m>F`r-Fbr6%-Fer6$-Fer6$&FgpF`cwFacwFbcwFj^
mFdcw>Feyf*F\^lF$FboF$FbfvF$F$6&FdvFcqF,Fc^m>F\\mf*F\^lF$FboF$F]gvF$F$Ff`z>Fe\m
-F_^l6#-Fer6$,&F\fpF:-F\\mF\^lFEF``l>F[jr*&F\dmF:Fe\mF:>F_]m-F^jm6(F`jm/Fbjm;-F
f_l6$FE-F\\mF9-F_^l6$F:F\^q/Fcjm;-Ff_l6$FE-F\\mFB-F_^l6$F:Fd^q/Fdjm;-Ff_l6$FE-F
\\mFR-F_^l6$F:F\_qF]\nF_\n-Fc\n6#Q<Det~parametriserede~omr|`yde:F$-Fc\n6#-Fbr6'
7$F`rF[]nFg^mF]]nF`]nFj^m-Fc\n6#-Fbr6'F_`qF]]nF`]nFg^mFj^mFh_q-Fc\n6#-Fbr6'7$F_
]mF[]nFg^mF]]nF`]nFj^mF$F^_xF__nFb_nFe_n-Fc\n6#QYBeregning~af~fluxene~UD~gennem
~de~respektive~sideflader:F$Fe_n-F\_n6%Q:Del-flux~[1,1]:~%a~=~%a~|+F$,$FcjyFE-F
[gl6#-F<6#Fadz-F\_n6%Q:Del-flux~[1,2]:~%a~=~%a~|+F$Fdjy-F[gl6#-F<6#Fdjy-F\_n6%Q
:Del-flux~[2,1]:~%a~=~%a~|+F$,$FejyFE-F[gl6#-F<6#F`ez-F\_n6%Q:Del-flux~[2,2]:~%
a~=~%a~|+F$Ffjy-F[gl6#-F<6#Ffjy-F\_n6%Q:Del-flux~[3,1]:~%a~=~%a~|+F$,$FgjyFE-F[
gl6#-F<6#F_fz-F\_n6%Q:Del-flux~[3,2]:~%a~=~%a~|+F$Fhjy-F[gl6#-F<6#FhjyFe_n-Fc\n
6#QSDen~totale~flux~UD~igennem~sidefladerne~er~derfor:F$Fe_n-Fc\n6#/-F[gl6#-Fa`
n6#Fg\l-F[gl6#-F<Fdgz6#Q_rFinder~totale~UDadg|`yende~flux~gennem~de~afgr|aynsen
de~side-flader~for~et~rumligt~omr|`yde~med~parameterfremstillingen~r.F$F$F$%*fl
adeKmGoGf*F^foF\gtF[goF$C1>Fc^mF]dn>F__mFddn>Fffmf*FeqF$FboF$FgjnF$F$Fhit>Fdblf
*FeqF$FboF$F[jtF$F$F^jt>Fe\rf*FeqF$FboF$Fa[uF$F$F[\u>Fdfrf*FeqF$FboF$F`\uF$F$Fb
\u>Fagrf*FeqF$FboF$Fe\uF$F$Fb\u>Fh\uf*FeqF$FboF$Fj\uF$F$Fb\u>FifpF]]u>F^fpFe]u>
Fd`mF]^u>FfaoFe^u>FbelF[_u>FeimFa_u>Fg_uFh_u6#QfoFinder~kraftmomentet~(eng.:~to
rque)~om~punkt~af~vektorfelt~paa|+v|aygtet~fladeF$F$F$%/kurveIntApproxGf*F[cn6D
F]vF^vF_v%*intEksaktG%'intAppGF/F^wF`vFfcnF`csFc]oFccnFdcnF,FhwFdvF.FfwFgwFhyFf
xFjyF[zFbvF]zF^zF_zFd^oFgvF`foFevFfvFciuFdiu6#%HRev.~27.12.03~(S.Markvorsen@mat
.dtu.dk)GF$C<@%FazC$>Fe\mFfz@%-Fjz6$&Fe\mF9F][l>F_]mFd[lC$>F_]mF%>Fe\mF\hoC$>F_
]mF%>Fe\mF\ho>FezFjal>&F__mF9-F<6#*&F?F:FbjzFE>Fc^m7#-Fer6$-F<6#-F\p6#,&FDF:*&F
btF:F\[[lF:F:/Fbt;F:Fbjz>Fffm-Fjz6$-F[al6$-F[al6$.-F<6#&&Fc^mFh^lF\^l/FdalF[\[l
FgalF][l>Fdbl-Fjz6$-F[al6$.-F<6#-FG6#-F`cl6#&Fc^mF]]p/F`]pF[\[lF][l@%FdblC$>Fc[
l-F_^l6#-Fer6$F_][l/F^]pF[\[l>Fhjl-Ff_lF[^[lF$>F8F[^p>F]of*F[flF$FboF$Fb^pF$F$F
_gs>Fgpf*F[flF$FboF$FbgsF$F$Fegs>Fcq-F[gl6#-F^hl6$-FgpFiclFbhl>F`r-%$SumG6$*(-F
G6#-F`clFe[[lF:-F]oFg[[lF:F\[[lF:/Fc[m;Fcs,&FbjzF:FEF:@%33FffmFezF_]mC.>Fdjt-F\
am6(F_^pFfcp/Fe_m7$,$FbjzFjnFjnFjcpFg^mFj^m>F[en-F`cl6$F:-F`cl6$F:Fdjt>Ffyf*F\^
lF$FboF$FddpF$F$6&Fd^oF[enF,Fe\m>Fgyf*F\^lF$FboF$FdepF$F$Fi`[l>Fhen-F_^l6#-Fer6
$,&-FfyF\^lF:-FgyF\^lFEF``l>Faco-F\am6(F_^pFfcpF_`[l/Fcam,$FhenFefpFg^mFj^m>Fgb
m-Fdel6$-F[gl6#-Fgel6$F[aqF[pF[p@%2-F<6#,&Fc[lF:FhjlFEFciw>Fg\lf*FiclF$FboF$-F\
am6)-Fecm6#,&FjjoF:*&F`sF:Fe\qF:F:/F`s;FcsFefw/Fcam,$F]dm#F:"#!*/Fe_m7$FCFgn/Fc
p%%cyanGFg^mFj^mF$F$6*F*F\pF^wFgbmF.F__mFgvFhen>Fg\lf*FiclF$FboF$-F\am6)Fib[lF]
c[lF_c[lFcc[l/Fcp-F<6#*&,&-Fcdo6#-F`cl6#Fi\mF:FhewFEF:,&T/F:FhewFEFEFg^mFj^mF$F
$60F*F\pF^wFgbmF.F__mFgvFhenF\cnFGFdiuFhjlFciuFc[l>F\z-Ffo6&F[aqFbhl/%*numpoint
sG"$+&Fgdm>Fj]l-Fer6$-Fg\lFg[[l/FbtFf_[lF`_q-Fc\n6#-Fbr6%7$FacoF[]nFg^mFj^mF$@%
33Fj_[lFdblF_]mC'>Fc`l-F_^l6#-Fer6$-Fc_t6#-F<6#-FG6#-F`cl6#-F\p6#,&FDF:Fi[[lF^`
t/Fbt;FcsFa`[l>Fgdo-F\am6*F_^pFfcpF_`[l/Fcam,$*(FhenF:Fg`tF:Fc`lFEF^`tF]atFgdmF
g^mFj^m>Fh^o-F\am6*F_^pFfcpF_`[l/Fcam,$*(FhenF:FeatF:Fc`lFEFfatF]atFegpFg^mFj^m
-Fc\n6#Q_oKurven~med~funktionens~v|ayrdier~antydet~langs~kurven~(tube-tykkelse)
:F$-Fc\n6#-Fbr6%7&FacoFh^oFgdoF[]nFg^mFj^mF$-Fc\n6#Q[oApproksimation,~net-finhe
d~og~antydede~funktionsv|ayrdier~(farve):F$-Fc\n6#-Fbr6'7%Fj]lF\zF[]nFg^mF]]nF`
]nFj^mFh`qF__n@%Faz-F\_n6$F_aqFe\mF$-F\_n6$Q:Funktion:~f(x,y,z)~=~%a~|+F$F`jmF[
ctF^ct-F\_n6#Q>Approksimerede~kurveintegral:F$-Fc\n6#/F`r-F<6#F`r-F\_n6#Q6Eksak
t~kurveintegral:F$-Fc\n6#/Fcq-F<6#Fcq6#QfrFinder~og~viser~(hvis~muligt)~|+kurve
integralet~af~funktionen~f(x,y,z)~|+langs~en~given~rumkurve~~med~parameterfrems
tilling~rF$F$F$%*fladeCmGoGf*6%F*F+F\cn6EFfxFfwFgwFhyFhwF.F/F0F_cnF`cnF2F^cnFhv
FivFgcnFevFfvFgvFccnFdcnFc]oF,FdvFacnFfcnFecnF^wF_wF`wF^vFicnF]wFjcnFjyF[zFayF$
C)>Fc`lF]dn>F[enFddn>Fhenf*FeqF$FboF$FgjnF$F$6&F^wFc`lF_wF[en>Fdjtf*FeqF$FboF$-
F[gl6#F^[oF$F$6$F`wFhen>Fgyf*F`dlF$FboF$FafnF$F$6*F*F\pF\cnFGF^vFdjtF+FA>Fhjl-F
[gl6#-F^hl6$-FgyF`dlF\il>F\jm7%-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(FghnF:Fhh
nF:-FdjtFeqF:FbhlF\ilF:FhjlFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(FiinF:FhhnF
:Fh\\lF:FbhlF\ilF:FhjlFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(&Fc_mFRF:FhhnF:F
h\\lF:FbhlF\ilF:FhjlFE6#QbqFinder~fladeintegralet~af~'masset|aytheden'~over~giv
et~fladestykke~og~giver~massemidtpunktets~koordinaterF$F$F$%(fluxIntGf*Fiu6UFdv
FevFfvFgvF.F/F0F2FhvFivFjvF,F[wF\wF]wF^wF_wF`wFawF^vFbwFcwFdwFewF_vF[fqFfwFgwFh
wFiwFjwF[xF\xF]xF^xF_xF`xFaxFbxFfxFgxFhxFixFjxF[yF\fqF\yF]yF^yF_yF`yF3F$CN@%Faz
C$>FeyFfz@%Fdfq>FhenFd[lC$>FhenF%>FeyFh[lC$>FhenF%>FeyFh[l>F_rFagq>FgsFegq>F8Fi
gq@%F[iqC$>F[enFd[l>Fc`lFhiqC$>F[enF%>Fc`lF%>Fc^mF]dn>F__mFddn>Fffmf*FeqF$FboF$
FgjnF$F$Fhit>Fdblf*FeqF$FboF$F_[rF$F$F^jt>F\\mf*FeqF$FboF$FfblF$F$Ffcl@%Fbjt>Fd
jtFejt>FdjtF%>Fd`mf*FeqF$FboF$Fg[oF$F$6(F`wFffmF*F\pFjuFG>Fgdof*FeqF$FboF$FafmF
$F$6$FhxFd`m>F_]mf*F`dlF$FboF$-F[gl6#-F^hl6$Fj^rF]pF$F$6&FhxFd`mF+FA>Fh^of*F`dl
F$FboF$Fh^pF$F$6$F]wF_]m>F^fp-F[gl6#-F^hl6$-F_]mF`dlF\il@%3F[auFdjtC:>F]of*F\^l
F$FboF$F^^lF$F$F]ar>Fgpf*F\^lF$FboF$Fe_lF$F$F]ar>FcqFaar>Fe\mFcau>F\zf*F`dlF$Fb
oF$F^\mF$F$Fep>Fgbmf*FiclF$FboF$Fg\mF$F$Fep>FbhoF`cr>FicrFjcr>Fj_mFcq>F_jrFcbu>
Fj]lFjdr>Fc_lF^er>Ffaof*FeqF$FboF$-F\am6)Fier/F`s;Fcs*&F\dmF:F]dmF:/Fcam*&F`dmF
:F]dmF:FbdmFddmFg^mFgdmF$F$6*F*F\pFjuFGFaxF_jrFbwFj_m>Fbelf*FeqF$FboF$-F\am6'Fi
er/F`s;*&F\dmF:F]dmF:*&FcemF:F]dmF:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F`sFEF:FbdmFi
emF$F$Fjc\l>Feimf*FeqF$FboF$F^fmF$F$6&FixFfaoFjxFbel>Fh_l-Fbr6$7#-Fer6$-Fer6$-F
eimF]hrF`drFfcrFg^m>FfyFijrFc\sFf\sFh_qFjfuF`]sFc]sFh]sF$F[_nF__n@%FhenF[_sF$F]
_sFiaqF`_sFc_sFe_sFehuF]`s-Fc\n6#/Fc`s-F[gl6#-FffmFeq-F\_n6#QAFluxen~af~V~genne
m~fladestykket:F$-Fc\n6#/.-F\bs6$-F\bx6$FG%"nGF^bs-F[gl6#F^fp-Fc\n6#/Faf\l-F<Fh
f\l-F\_n6#Q5Beregning~af~fluxen:F$-Fc\n6#/Faf\l-F\bs6$-F\bs6$*&Fdf\lF:FhhuF:Fbh
lF\il-Fc\n6#/Faf\l-F\bs6$-F\bs6$-F\bx6$FGFe`sFbhlF\il-Fc\n6#/Faf\l-F\bs6$-F\bs6
$-Fd`mFeqFbhlF\ilF_ex-Fc\n6#/-F\bs6$-FgdoFeqFbhl&7#-F^hl6$F^i\lF[pF\fx-Fc\n6#/F
`fx-F^hlF]i\l-Fc\n6#/-F\bs6$F\i\lF\il-F\bs6$-Fh^oF`dlF\il-Fc\n6#/F\j\l&7#-F^hl6
$F^j\lF`oFagx-Fc\n6#/F`fx-F^hlF]j\lFjhxF^f\lF]ix6#Q\pFinder~og~viser~(hvis~muli
gt)~fluxen~af~givet~vektorfelt~gennem~givet~fladestykkeF$F$F$%'base2dGf*6$FbjmF
cjm6&%&e1_2dG%)mark1_2dG%&e2_2dG%)mark2_2dG6#%HRev.~30.12.03~(S.Markvorsen@mat.
dtu.dk)GF$C'>F8-Fdo6$7$F:FcsFb\m>Fgp-Fdo6$7$FcsF:F[]m>F]of*F`ewF$FboF$-%)textpl
otG6%7%$"#7FE,$FfemFEF\pFgdm/%%fontG7%%&TIMESG%'ITALICGFbcuF$F$F$>Fcqf*6#FcjmF$
FboF$-Fb\]l6%7%,$FfemFEFe\]lF\pFgdmFh\]lF$F$F$-Fbr6#7&F8-F]oFj^lFgp-FcqFegl6#Q.
2d~basis~plotF$F$F$%)fladeIntGf*F[cn6JFfxFfwFgwFhyFhwF.F/F0F_cnF`cnF2F^cnFhvFiv
FgcnFevFfvFgvFccnFdcnFc]oF,FdvF_vF[fqFciuFdiuFacnFfcnFecnF^wF_wF`wF^vFicn%-flad
eDisplayGF]wFjcnFjyF[zFayF$CH@%FazC$>FhgoFfz@%3-Fjz6$Fb[yF][l-Fjz6$Fg[yF][l>F`r
Fd[lC$>F`rF%>FhgoFh[lC$>F`rF%>FhgoFh[l>FgpF[iq>Fj`m-F<6#*&F?F:Fb[yFE>Fham-F<6#*
&FMF:Fg[yFE>Fgho7#-Fer6$-Fer6$-F<6#-F\p6$,&FDF:*&FbtF:Fj`mF:F:,&FQF:*&F]sF:Fham
F:F:/Fbt;F:Fb[y/F]s;F:Fg[y>F]o-Fjz6$-F[al6$-F[al6$Fjio/Fdal;F:*&Fb[yF:Fg[yF:Fga
lF][l>Fcq-Fjz6$-F[al6$Fe\p/F`]pFca]lF][l@%FcqC$>Fh^o-F_^l6#-Fer6$Ff\p/F^]pFca]l
>Fc`l-Ff_lF_b]lF$>FfyF]dn>FgyFddn>Fc[lf*FeqF$FboF$FgjnF$F$6&F^wFfyF_wFgy>Fhjlf*
FeqF$FboF$F^[\lF$F$6$F`wFc[l>Facof*FeqF$FboF$-F[gl6#*&F]clF:FfglF:F$F$6(F*F\pF\
cnFGF^vFhjl>F[jrf*F`dlF$FboF$FafnF$F$6*F*F\pF\cnFGF^vFhjlF+FA>Fgdof*F`dlF$FboF$
Fh^pF$F$6$F]wF[jr>F]]r-F[gl6#-F^hl6$-F[jrF`dlF\il@%3F`cpF`rC;>Fe\m-Fa_m6'Fc_mFb
hlF\il/Fe_m7$,$Fb[yFjn,$Fg[yFjnFg^m>F_]m-F`cl6$F:-F`cl6$F:Fe\m>Fc^mf*F\^lF$FboF
$-F_^l6#-Fer6$Fhdp/F^]p;F:,$F`_lFjnF$F$6&FgcnF_]mF,Fhgo>F__mf*F\^lF$FboF$-Ff_lF
ae]lF$F$Fge]l>Fffm-F_^l6#-Fer6$,&-Fc^mF\^lF:-F__mF\^lFEF``l>Fj]lf*F`dlF$FboF$-%
%plotG6$7%F[pF\pF]ivFb\mF$F$F_iv>Fc_lf*FiclF$FboF$-Fff]l6$7%F\pF`oFeivF[]mF$F$F
_iv>F]\l-Fht6#-Fer6$-Fj]l6#Fe`]l/F]s;FcsFg[y>Fb\l-Fht6#-Fer6$-Fc_l6#Fc`]l/Fbt;F
csFb[y>Fgbmf*F`dlF$FboF$F^\mF$F$Fep>Fg\lf*FiclF$FboF$Fg\mF$F$Fep>Fads-Fbr6#-Fer
6$-FgbmFdg]lFeg]l>&Fh_lFB-Fbr6#-Fer6$-Fg\lF]h]lF^h]l>F\\m-Fbr6%7$FadsF[i]lFg^mF
j^m>F[enf*FeqF$FboF$7&Fbcl,&FbclF:-F<6#*&FbfmF:FefwF:F:,(FbclF:Fii]lF:-F<6#*&Fd
]rF:F_dwF:F:,&FbclF:F]j]lF:F$F$6*F*F\pF^wFfyF.F\zF_wFgy@%2,&Fh^oF:Fc`lFEFciw>Fd
jtf*FeqF$FboF$-Fgiw6%FafmF`jwFg^mF$F$6$FacnF[en>Fdjtf*FeqF$FboF$-Fgiw6%Fafm/Fcp
-Fbjw6$Fdjw-F<6#*&,&F[\oF:F]dmFEF:Ff[xFEFg^mF$F$6,FacnF[enF*F\pF\cnFGFdiuFc`lFc
iuFh^o>Fhen-Fbr6$7#-Fer6$-Fer6$-FdjtFb`]l/Fbt;Fcs,&Fb[yF:FEF:/F]s;Fcs,&Fg[yF:FE
F:Fg^m>F^ir-Fa_m6'Fc_mFbhlF\il/Fe_m7$,$Fb[yFS,$Fg[yFSFg^mF]]x-Fc\n6#-Fbr6%7%F]\
lFb\l-F\[]lFeqFg^mF]]n-Fc\n6#QCDet~parametriserede~billed-omr|`yde:F$-Fc\n6#-Fb
r6'7%FadsF[i]lF[]nFg^mF]]nF`]nFj^m-Fc\n6#-Fbr6'7$F^irF[]nFg^mF]]nF`]nFj^m-Fc\n6
#QZApproksimation,~net-finhed,~og~antydede~funktionsv|ayrdier:F$-Fc\n6#-Fbr6'7%
FhenF\\mF[]nFg^mF]]nFj^mF`]nF$F[_nF__n-F\_n6$F_aqFhgo-F\_n6$QBAngivet~funktion:
~f(x,y,z)~=~%a~|+F$F`jmFe_s-Fc\n6#/.-FhjlFeqF`_^l-F\_n6#Q>Fladeintegralet~af~fu
nktionenF$-Fc\n6#/.-F\bs6$FbbqF^bs-F[gl6#F]]r-Fc\n6#/Fg_^l-F<F[`^l-F\_n6#Q?Bere
gning~af~flade-integralet:F$-Fc\n6#/Fg_^l-F\bs6$-F\bs6$*&FhhnF:F__^lF:FbhlF\il-
Fc\n6#/Fg_^l-F\bs6$-F\bs6$*&FhhnF:F`_^lF:FbhlF\ilF_ex-Fc\n6#/-F\bs6$Fh`sFbhl&7#
-F^hl6$Fh`sF[pF\fx-Fc\n6#/F`fx-F^hlFga^l-Fc\n6#/-F\bs6$Ffa^lF\il-F\bs6$-FgdoF`d
lF\il-Fc\n6#/Feb^l&7#-F^hl6$Fgb^lF`oFagx-Fc\n6#/F`fx-F^hlFfb^lFjhxFd_^lF]ix6#Q]
oFinder~og~viser~fladeintegralet~af~funktion~over~givet~fladestykkeF$F$F$%)kurv
eIntGf*F[cnF_csFiizF$CA@%FazC$>Fg\lFfz@%Ffcs>Fj]lFd[lC$>Fj]lF%>Fg\lF\hoC$>Fj]lF
%>Fg\lF\ho>F\\mFjal>FadsFbds>Fc_lFfds>FeyFbes>Fe\mF_fs>F8F[^p>F]of*F[flF$FboF$F
b^pF$F$F_gs>Fgpf*F[flF$FboF$FbgsF$F$Fegs>Fc^mFg^[l@%FejsC*>FacoFijs>FhgoF_[t>Fg
dof*F\^lF$FboF$FddpF$F$Fe[t>Fh^of*F\^lF$FboF$FdepF$F$Fe[t>FghoFi[t>FffmFa\tF`_q
-Fc\n6#-Fbr6&Fh^tFg^mF`]nFj^mF$@%Fj^tC'>Fj_mF^_t>Fez-F\am6+F_^pFfcpF[[tFd`tF]at
FgdmF`]nFg^mFj^m>Fdbl-F\am6+F_^pFfcpF[[tFbatF]atFegpF`]nFg^mFj^m-Fc\n6#QUKurven
~med~funktionens~v|ayrdier~antydet~langs~kurven:F$F`btF$Fh`qF__n@%FazFfbtF$F]i[
lF[ctF^ct-F\_n6#Q1Kurveintegralet:F$-Fc\n6#/-F\bs6$Fbbq/F_bs;&%"KGFj^lFcbsFc^m-
Fc\n6#/F_g^lF`hs-F\_n6#Q>Beregning~af~kurveintegralet:F$-Fc\n6#/.F_g^l-F\bs6$-F
\bx6$F`jmFactFbhl-Fc\n6#/F^h^l-F\bs6$-F\bx6$FcbqFactFbhl-Fc\n6#/F^h^l-F\bsFj^[l
F_ex-Fc\n6#/F]i^l&7#-F^hl6$F[_[lF[pF\fx-Fc\n6#/F`fxFi^[lFjhx-Fc\n6#/F^h^lFc^mF]
ixF`j[lF$F$%/traadModelPlanGf*F)6(F.FbjmFcjmF/F0F2FayF$C+>F7F;>FIFJ>F]of*FeqF$F
boF$&FbclF9F$F$6$F*F\p>Fgpf*FeqF$FboF$&FbclFBF$F$Fdj^l>Fcqf*F`dlF$FboF$-Fff]l6$
7%Fj^r-F__lFa\m/F[p;&F\\oF9&F\\oFBFb\mF$F$6(FbjmF]oFcjmFgpF+FA>F`rf*FiclF$FboF$
-Fff]l6$7%-Fg^lFj\m-F__lFj\m/F`o;&F\\oFR&F\\oFOF[]mF$F$Fb[_l>Fj`m-Fbr6#-Fer6$-F
cqF]ftFas>Fham-Fbr6#-Fer6$-F`rFdftFct-Fbr6#7$Fj`mFham6#QAViser~tr|`ydmodel~af~p
lant~omr|`yde.F$F$F$%,rumTotalRotGf*Fiu6]pFdvFevFfvFgvF.F/F0F2FhvFivFjvF,Ff]oFd
]oF1Fi]oF^cnFe]oF`cnF_cn%.feltGlobalRotG%%rotLG%$rotG%%rotCG%%rotBG%%rotAG%$mmC
G%$mmBG%$mmAGFjfoFifoFhfo%-rotIntegrandG%&rotatGFgcnFh]oF[wF\wF]wF^wF_wF`wFawF^
vFbwFcwFdwFewF_vF[fqFfwFgwFhwFiwFjwF[xF\xF]xF^xF_xF`xFaxFbxF`eqFaeqF_eqFbfoFcfo
FdfoFhyF]^oFfxFgxFhxFixFjxF[yF\fqFafoFefo%*torqueIntGF]fqF$CH@%FazC$>FeyFfz@%3F
dfq-Fjz6$&FeyFRF][l>FgboFd[lC$>FgboF%>FeyF^\nC$>FgboF%>FeyF^\n>F_rFagq>FgsFegq>
Fft-F<6#*&FZF:Ff^_lFE@%33F[iqFc\vFg\vC%>F\fmFd[l>F87#-Fer6$-Fer6$-Fer6$-F<6#-F\
p6%FbhqFdhq,&FhnF:*&FaoF:FftF:F:FfhqFhhq/Fao;F:Ff^_l>F[em-Fjz6$-F[al6$-F[al6$.F
^jq/Fdal;F:*(F]\lF:Fb\lF:Ff^_lF:FgalF][l>F\fmF%>F[jrf*F`_oF$FboF$Fd_yF$F$Ffcl>F
dev-Fjz6$-F[al6$-F[al6$-F[al6$-F[jr6%FbhqFdhq,&FhnF:*&Fi`yF:FftF:F:Fa\rFb\r/F]a
yFh`_lF][l>FifpFi^o>F^fpFb_o>F^ir-Fdel6&-F[gl6#F[`oF[pF`oFao>F\jmf*F`_oF$FboF$F
_`oF$F$6(F^wFifpF_wF^fpFh]oF^ir>Fbelf*F`_oF$FboF$Fj`oF$F$6$FgcnF\jm>Fhjl-Fdel6&
7%,&-Fgel6$&F`jmFRFcjmF:-Fgel6$&F`jmFBFdjmFE,&-Fgel6$&F`jmF9FdjmF:-Fgel6$Fic_lF
bjmFE,&-Fgel6$F\d_lFbjmF:-Fgel6$F`d_lFcjmFEFbjmFcjmFdjm>Fc[lf*F`_oF$FboF$-F[gl6
#-F__lFbdvF$F$6&F*F\pF\^_lFhjl>Fgyf*F`_oF$FboF$*&&Fc`oF9F:Fe`oF:F$F$6&F[^_lFc[l
F^vFbel>Ffyf*F`_oF$FboF$*&&Fc`oFBF:Fe`oF:F$F$Fbe_l>Fdjtf*F`_oF$FboF$*&&Fc`oFRF:
Fe`oF:F$F$Fbe_l>8]o-Fdel6%-F[gl6#-F^hl6$-FgyF`_oFbhlF`oFao>8^o-Fdel6%-F[gl6#-F^
hl6$-FfyF`_oFbhlF`oFao>8_o-Fdel6%-F[gl6#-F^hl6$-FdjtF`_oFbhlF`oFao>Fhen-Fdel6$-
F[gl6#-F^hl6$-F\f_lF_oF\ilFao>F[en-Fdel6$-F[gl6#-F^hl6$-Fef_lF_oF\ilFao>Fc`l-Fd
el6$-F[gl6#-F^hl6$-F^g_lF_oF\ilFao>Fh^o-F[gl6#-F^hl6$-FhenFidoFheo>FgdoFeev>Fac
o-F[gl6#-F^hl6$-Fc`lFidoFheo>Fgho7%Fh^oFgdoFaco>Fhgo-F`fl6#,(*$&FghoF9FCF:*$&Fg
hoFBFCF:*$&FghoFRFCF:@%333F[emF\fmFdevFgboCB>F]of*F\^lF$FboF$FbfvF$F$F]ar>Fgpf*
F\^lF$FboF$F]gvF$F$F]ar>FcqFaar>FfbuFijr>Fj_m-F^jm6(-FhjlFajmF[[sFc[sF[\sF]\nF_
\n>Fdblf*F_oF$FboF$FjhvF$F$F_iv>Fezf*FipF$FboF$FbivF$F$F_iv>Fffmf*FeqF$FboF$Fii
vF$F$F_iv>F\[[l-Fbr6#-Fer6$-Fer6$-Fdbl6$Fdhq,&FhnF:*&F`sF:FftF:F:Ffcr/F`s;FcsFf
^_l>&F__mFB-Fbr6#-Fer6$-Fer6$-Fez6$FbhqFe\`lF`drFg\`l>&F__mFR-Fht6#-Fer6$-Fer6$
-FffmFahqF`drFfcr>Fc^m-Fbr6%7%F\[[lFj\`lFd]`lFg^mFj^m>F\zf*F_oF$FboF$F\]wF$F$Fe
p>Fgbmf*FipF$FboF$F`]wF$F$Fep>F_]mf*FeqF$FboF$Fd]wF$F$Fep>Fbho-Fbr6#-Fer6$-Fer6
$-F\zFd\`lFfcrFg\`l>Ficr-Fbr6#-Fer6$-Fer6$-FgbmFb]`lF`drFg\`l>&Fg\lFR-Fht6#-Fer
6$-Fer6$-F_]mFahqF`drFfcr>Fj]l-Fbr6#7%FbhoFicrFg_`l>&Fe\mFg_w-Fa_m6(Fj_wF\ilFhe
oFb\m/Fe_m7$,$Fb\lFS,$Ff^_lFSFg^m>&Fe\mFb`w-Fa_m6(Fe`wF\ilFheoFb\mFg``lFg^m>&Fe
\mFi`w-Fa_m6(F\awFbhlFheoF[]m/Fe_m7$,$F]\lFSFj``lFg^m>&Fe\mFcaw-Fa_m6(FfawFbhlF
heoF[]mFca`lFg^m>&Fe\mFjaw-Fa_m6(F]bwFbhlF\ilFegp/Fe_m7$Fea`lFi``lFg^m>&Fe\mFcb
w-Fa_m6(FfbwFbhlF\ilFegpF^b`lFg^m>F\\m-Fbr6%-Fer6$-Fer6$&Fe\mF`cwFacwFbcwFj^mF]
]nFjbz-Fc\n6#-Fbr6'Fg_qFg^mFj^mF`]nF]]n-Fc\n6#Q-VektorfeltetF$-Fc\n6#-Fbr6'7%Ff
buF[]nFj]lFg^mFj^mF`]nF]]n-Fc\n6#QBVektorfeltets~rotationsvektorfeltF$-Fc\n6#-F
br6'7%Fj_mF[]nFj]lFg^mFj^mF`]nF]]nF$F^_xF__nFb_nFf_x-Fc\n6#/.FgexFgex-F\_n6#Q8R
otations-Vektorfeltet:F$-Fc\n6#/.-Fgho6#FGFe[`l-F\_n6#QWTotale~rotation~af~vekt
orfeltet~over~det~givne~omr|`yde:F$-Fc\n6#/.-F\bs6$F[e`lF^bx-F[gl6#Fgho6#QcpFin
der~og~viser~totale~rotation~(eng.:~total~vorticity)~af~vektorfelt~paa|+rumligt
~omr|`ydeF$F$F$%1fladeIntApproxGoGf*F[cn6KFfxFfwFgwFhyFhwF.F/F0F_cnF`cnF2F^cnFh
vFivFgcnFevFfvFgvFccnFdcnFc]oF,FdvF_vF[fqFciuFdiuFacnFfcnFecnF^wF_wF`wF^vFicnFg
izFhizF]wFjcnFjyF[zFayF$C*>FhgoFfz>Fj`mFd_]l>FhamFh_]l>FfyF]dn>FgyFddn>Fc[lf*Fe
qF$FboF$FgjnF$F$Fib]l>Fhjlf*FeqF$FboF$F^[\lF$F$F\c]l>F[jr-F^_[l6$-F^_[l6$**-FG6
#-F`clF``]lF:-FhjlFb`]lF:Fj`mF:FhamF:/Fc[mF`\^l/Fe[mFc\^l6#Q_pFinder~og~viser~a
pproksimation~af~fladeintegralet~af~funktion~over~givet~fladestykkeF$F$F$%/flad
eIntApproxGf*F[cnF\f`lFayF$C>@%FazC$>FhgoFfz@%Fd^]l>F`rFd[lC$>F`rF%>FhgoFh[lC$>
F`rF%>FhgoFh[l>FgpF[iq>Fj`mFb_]l>FhamFf_]l>FghoFj_]l>F]oF\a]l>FcqFfa]l@%FcqC$>F
h^oF^b]l>Fc`lFdb]lF$>FfyF]dn>FgyFddn>Fc[lf*FeqF$FboF$FgjnF$F$Fib]l>Fhjlf*FeqF$F
boF$F^[\lF$F$F\c]l>Facof*FeqF$FboF$F_c]lF$F$Fbc]l>F_jrf*F`dlF$FboF$FafnF$F$Fec]
l>Fgdof*F`dlF$FboF$Fh^pF$F$6$F]wF_jr>F^ir-F[gl6#-F^hl6$-F_jrF`dlF\il>F[jrFhf`l@
%F`d]lC9>Fe\mFcd]l>F_]mFjd]l>Fc^mf*F\^lF$FboF$F`e]lF$F$Fge]l>F__mf*F\^lF$FboF$F
je]lF$F$Fge]l>FffmF\f]l>Fj]lf*F`dlF$FboF$Fef]lF$F$F_iv>Fc_lf*FiclF$FboF$F[g]lF$
F$F_iv>F]\lF_g]l>Fb\lFhg]l>Fgbmf*F`dlF$FboF$F^\mF$F$Fep>Fg\lf*FiclF$FboF$Fg\mF$
F$Fep>FadsFeh]l>F[i]lF\i]l>F\\mFbi]l>F[enf*FeqF$FboF$Fgi]lF$F$Faj]l@%Fcj]l>Fdjt
f*FeqF$FboF$Fgj]lF$F$Fij]l>Fdjtf*FeqF$FboF$F\[^lF$F$Fe[^l>FhenFg[^lF]]xF\]^lFb]
^lFe]^l-Fc\n6#QYApproksimation,~net-finhed~og~antydede~funktionsv|ayrdier:F$-Fc
\n6#-Fbr6'Ff^^lFg^mF]]nF`]nFj^mF$F[_nF__nFg^^lFi^^lFe_sF\_^l-F\_n6#QCApproksima
tion~af~fladeintegralet:F$-Fc\n6#/F[jr-F<6#F[jr-F\_n6#Q6Eksakt~fladeintegral:F$
-Fc\n6#/F^ir-F<6#F^irFcg`lF$F$%(rumKmGoGf*F^fo6`pFdvFevFfvFgvF.F/F0F2FhvFivFjvF
,F[wF\wF]wF^wF_wF`wFawF^vFbwFcwFdwFewF_vF[fqFfwFgwFhwFiwFjwF[xF\xF]xF^xF_xF`xFa
xFbxF`eqFaeqF_eqFbfoFcfoFdfoFhyF]^oFfxFgxFhxFixFjxF[yF\fqFafoFefoFffoFgfoFhfoFi
foFjfoF\yF]yF^yF_yF`yFe_pFg_pFccnFdcnFc]oFgcnFj]_lFi]_lFh]_lF_cnF`cnFe]oF^cnFf]
oFh]oFi]oF1Fd]oF[goF$C5>Fc^mFi^o>F__mFb_o>8^pFhb_l>8eof*F`_oF$FboF$F_`oF$F$6(F^
wFc^mF_wF__mFh]oF]^al>Fdblf*F`_oF$FboF$Fj`oF$F$6$FgcnF_^al>Fe\rf*F`_oF$FboF$-F[
gl6#-Fhjn6$-Fecm6#,&*&Fg^lF:Fe[xF:F:Fi[uFE-FcdoFb[xF$F$F[\u>Fdfrf*F`_oF$FboF$*(
FadvF:&Ff`oF9F:FedvF:F$F$Fb\u>Fagrf*F`_oF$FboF$*(FadvF:&Ff`oFBF:FedvF:F$F$Fb\u>
Fh\uf*F`_oF$FboF$*(FadvF:&Ff`oFRF:FedvF:F$F$Fb\u>Fifp-Fdel6#-F[gl6%-F^hl6$-Fdfr
F`_oFbhlF`oFao>F^fp-Fdel6#-F[gl6%-F^hl6$-FagrF`_oFbhlF`oFao>Fd`m-Fdel6#-F[gl6%-
F^hl6$-Fh\uF`_oFbhlF`oFao>8fo-Fdel6#-F[gl6$-F^hl6$-FifpF_oF\ilFao>8go-Fdel6#-F[
gl6$-F^hl6$-F^fpF_oF\ilFao>8ho-Fdel6#-F[gl6$FegxFao>Ffao-F[gl6#-F^hl6$-FeaalFid
oFheo>Fbel-F[gl6#-F^hl6$-F^balFidoFheo>Feim-F[gl6#-F^hl6$-FgbalFidoFheo>Fg_uFh_
uFbizF$F$%1rumRandTotalVridGf*Fiu6IFeixFfixFd]oFdvFf]oFjvFgixFhixFcwFdwFewFevFf
vFgvFjwF,FfwFhwFgwFiwF[wF.F/F0F1F2%)vridningGFiixFgxFhvF[xF\xF]xF^xF_xF`xFaxFbw
FjixF[jxF$CR@%FazC$>Fc^mFfz@%Fbjx>FffmFd[lC$>FffmF%>Fc^mF^\nC$>FffmF%>Fc^mF^\n>
Fb[yFc[y>Fg[yFh[y>F\\yF]\y@%Fa\y>FezFd[l>FezF%>FcqFj\y>F__mFe^y>Fj_mf*F`_oF$Fbo
F$Fd_yF$F$Ffcl>FdblFi_y>F_ayF`ay>FbayF`ay>FdayFeay>FgayFeay>FiayFjay>F\byFjay>F
^byf*F_oF$FboF$F`byF$F$F]dl>&F[enFg_wFdby>Fjbyf*F_oF$FboF$F\cyF$F$F]dl>&F[enFb`
wF`cy>Ffcyf*FipF$FboF$FhcyF$F$F]dl>&F[enFi`wF\dy>Fbdyf*FipF$FboF$FddyF$F$F]dl>&
F[enFcawFhdy>F^eyf*FeqF$FboF$F`eyF$F$F]dl>&F[enFjawFdey>Fjeyf*FeqF$FboF$F\fyF$F
$F]dl>&F[enFcbwF`fy>Fc`lf*Fgfy6-F^wF_wF`wFhxF^vF`eqFaeqF_eqF\eqF]eqF^eqF]fqF$C.
>F8F]dn>F]oFddn>Fgpf*FeqF$FboF$FgjnF$F$F[[o>F`rf*FeqF$FboF$F_[rF$F$Fd[o>Fcqf*Fe
qF$FboF$-F[gl6#-Fecm6#*&F]hyF:-Fhjn6$FafmFchyF:F$F$Fhhy>F\zf*F`dlF$FboF$Fe^rF$F
$6&FhxFcqFifyFG>Fgbmf*F`dlF$FboF$F^_rF$F$F]ial>Fg\lf*F`dlF$FboF$Fe_rF$F$F]ial>F
j]l-F[gl6#-F^hl6$-F\zF`dlFdiy>Fc_l-F[gl6#-F^hl6$-FgbmF`dlFdiy>Fh_l-F[gl6#-F^hl6
$-Fg\lF`dlFdiy-F[gl6#7%Fj]lFc_lFh_l6#QioFinder~vridningsvektoren~for~de~6~sidef
lader~mht.~den~udadrettede~normalvektorF$F$F$>Fgbmf*F`cwF$FboF$F[jyF$F$6,FeixF8
FiixF[enFfixF]oFjuFGFbdalFc`l>Fg\lFbjyFg\l@%FjjyC:>Fghof*F_oF$FboF$F\]wF$F$Fep>
Facof*FipF$FboF$F`]wF$F$Fep>Fgdof*FeqF$FboF$Fd]wF$F$Fep>FchpFe[z>FjhpFd\z>F^ipF
_]z>FdjtFh]z>F\^zF]^z>Fd^zFe^z>Fh^zFi^z>F__zF`_z>Fc_zFd_z>Fi_zFj_z>F`rF]`z>Feyf
*F\^lF$FboF$FbfvF$F$Ff`z>F\\mf*F\^lF$FboF$F]gvF$F$Ff`z>Fe\mFj`z>F_jr*&F\dmF:Fe\
mF:>F_]mFcazFjbzF]cz-Fc\n6#-Fbr6'7$FdjtF[]nF]]nF`]nFg^mFj^mFh_qFfczF$F^_xF__nFb
_nFe_n-Fc\n6#QenBeregning~af~vridningen~ved~V~af~de~respektive~sideflader:F$Fe_
nF^dzFfdzF]ezFeezF\fzFdfzFe_n-Fc\n6#QODen~totale~vridning~af~sidefladerne~er~de
rfor:F$Fe_n-Fc\n6#/Fagz-F<6#Fagz6#QgrFinder~totale~vridning~ved~vektorfeltet~V~
af~de~afgr|aynsende~side-flader~for~et~rumligt~omr|`yde~med~parameterfremstilli
ngen~r.F$F$F$%(kurveImGf*F_]o6@F]vF^vF_vF,FhwFdvF.FfwFgwFhyF^^oF]^oFfxFjyF[zFbv
F]zF^zF_^oF`^oF_zFa^oFb^oFc^oFd^oFe^oFgvF`foFevFfv6#%HRev.~28.12.03~(S.Markvors
en@mat.dtu.dk)GF$CJ@%F_goC$>FcqFbgo@%-Fjz6$FjglF][l>F`rFd[lC$>F`rF%>FcqF\hoC$>F
`rF%>FcqF\ho>Fj]lFjal>Ff]t-F<6#*&F?F:FjglFE>F\z7#-Fer6$-F<6#-F\p6#,&FDF:*&FbtF:
Ff]tF:F:/Fbt;F:Fjgl>Fg\l-Fjz6$-F[al6$-F[al6$F\_v/FdalF]`blFgalF][l>Fc_l-Fjz6$-F
[al6$Fbav/F`]pF]`blF][l@%Fe_v>FeyFd[l>FeyF%@%F][p>Fh_lFd[l>Fh_lF%>F8F[^p>F]of*F
[flF$FboF$-F`fl6#-F[glFe^pF$F$F_gs>FdblFgao>Facof*F[flF$FboF$FbcvF$F$6&F`]oFfzF
`^oFdbl>Fezf*F[flF$FboF$,&FjjoF:F`pFEF$F$Febo>Fh^of*F[flF$FboF$,&Ff^pF:*$-Fibl6
%Fh^pF__lFdclFCFEF$F$6&F_^oFezF`^oFdbl>Fgpf*F[flF$FboF$*(FfjoF:F^`pF:F_`pF:F$F$
6*F*F\pF\cnFGFe^oFh^oF^vF]o>FhgoFigs@%33333Fg\lFj]lFc_lFeyFh_lF`rC0>F[en-F\am6(
F_^pFfcp/Fe_m7$,$FjglFjnFjnFjcpFg^mFj^m>Fgdo-F`cl6$F:-F`cl6$F:F[en>Fhenf*F\^lF$
FboF$FddpF$F$6&Fd^oFgdoF,Fcq>Fdjtf*F\^lF$FboF$FdepF$F$F_dbl>Fc`l-F_^l6#-Fer6$,&
-FhenF\^lF:-FdjtF\^lFEF``l>Fc^m-F\am6(F_^pFfcpFecbl/Fcam,$Fc`lFefpFg^mFj^m>Fj_m
-F_^l6#-Fer6$-Fc_t6#-F<6#-FG6#-F`cl6#-F\p6#,&FDF:F[`blF^`t/Fbt;FcsFgcblF`_q-Fc\
n6#-Fbr6%7$Fc^mF[]nFg^mFj^m>Fgho-F\am6*-FacoF[fl/F`s;,$-F_^l6$F_hvFc`lFEF]gblF`
hv/Fcam-F_^l6$Fehv,$Fc`lFghvFiemFb\mFg^mFj^m>F__m-F\am6*F_^pFfcpFecbl/Fcam,$*(F
c`lF:Fg`tF:Fj_mFEFigpF]atFgdmFg^mFj^m>Fffm-F\am6*F_^pFfcpFecbl/Fcam,$*(Fc`lF:Fe
atF:Fj_mFE#FEFjgpF]atFegpFg^mFj^m-Fc\n6#QOKurven~(med~v|aygtfunktion)~og~omdrej
ningsaksen:F$-Fc\n6#-Fbr6%7'Fc^mFffmF__mFghoF[]nFg^mFj^mF$Fh`qF__n@%F`r-F\_n6$F
_aqFcqF$F^`xFa`xFhbtF[ctF^ct-F\_n6#Q<Afstand~til~aksen~fra~r(u):F$-Fc\n6#/.-F\a
xFicl-Fh^oFicl-F\_n6#QHKurvens~inertimoment~om~den~givne~akse:F$-Fc\n6#/.-F\bs6
$F[bxFag^l-F[gl6#Fhgo-Fc\n6#/Fjibl-F<F^jbl-F\_n6#Q=Beregning~af~inertimomentet:
F$-Fc\n6#/Fjibl-F\bs6$-F\bx6%Febq.F\axFactFbhl-Fc\n6#/Fjibl-F\bs6$-F\bx6$*&Febq
F:FciblF:FactFbhl-Fc\n6#/FjiblF]i^lF_exF^i^lFei^lFjhxFgiblF]ix6#QhuFinder~og~vi
ser~(hvis~muligt)~|+inertimomentet~af~rumkurve~K~med~v|aygtfunktion~f(x,y,z)~|+
omkring~given~linje~gennem~pt~med~retningsvektor~e~|+(ikke~n|czdvendigvis~enhed
svektor).~F$F$F$%.rumTotalRotGoGf*F^jnFa]_lF]fqF$C:>F_rFagq>FgsFegq>FftFa__l>Fi
fpFi^o>F^fpFb_o>F^irFhb_l>F\jmf*F`_oF$FboF$F_`oF$F$F^c_l>Fbelf*F`_oF$FboF$Fj`oF
$F$Fac_l>FhjlFcc_l>Fc[lf*F`_oF$FboF$Fjd_lF$F$F]e_l>Fgyf*F`_oF$FboF$F`e_lF$F$Fbe
_l>Ffyf*F`_oF$FboF$Fee_lF$F$Fbe_l>Fdjtf*F`_oF$FboF$Fie_lF$F$Fbe_l>F\f_lF]f_l>Fe
f_lFff_l>F^g_lF_g_l>FhenFgg_l>F[enF_h_l>Fc`lFgh_l>Fh^oF_i_l>FgdoFeev>FacoFfi_l>
FghoF\j_lFfe`l6#QjoFinder~totale~rotation~(eng.:~total~vorticity)~af~vektorfelt
~paa|+rumligt~omr|`ydeF$F$F$%*kurveFlowGf*6'F*F+Fju%"TGF[v6hnFgwFfwFhyFhw%'ques
tTG%#VTG%&sysV1G%%sol1G%)orderSolG%(deformVG%)initBulkG%*integralKG%$defG%&AFde
fG%(maxidefG%(minidefG%%sizeG%(flowboxG%$boxGFfxFdvFevFfvFgvFjwF,FdcnFccn%$ptsG
%'cendefG%'ptsdefGFbjmFcjmFdjmFd^o%+extension1GFi^l%$rrrG%(defAnimG%,flowboxAni
mG%(questTTGF`foFbv%'kurve2GF[wFiwF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFi]oFhvFgcn
6#%HRev.~14.01.05~(S.Markvorsen@mat.dtu.dk)GF$C<@%/FbzFjnC$>Fh^oFjao@%-Fjz6$Fch
pF][l>FcqFd[lC$>FcqF%>Fh^oF\hoC$>FcqF%>Fh^oF\ho>F8Fjal>Fgbp-F<6#*&F?F:FchpFE>Fj
_m7#-Fer6$-F<6#-F\p6#,&FDF:*&,&FbtF:FEF:F:FgbpF:F:/Fbt;F:,&FchpF:F:F:>F]o-Fjz6$
-F[al6$-F[al6$.-F<6#&&Fj_mF]]alF\^l/.F[jr;F:FchpFgalF][l>Feimf*FiclF$FboF$FbjoF
$F$Ffcl>Fcjl-Fjz6$-%%evalG6#-F[al6$-Feim6#,&FDF:*&FbtF:FgbpF:F:/Fc[mF\cclF][l>F
`rFf_v>F^fp331Fi_vFcs1FcsF^`v2Fi_vF^`v@%4F^fp-Fc\n6#QcpNB:~Tidsintervallet~skal
~indeholde~t~=~0~s|`y~integralkurverne~starter~p|`y~den~givne~kurve!F$F$>F\z-F]
bp6&/Fgy-FgyF[fl/Fc[l-Fc[lF[fl/Fhjl-FhjlF[fl-FG6%FgyFc[lFhjl>Fgbm6%/-Fgel6$F\ec
lF`sFj`m/-Fgel6$F^eclF`sFham/-Fgel6$F`eclF`sF]bm>Fg\l-%'dsolveG6$7&Fgbm/-Fgy6#F
csFbiu/-Fc[lFefcl%"bG/-FhjlFefcl%"cG7%F\eclF^eclF`ecl>Fj]l-Fdel6'-F]bp6$Fg\lF\g
clFbiuFhfclF[gclF`s>Fc_l-Fdel6%-Fj]l6$FgbqF`sF[pF`s>Fh_l7#-Fer6$7%/Fdfcl&&Fj_mF
]]pF9/Fgfcl&F^hclFB/Fjfcl&F^hclFR/F^]p;F:-%%nopsG6#Fj_m@%3333F\avFcjlFcqF`rF^fp
C>>Fey-Fbr6#-Fer6$-Ffo6%-Fc_l6$FicclF`s/F`s;F[`vF``vFgdm/Fbt;FcsFchp>F\\m-Fdel6
$-Fc_l6$F[pF``vF[p>Fe\m7#-Fer6$-Fer6$-F<6#-Fc_l6$Ficcl,&F[`vF:*&,&F``vF:F[`vFEF
:%#ttGF:FghvFiicl/F][dl;FcsFaam>F_]mf*F\^lF$FboF$-F_^l6#-Fer6$&&Fg^l6#F__lFj^l/
F__l;F:-Ffhcl6#Fg^lF$F$6&F[_clFe\mFi^lF[jr>Fc^mf*F\^lF$FboF$-Ff_lFc[dlF$F$F]\dl
>Faco,$-F_^l6#-Fer6$,&-F_]mF\^lF:Faf]lFEF``lFacu>Fd`m-F\am6(F_^pFfcp/Fe_m7$,$Fc
hpFjnFjnFjcpFg^mFj^m>F\jm-F`cl6$F:-F`cl6$F:Fd`m>Fhgof*F\^lF$FboF$-F_^l6#-Fer6$-
Fer6$&Fg^l6%F__lF^]pF\p/F__l;F:,$FefwFjnF_epF$F$6(Fd^oF\jmFi^lF[jrF,Fh^o>Fghof*
F\^lF$FboF$-Ff_lFg]dlF$F$Fa^dl>F^ir-F_^l6#-Fer6$,&-FhgoF\^lF:-FghoF\^lFEF``l>Fb
el-F\am6(F_^pFfcpF\]dl/Fcam,$F^irFefpFg^mFj^m>Ffao-F\am6(F_^pFfcpF\]dl/Fcam,$Fa
coF^`tFg^mFj^m?(F[jrF:F:"#@Fd[lC$>&F_jrF]]alf*FiclF$FboF$-Fggl6$F\p,&F_elF:*&,&
FcdlF:F_elFEF:,&F__lF:FEF:F:FacuF$F$6(F`^clFfzFi^lF[jrFg^clFc_l>&F]]rF]]al-F\am
6(-F[`dlFiclFbhlF\]dlFe_dlFg^mFj^m>F__m7%;,&FacoFE-Ff_l6$Fcs-Fc^mF9F:,&-F_^l6$F
:-F_]mF9F:FacoF:;,&FacoFE-Ff_l6$Fcs-Fc^mFBF:,&-F_^l6$F:-F_]mFBF:FacoF:;,&FacoFE
-Ff_l6$Fcs-Fc^mFRF:,&-F_^l6$F:-F_]mFRF:FacoF:>Fgdo-F^jm6(Faecl/FgyF[adl/Fc[lFda
dl/FhjlF]bdlF]\nF_\n>Fifp7#-Fer6$-Fbr6%7&-F\]nFbeclFfaoFeyFe`dl/%%viewGF__mF]]n
/F[jr;F:Fh_dl>FezFfao-Fc\n6#Q;Den~parametriserede~kurve:F$-Fc\n6#-Fbr6'7$FbelFc
cdlFg^mFj^mF`]nF]]nFc]n-Fc\n6#-Fbr6'7%FgdoFfaoFccdlFg^mFj^mF`]nF]]n-Fc\n6#QUDe~
tilh|czrende~integralkurver~gennem~den~givne~kurve:F$-Fc\n6#-Fbr6(7%FccdlFfaoFe
yFg^mFj^mF`]nF]]nFdcdl-Fc\n6#QVDen~fremadrettede~flow-deformation~af~den~givne~
kurveF$-Fc\n6#-Fbr6(7&FccdlFeyFfao&F]]r6#Fh_dlFg^mFj^mF`]nF]]nFdcdl-Fc\n6#Q\oAn
imation~af~deformationen~via~integralkurverne~for~vektorfeltet:F$-Fc\n6#-Fbr6(F
ifp/%+insequenceGFd[lFg^mFj^mF`]nF]]nF$-F\_n6$QNParameterfremstilling~for~kurve
n:~r(u)~=~%a~|+F$F[aqF__n@%Fcq-F\_n6$F_aqFh^oF$-F\_n6$Fd_nFaecl-F\_n6$Q3Tidsint
erval:~%a~|+F$Ffz-F\_n6#QEIntegralkurverne~for~vektorfeltet:~|+F$-Fc\n6#/.F\ecl
&-Fj]l6&FbiuFhfclF[gclF`sF9-Fc\n6#/.F^ecl&FdgdlFB-Fc\n6#/.F`ecl&FdgdlFR6#QbsFin
der~og~viser~V~over~given~kurve~samt~de~|+tilh|czrende~integralkurver~i~tidsint
ervallet~T~og~den~tilh|czrende~deformation~af~grundkurven.F$F$F$%1kurveIntAppro
xGoGf*F[cnFfizFiizF$C)>Fe\mFfz>F\[[lF][[l>F8F[^p>F]of*F[flF$FboF$Fb^pF$F$F_gs>F
gpf*F[flF$FboF$FbgsF$F$Fegs>FcqFg^[l>F`rF]_[lF`j[lF$F$%(planIntGf*F[cn6GFfxFfwF
gwFhyFhwFbjmFcjmF^wF_wF.F2F/F0F^cnF_cnF]wF`cnFacnFbcnFccnFdiuFciuFdcnF,FdvF_vF[
fqFecnFhvFfcnFgcnFhcnF^vFicnFjcnFjyF[zFayF$CH@%FazC$>FacoFfz@%3-Fjz6$Fg]wF][l-F
jz6$F`^wF][l>F`rFd[lC$>F`rF%>FacoFh[lC$>F`rF%>FacoFh[l>FgpF[iq>&Fc_lF9-F<6#*&F?
F:Fg]wFE>&Fc_lFB-F<6#*&FMF:F`^wFE>Fgdo7#-Fer6$-Fer6$-F<6#-F\p6$,&FDF:*&FbtF:Fdj
dlF:F:,&FQF:*&F]sF:FijdlF:F:/Fbt;F:Fg]w/F]s;F:F`^w>F]o-Fjz6$-F[al6$-F[al6$.-F<6
#&&FgdoFh^lF\^l/Fdal;F:*&Fg]wF:F`^wF:/FhalFccwF][l>Fcq-Fjz6$-F[al6$.-F<6#-FG6#-
F`cl6#&FgdoF]]p/F`]pF\]elF][l@%FcqC$>Fhgo-F_^l6#-Fer6$Fe]el/F^]pF\]el>Fj_m-Ff_l
Fa^elF$>Fg\lF]dn>Fj]lFddn>Fgyf*FeqF$FboF$F]enF$F$Ffen>Fc[lf*FeqF$FboF$FjenF$F$6
$FhcnFgy>Fh^of*FeqF$FboF$F_c]lF$F$6(F*F\pF\cnFGF^vFc[l>Fc^mf*F`dlF$FboF$FafnF$F
$6*F*F\pF\cnFGF^vFc[lF+FA>Fc`lf*F`dlF$FboF$Fh^pF$F$6$F]wFc^m>F^irFdgn@%F`d]lC6>
F\zf*FeqF$FboF$Fcj^lF$F$Fdj^l>Fgbmf*FeqF$FboF$Fgj^lF$F$Fdj^l>F_]mf*F`dlF$FboF$F
ef]lF$F$F_iv>F__mf*FiclF$FboF$F[g]lF$F$F_iv>Fbjz-Fbr6#-Fer6$-F_]m6#Fi[el/F]s;Fc
sF`^w>&Fe\mFB-Fbr6#-Fer6$-F__m6#Fg[el/Fbt;FcsFg]w>Feyf*F`dlF$FboF$Fjj^lF$F$6(Fb
jmF\zFcjmFgbmF+FA>F\\mf*FiclF$FboF$Fe[_lF$F$Fgael>Fads-Fbr6#-Fer6$-FeyFh`elFi`e
l>F[i]l-Fbr6#-Fer6$-F\\mFbaelFcael>Fhen-Fbr6#Fdi]l>Fffmf*FeqF$FboF$7&F`fm,&F`fm
F:-F<6#*&FfglF:&F]dmF9F:F:,(F`fmF:F]celF:-F<6#*&-FedwFcclF:&F]dmFBF:F:,&F`fmF:F
bcelF:F$F$6,FbjmF\zFcjmFgbmF^wFg\lF.Fc_lF_wFj]l@%2,&FhgoF:Fj_mFEFciw>F[enf*FeqF
$FboF$-%,polygonplotGFhj]lF$F$6$FacnFffm>F[enf*FeqF$FboF$-F_delF][^lF$F$6,FacnF
ffmF*F\pF\cnFGFdiuFj_mFciuFhgo>Fdjt-Fbr6$7#-Fer6$-Fer6$-F[enFf[el/Fbt;Fcs,&Fg]w
F:FEF:/F]s;Fcs,&F`^wF:FEF:Fg^mF]]x-Fc\n6#-Fbr6%7%FbjzF\aelFa]^lFg^mF]]nFb]^l-Fc
\n6#-Fbr6%7%FadsF[i]l-F\[]l6$F\zFgbmFg^mF]]n-Fc\n6#QIApproksimation~med~den~giv
ne~net-finhed:F$-Fc\n6#-Fbr6%7%FdjtFhenF^felFg^mF]]nF$F[_nF__n-F\_n6$F_aqFaco-F
\_n6$Q@Angivet~funktion:~f(x,y)~=~%a~|+F$-FGF_felFe_s-Fc\n6#/.-Fc[lFeqFbgel-F\_
n6#Q=Planintegralet~af~funktionenF$-Fc\n6#/Fg_^l-F[glFe]al-Fc\n6#/Fg_^lFd]al-F\
_n6#Q=Beregning~af~planintegralet:F$-Fc\n6#/Fg_^l-F\bs6$-F\bs6$*&FhhnF:FagelF:F
bhlF\il-Fc\n6#/Fg_^l-F\bs6$-F\bs6$*&FhhnF:FbgelF:FbhlF\ilF_ex-Fc\n6#/-F\bs6$F\`
sFbhl&7#-F^hl6$F\`sF[pF\fx-Fc\n6#/F`fx-F^hlFdiel-Fc\n6#/-F\bs6$FcielF\il-F\bs6$
-Fc`lF`dlF\il-Fc\n6#/Fbjel&7#-F^hl6$FdjelF`oFagx-Fc\n6#/F`fx-F^hlFcjelFjhxFfgel
F]ix6#Q\pFinder~og~viser~(hvor~muligt)~planintegralet~af~funktion~over~givet~pl
ant~omr|`yde.F$F$F$F\]nf*Fajm6(%&e1_3dG%)mark1_3dG%&e2_3dG%)mark2_3dG%&e3_3dG%)
mark3_3dG6#%HRev.~27.07.04~(S.Markvorsen@mat.dtu.dk)GF$C)>F8-Fdo6&7%F:FcsFcs/%&
shapeG%2cylindrical_arrowG/%'fringeG.FhdmFb\m>Fgp-Fdo6&7%FcsF:FcsFa\flFd\flF[]m
>F`r-Fdo6&7%FcsFcsF:Fa\flFd\flFa[q>F]of*F`ewF$FboF$-%+textplot3dG6%7&Fe\]lFcsFc
sF\pFgdmFh\]lF$F$F$>Fcqf*F_]]lF$FboF$-Fb]fl6%7&FcsFe\]lFcsF\pFgdmFh\]lF$F$F$>F\
zf*6#FdjmF$FboF$-Fb]fl6%7&FcsFcsFe\]lF\pFgdmFh\]lF$F$F$-Fbr6$7(F8Fg]]lFgpFh]]lF
`r-F\zF\e`lF`]n6#Q.3d~basis~plotF$F$F$%+fladeIntGoGf*Fdj[lFej[lFayF$C(>Fc`lF]dn
>F[enFddn>Fhenf*FeqF$FboF$FgjnF$F$F[[\l>Fdjtf*FeqF$FboF$F^[\lF$F$F`[\l>Fgyf*F`d
lF$FboF$FafnF$F$Fc[\l>FhjlFe[\l6#QZFinder~fladeintegralet~af~funktion~over~give
t~fladestykkeF$F$F$%'divIntGf*Fiu6RFgwFfwFhyFhwFfxFdvFevFfvFgvFdcnF\cnFjwF,F_vF
[fqFeiuF[wFiwFccnFc]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]o%$mmmGFg]oF]wF^wF_wFh]oF
i]oFhvFgcnF^vFecnFfcnFj]oF[^oFjyF[zF\^oF3F$CU@%FazC$>F\\mFfz@%33-Fjz6$&F\\mF9F]
[l-Fjz6$&F\\mFBF][l-Fjz6$&F\\mFRF][l>FcqFd[lC$>FcqF%>F\\mF^\nC$>FcqF%>F\\mF^\n>
Ffgo-F<6#*&F?F:Fa`flFE>&Fj_mFB-F<6#*&FMF:Fd`flFE>&Fj_mFR-F<6#*&FZF:Fg`flFE@%Fe_
_l>F8Fd[l>F8F%>F\z7#-Fer6$-Fer6$-Fer6$-F<6#-F\p6%,&FDF:*&FbtF:FfgoF:F:,&FQF:*&F
]sF:FdaflF:F:,&FhnF:*&FaoF:FiaflF:F:/Fbt;FcsFa`fl/F]s;FcsFd`fl/Fao;FcsFg`fl>F]o
-Fjz6$-Fcccl6#-F[al6$-F[al6$F\_v/Fdal;F:*(Fa`flF:Fd`flF:Fg`flF:FgalF][l>F__mf*F
`_oF$FboF$Fd_yF$F$Ffcl>Fffm-Fjz6$-Fcccl6#-F[al6$-F[al6$-F[al6$-F__m6%F\cflF^cfl
,&FhnF:*&Fi`yF:FiaflF:F:/Fc[m;F:Fa`fl/Fe[m;F:Fd`fl/F]ay;F:Fg`flF][l>F\jmFi^o>F^
irFb_o>F[jrFhb_l>Fifpf*F`_oF$FboF$F_`oF$F$6(F^wF\jmF_wF^irFh]oF[jr>F^fpf*F`_oF$
FboF$Fj`oF$F$6$FgcnFifp>Fh_l-Fdel6&-_Fjbl%(divergeG6$F`jm7%FbjmFcjmFdjmFbjmFcjm
Fdjm>Fgy-F[gl6#*&-Fh_lFbdxF:-F^fpF`_oF:>Fe\mf*F`_oF$FboF$Fg^lF$F$6$Fg_flFgy>Fc[
l-F[gl6#-F^hl6$FgyFbhl>F_]mf*F_oF$FboF$Fg^lF$F$6$Fg]oFc[l>Fhjl-F[gl6#-F^hl6$Fc[
lF\il>Fc^mf*FidoF$FboF$Fg^lF$F$6$F]wFhjl>Fcjl-F[gl6#-F^hl6$FhjlFheo@%33F\avFffm
FcqC@>Fgbmf*F\^lF$FboF$FbfvF$F$6&FdvF\zF,F\\m>Fg\lf*F\^lF$FboF$F]gvF$F$F`ifl>Fj
]lF_gv>Fey-F^jm6(F`jm/Fbjm;-Ff_l6$FE-Fg\lF9-F_^l6$F:-FgbmF9/Fcjm;-Ff_l6$FE-Fg\l
FB-F_^l6$F:-FgbmFB/Fdjm;-Ff_l6$FE-Fg\lFR-F_^l6$F:-FgbmFRF]\nF_\n>Fh^of*F_oF$Fbo
F$FjhvF$F$F_iv>Fc`lf*FipF$FboF$FbivF$F$F_iv>F[enf*FeqF$FboF$FiivF$F$F_iv>&FdjtF
9-Fbr6#-Fer6$-Fer6$-Fh^o6$F^cfl,&FhnF:*&F`sF:FiaflF:F:Fdcfl/F`sFgcfl>&FdjtFB-Fb
r6#-Fer6$-Fer6$-Fc`l6$F\cflF_\glFbcflFa\gl>&FdjtFR-Fht6#-Fer6$-Fer6$-F[en6$F\cf
lF^cflFbcflFdcfl>F_jr-Fbr6%7%Ff[glFc\glF]]glFg^mFj^m>Fghof*F_oF$FboF$F\]wF$F$Fe
p>Facof*FipF$FboF$F`]wF$F$Fep>Fgdof*FeqF$FboF$Fd]wF$F$Fep>&FhenF9-Fbr6#-Fer6$-F
er6$-FghoF^\glFdcflFa\gl>&FhenFB-Fbr6#-Fer6$-Fer6$-FacoF[]glFbcflFa\gl>&FhenFR-
Fht6#-Fer6$-Fer6$-FgdoFe]glFbcflFdcfl>F]]r-Fbr6#7%Fa^glFj^glFc_gl>&FhgoFg_w-Fa_
m6(Fj_wF\ilFheoFb\m/Fe_m7$,$Fd`flFS,$Fg`flFSFg^m>&FhgoFb`w-Fa_m6(Fe`wF\ilFheoFb
\mFc`glFg^m>&FhgoFi`w-Fa_m6(F\awFbhlFheoF[]m/Fe_m7$,$Fa`flFSFf`glFg^m>&FhgoFcaw
-Fa_m6(FfawFbhlFheoF[]mF_aglFg^m>&FhgoFjaw-Fa_m6(F]bwFbhlF\ilFegp/Fe_m7$FaaglFe
`glFg^m>&FhgoFcbw-Fa_m6(FfbwFbhlF\ilFegpFjaglFg^m>Ffy-Fbr6%-Fer6$-Fer6$&FhgoF`c
wFacwFbcwFj^mF]]nFjbzFjfu-Fc\n6#-Fbr6'7$F]]rF[]nFg^mFj^mF`]nF]]nFc]n-Fc\n6#-Fbr
6'7$FeyF[]nFg^mFj^mF`]nF]]nF$F^_xF__nFb_nFf_x-Fc\n6#/.FagflFagfl-F\_n6#QFDiverg
ensen~af~V~(punktvis~i~rummet):F$-Fc\n6#/.-%$divGF\e`l-Fh_lFajm-F\_n6#QEDiverge
nsen~af~V~i~punktet~r(u,v,w):F$-Fc\n6#/F\dglF`gfl-F\_n6#Q6Divergens-integralet:
F$-Fc\n6#/.-F\bs6$F]dglF^bx-F[glFebn-Fc\n6#/F\eglFfbn-F\_n6#QCBeregning~af~den~
totale~divergens:F$-Fc\n6#/F\egl-F\bs6$-F\bs6$-F\bs6$*&F]dglF:FecglF:FbhlF\ilFh
eo-Fc\n6#/F\egl-F\bs6$-F\bs6$-F\bs6$*&F]dglF:FagflF:FbhlF\ilFheo-Fc\n6#/F\egl-F
\bs6$-F\bs6$-F\bs6$*&-Fh_lF`_oF:FagflF:FbhlF\ilFheoF_ex-Fc\n6#/-F\bs6$-Fe\mF`_o
Fbhl&7#-F^hl6$FjgglF[pF\fx-Fc\n6#/F`fx-F^hlFiggl-Fc\n6#/-F\bs6$FhgglF\il-F\bs6$
-F_]mF_oF\il-Fc\n6#/Fhhgl&7#-F^hl6$FjhglF`oFagx-Fc\n6#/F`fx-F^hlFihgl-Fc\n6#/-F
\bs6$FfhglFheo-F\bs6$-Fc^mFidoFheo-Fc\n6#/F[jgl&7#-F^hl6$F]jglFaoFehx-Fc\n6#/F`
fx-F^hlF\jglFjhxFidglF]ix6#Q_qFinder~og~viser~rumintegralet~af~divergensen~af~V
~over~givet~rumligt~omr|`yde,~eksakt~og~approksimeretF$F$F$%(rumFlowGf*F_^cl6ZF
gwFfwFhyFhwFb^clFc^clFd^clFe^clFf^clFg^clFh^clFi^clFj^clF[_clF\_clF]_clF^_clF__
clF`_clFfxFdvFevFfvFgvFjwF,FdcnFccnFa_clFb_clFc_clFbjmFcjmFdjm%&siderGFi^lFe_cl
Ff_clFg_clFh_clF[wFiwF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFi]oFhvFgcnFj_clF$C>@%F^
`clC$>Fh^oFjao@%33Fb`cl-Fjz6$FjhpF][l-Fjz6$F^ipF][l>FcqFd[lC$>FcqF%>Fh^oF^\nC$>
FcqF%>Fh^oF^\n>&FfaoF9F]acl>&FfaoFB-F<6#*&FMF:FjhpFE>&FfaoFR-F<6#*&FZF:F^ipFE@%
Fe__l>F8Fd[l>F8F%>Fj_m7#-Fer6$-Fer6$-Fer6$-F<6#-F\p6%,&FDF:*&FbtF:Fb\hlF:F:,&FQ
F:*&F]sF:Fd\hlF:F:,&FhnF:*&FaoF:Fi\hlF:F:Fiicl/F]s;FcsFjhp/Fao;FcsF^ip>F]o-Fjz6
$-Fcccl6#-F[al6$-F[al6$.-F<6#&&Fj_mFe]alF\^l/.F^ir;F:*(FchpF:FjhpF:F^ipF:FgalF]
[l>F^fpf*F`_oF$FboF$Fd_yF$F$Ffcl>Fd`m-Fjz6$-Fcccl6#-F[al6$-F[al6$-F[al6$-F^fp6%
F\^hlF^^hl,&FhnF:*&Fi`yF:Fi\hlF:F:F[dcl/Fe[m;F:Fjhp/F]ay;F:F^ipF][l>F`rFf_v>Fif
pF^dcl@%4Fifp-Fc\n6#QKNB:~Tidsintervallet~skal~indeholde~t~=~0~!F$F$>F\zFidcl>F
gbmFdecl>Fg\lF_fcl>Fj]lF^gcl>Fc_l-Fdel6'-Fj]l6$FcdxF`sF[pF`oFaoF`s>Fh_lFhgcl@%3
333F\avFd`mFcqF`rFifpCL>Fhgof*F\^lF$FboF$-F_^l6#-Fer6$Ff[dl/F__l;F:*(FefwF:F_dw
F:FfdwF:F$F$6(FdvFj_mFi^lF^irF,Fh^o>Fghof*F\^lF$FboF$-Ff_lFgbhlF$F$F]chl>Feimf*
F_oF$FboF$-Ffo6&FhoF]pFgdm/%*thicknessGFCF$F$Fep>Fcjlf*FipF$FboF$-Ffo6&F\qF^qFg
dmFechlF$F$Fep>Fiyf*FeqF$FboF$-Ffo6&FhqFjqFgdmFechlF$F$Fep>&F[emF9-Fbr6#-Fer6$-
Fer6$-Feim6$F^^hl,&FhnF:*&F`sF:Fi\hlF:F:Fb^hl/F`sFe^hl>&F[emFB-Fbr6#-Fer6$-Fer6
$-Fcjl6$F\^hlFidhlFiiclF[ehl>&F[emFR-Fht6#-Fer6$-Fer6$-Fiy6$F\^hlF^^hlFiiclFb^h
l>Ffbu-Fbr6#7%F`dhlF]ehlFgehl>F\jmf*F)6#Fd]oF`jnF$C)>F^by-Fa_m6(Fj_wF\ilFheoFb\
m/Fe_m7$,$FTFS,$F[oFSFg^m>Fjby-Fa_m6(Fe`wF\ilFheoFb\mF[ghlFg^m>Ffcy-Fa_m6(F\awF
bhlFheoF[]m/Fe_m7$,$FFFSF^ghlFg^m>Fbdy-Fa_m6(FfawFbhlFheoF[]mFeghlFg^m>F^ey-Fa_
m6(F]bwFbhlF\ilFegp/Fe_m7$FgghlF]ghlFg^m>Fjey-Fa_m6(FfbwFbhlF\ilFegpF^hhlFg^m-F
br6%-Fer6$-Fer6$&F86$Fg^lF^]p/Fg^lFa`lFbcwFj^mF]]n6#QeoFinder~og~viser~de~6~afg
r|aynsende~sideflader~for~det~rumlige~billede~r(B)~.F$F$6$Fi^lF^ir>Fgbo-F\jm6%F
\pFAFh^o>Fey-Fbr6#-Fer6$-Fer6$-Fer6$-Ffo6%-Fc_l6&F\^hlF^^hl,&FhnF:*&%"dGF:Fi\hl
F:F:F`sFgiclFgdmFiiclFb^hl/FajhlFe^hl>F\\m-Fdel6&-Fc_l6&F[pF`oFaoF``vF[pF`oFao?
(F^irFcsF:FbcuFd[lC$>&F[jrFe]alf*F`_oF$FboF$-Fggl6&F\pFAFG,&F_elF:*&Fa`dlF:F__l
F:FacuF$F$6(F`^clFfzFi^lF^irFg^clFc_l>&F_jrFe]al-%+sideFladerG6%F[[ilFAFh^o>Fe\
m7#-Fer6$-Fer6$-Fer6$-Fer6$-F<6#-Fc_l6&F\^hlF^^hlF`^hlFjjclFiiclFb^hlFd^hlF^[dl
>F_]mf*F\^lF$FboF$Fb[dlF$F$6&F[_clFe\mFi^lF^ir>Fc^mf*F\^lF$FboF$F`\dlF$F$Fg\il>
FacoFb\dl>F__m7%;,&FacoFE-Ff_l6%Fcs-FghoF9F_adlF:,&-F_^l6%F:-FhgoF9FcadlF:FacoF
:;,&FacoFE-Ff_l6%Fcs-FghoFBFhadlF:,&-F_^l6%F:-FhgoFBF\bdlF:FacoF:;,&FacoFE-Ff_l
6%Fcs-FghoFRFabdlF:,&-F_^l6%F:-FhgoFRFebdlF:FacoF:>Fgdo-F^jm6(Faecl/FgyF]]il/Fc
[lFf]il/FhjlF_^ilF]\nF_\n>F[_pf*F_oF$FboF$FjhvF$F$F_iv>Fbeof*FipF$FboF$FbivF$F$
F_iv>F`cmf*FeqF$FboF$FiivF$F$F_iv>&F\fmF9-Fbr6#-Fer6$-Fer6$-F[_pFhdhlFb^hlF[ehl
>&F\fmFB-Fbr6#-Fer6$-Fer6$-FbeoFeehlFiiclF[ehl>&F\fmFR-Fht6#-Fer6$-Fer6$-F`cmF_
fhlFiiclFb^hl>Fdev-Fbr6%7%Fe_ilF^`ilFg`ilFg^mFj^m>Fffm-F\jm6%F\\mFAFh^o>F]]r7#-
Fer6$-Fbr6%7&FccdlFfbuFeyFc[ilFdcdlF]]n/F^ir;FcsFbcu>FezFfbuFjbz-Fc\n6#-Fbr6'7$
FgboFccdlFg^mFj^mF`]nF]]n-Fc\n6#-Fbr6'7$FfbuFccdlFg^mFj^mF`]nF]]nFc]n-Fc\n6#-Fb
r6'7%FgdoFezFccdlFg^mFj^mF`]nF]]n-Fc\n6#-Fbr6'7%FgdoFgboFccdlFg^mFj^mF`]nF]]n-F
c\n6#QVDe~tilh|czrende~integralkurver~gennem~det~givne~omr|`yde:F$-Fc\n6#-Fbr6(
7%FccdlFfbuFeyFg^mFj^mF`]nF]]nFdcdl-Fc\n6#QWDen~fremadrettede~flow-deformation~
af~det~givne~omr|`ydeF$-Fc\n6#-Fbr6(7&FccdlFeyFgboFffmFg^mFj^mF`]nF]]nFdcdl-Fc\
n6#QQAnimation~af~deformationen~via~integralkurverne:F$-Fc\n6#-Fbr6(F]]rF_fdlFg
^mFj^mF`]nF]]nF$-F\_n6$QTParameterfremstilling~for~omr|`ydet:~r(u,v,~w)~=~%a~|+
F$F__oF__n@%FcqFefdlF$FgfdlFifdlF\gdlF_gdlFfgdlF[hdl6#Q\tFinder~og~viser~V~over
~givet~rumligt~omr|`yde~samt~de~|+tilh|czrende~integralkurver~i~tidsintervallet
~T~og~den~tilh|czrende~deformation~af~grundomr|`ydet.F$F$F$%0StokesRandIntGoGf*
F^jnF\vFayF$C/Fdy>F_rf*FiclF$FboF$FjclF$F$F]dl>Fgsf*F`dlF$FboF$FadlF$F$F]dl>Fft
f*FiclF$FboF$FfdlF$F$F]dl>F[elf*F`dlF$FboF$F]elF$F$F]dl>FbelFcel>F]of*F^flF$Fbo
F$F_flF$F$Fffl>Fgpf*FiflF$FboF$FjflF$F$Fhgl>FjglF[hl>FehlFfhl>F_ilF`il>FiilFjil
>Fcjl-F[gl6#Fdjl6#Q`qFinder~tangentielle~kurveintegral~af~givet~vektorfelt~lang
s~~|+totale~randkurve~for~given~fladestykke.F$F$F$%3traadModelPlanGrayGf*F)F]j^
lFayF$C+>F7F>>FIFL>F]of*FeqF$FboF$Fcj^lF$F$Fdj^l>Fgpf*FeqF$FboF$Fgj^lF$F$Fdj^l>
Fcqf*F`dlF$FboF$-Fff]l6$F\[_lFbpF$F$Fb[_l>F`rf*FiclF$FboF$-Fff]l6$Fg[_lFbpF$F$F
b[_l>Fj`mF_\_l>FhamFe\_lFj\_lF]]_lF$F$%*planIntGoGf*Fdj[lF]cnFayF$C(>Fg\lF]dn>F
j]lFddn>F[enf*FeqF$FboF$F]enF$F$Ffen>Fhenf*FeqF$FboF$FjenF$F$F^fn>Fc^mf*F`dlF$F
boF$FafnF$F$Fbgn>FgyFdgn6#QenFinder~planintegralet~af~funktion~over~givet~plant
~omr|`yde.F$F$F$%)divIntGoGf*F^jn6NFgwFfwFhyFhwFfxFdvFevFfvFgvFdcnF\cnFjwF,F[wF
iwFccnFc]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFg]oF]wF^wF_wFh]oFi]oFhvFgcnF^vFecn
FfcnFj]oF[^oFjyF[zF\^oFayF$C,>FfyFi^o>FgyFb_o>Fc[lFi_o>F^irf*F`_oF$FboF$F_`oF$F
$6(F^wFfyF_wFgyFh]oFc[l>F[jrf*F`_oF$FboF$Fj`oF$F$6$FgcnF^ir>Fh_lFeffl>-Fg_flF`_
o-F[gl6#*&F`gflF:-F[jrF`_oF:>-FhenF_o-F[gl6#-F^hl6$FjiilFbhl>-FdjtFido-F[gl6#-F
^hl6$F`jilF\il>Fd`m-F[gl6#-F^hl6$FfjilFheo6#QfpFinder~og~viser~rumintegralet~af
~divergensen~af~givne~vektorfelt~~over~givet~rumligt~omr|`ydeF$F$F$%1fladeTotal
VridGoGf*FiuF[eqF]fqF$C.>FhgoF]dn>FghoFddn>Facof*FeqF$FboF$FgjnF$F$F\[r>Fh^of*F
eqF$FboF$F_[rF$F$Fb[r>Fe\rf*FeqF$FboF$Fg\rF$F$F[]r>F__mf*F`dlF$FboF$Fe^rF$F$F[_
r>Fffmf*F`dlF$FboF$F^_rF$F$F[_r>Fc^mf*F`dlF$FboF$Fe_rF$F$F[_r>F\\mF[`r>Fe\mFa`r
>F_]mFdgnFdbsF[csF$F$%&rumKmGf*F^foFh]alF[goF$CQ@%F_goC$>FeyFbgo@%Fc^_l>FhenFd[
lC$>FhenF%>FeyF^\nC$>FhenF%>FeyF^\n>F_rFagq>FgsFegq>FftFa__l@%Fe__lC%>F[enFd[l>
F8Fj__l>Fc`lFj`_l>F[enF%@%F][p>FiyFd[l>FiyF%>Fcjl-Fjz6$-F[al6$Fdht/F`]pFba_lF][
l>F\\mf*F`_oF$FboF$Fd_yF$F$F[[p>Fdjt-Fjz6$-F[al6$-F[al6$-F[al6$-F\\mFab_lFa\rFb
\rFdb_lF][l>Fc^mFi^o>F__mFb_o>F]^alFhb_l>F_^alf*F`_oF$FboF$F_`oF$F$Fa^al>Fdblf*
F`_oF$FboF$Fj`oF$F$Fd^al>Fe\rf*F`_oF$FboF$Fg^alF$F$F[\u>Fdfrf*F`_oF$FboF$Fb_alF
$F$Fb\u>Fagrf*F`_oF$FboF$Ff_alF$F$Fb\u>Fh\uf*F`_oF$FboF$Fj_alF$F$Fb\u>Fifp-Fdel
6%-F[gl6#Fa`alF`oFao>F^fp-Fdel6%-F[gl6#Fi`alF`oFao>Fd`m-Fdel6%-F[gl6#FaaalF`oFa
o>Feaal-Fdel6$-F[gl6#FjaalFao>F^bal-Fdel6$-F[gl6#FcbalFao>Fgbal-Fdel6$-F[gl6#Fe
gxFao>FfaoF]cal>FbelFccal>FeimFical>Fg_uFh_u>FeerF^`u@%3333FbjtFdjtFhenFiyFcjlC
E>F]of*F\^lF$FboF$FbfvF$F$F]ar>Fgpf*F\^lF$FboF$F]gvF$F$F]ar>FcqFaar>Ffbu-Fdo6&F
\hpFibu/Fbgp-Ff_l6$FcqFeerFegp>F_cu,$FcqFigp>FdcuF\hp>FfcuFgcu>FfyFhfu>8iof*F_o
F$FboF$FjhvF$F$F_iv>8jof*FipF$FboF$FbivF$F$F_iv>8[pf*FeqF$FboF$FiivF$F$F_iv>&8\
pF9-Fbr6#-Fer6$-Fer6$-F[djlFd\`lFfcrFg\`l>&FedjlFB-Fbr6#-Fer6$-Fer6$-F^djlFb]`l
F`drFg\`l>&FedjlFR-Fht6#-Fer6$-Fer6$-FadjlFahqF`drFfcr>8_p-Fbr6%7%FddjlF^ejlFge
jlFg^mFj^m>F\zf*F_oF$FboF$F\]wF$F$Fep>Fgbmf*FipF$FboF$F`]wF$F$Fep>8`pf*FeqF$Fbo
F$Fd]wF$F$Fep>FbhoFg^`l>FicrF__`l>Fg_`l-Fht6#-Fer6$-Fer6$-FifjlFahqF`drFfcr>Fj]
lF```l>&8apFg_wFe``l>&FhgjlFb`wF]a`l>&FhgjlFi`wFaa`l>&FhgjlFcawFha`l>&FhgjlFjaw
F\b`l>&FhgjlFcbwFbb`l>8]p-Fbr6%-Fer6$-Fer6$&FhgjlF`cwFacwFbcwFj^mF]]nFjbz-Fc\n6
#-Fbr6'7$FdhjlF[]nFg^mFj^mF`]nF]]n-Fc\n6#QNVektorfeltet~og~kraftmomentet~om~giv
et~punkt:F$-Fc\n6#-Fbr6'7'FfyF[]nFj]lFfcuFfbuFg^mFj^mF`]nF]]n-Fc\n6#QfnDet~para
metriserede~omr|`yde~og~kraftmomentet~af~vektorfeltetF$-Fc\n6#-Fbr6'7&Fj]lF[]nF
fcuFfbuFg^mFj^mF`]nF]]nF$F^_xF__n-F\_n6$Fd_nFg]qFf_x-Fc\n6#/.-FdblF`_oFgjjlF`aq
FcaqFfaqFiaqF[bq-Fc\n6#/Fabq-F[gl6#FadxFibqF\cqF`_s-Fc\n6#/F_cq-FfzFbdxF_dqFjhu
6#QhpFinder~og~viser~kraftmomentet~(eng.:~torque)~om~punkt~af~vektorfelt~paa|+v
|aygtet~rumligt~omr|`ydeF$F$F$%)rumIntGoGf*Fdj[l6JFgwFfwFhyFhwFfxFdvFevFfvFgvFd
cnF,FccnFc]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFg]oF]wF^wF_wFh]oFi]oFhvFgcnF^vFe
cnFfcnFj]oF[^oFjyF[zF\^oFayF$C*>Fc`lFi^o>F[enFb_o>FhenFi_o>Fgyf*F`_oF$FboF$F_`o
F$F$6(F^wFc`lF_wF[enFh]oFhen>Fc[lf*F`_oF$FboF$Fj`oF$F$6$FgcnFgy>Fgdof*F_oF$FboF
$-F[gl6#-F^hl6$-F[gl6#*&FjcoF:F^doF:F]gnF$F$Fc_el>Fh^of*FidoF$FboF$FjdoF$F$6&Fg
]oFgdoF+FA>F_jr-F[gl6#-F^hl6$-Fh^oFidoFheo6#QenFinder~rumintegralet~af~funktion
~over~givet~rumligt~omr|`ydeF$F$F$%0planIntApproxGoGf*F[cn6GFfxFfwFgwFhyFhwFbjm
FcjmF^wF_wF.F2F/F0FciuFdiuF^cnF_cnF]wF`cnFacnFbcnFccnFdcnF,FdvFecnFhvFfcnFgcnFh
cnF^vFicnFjcnFjyF[zFhizFgizF3F$C*>FacoFfz>FdjdlFejdl>FijdlFjjdl>Fg\lF]dn>Fj]lFd
dn>Fdjtf*FeqF$FboF$F]enF$F$Ffen>Ffyf*FeqF$FboF$FjenF$F$6$FhcnFdjt>F^ir-F^_[l6$-
F^_[l6$**-FG6#-F`clFd[elF:-FfyFf[elF:FdjdlF:FijdlF:/Fc[mF_eel/Fe[mFbeelFjinF$F$
%*fladeFlowGf*F_^cl6[oFgwFfwFhyFhwFb^clFc^clFd^clFe^clFf^clFg^clFh^clFi^clFj^cl
F[_clF\_clF]_clF^_clF__clF`_clFfxFdvFevFfvFgvFjwF\wFbwFaxFivFbjmFcjmFdjmF,FdcnF
ccnFa_clFb_clFc_clFd^oFd_clFi^lFe_clFf_clFg_clFh_clF`foFbvFi_clF[wFiwF.Fd]oF/F0
F1F_cnF`cnFe]oF2F^cnFi]oFhvFgcn6#%HRev.~15.01.05~(S.Markvorsen@mat.dtu.dk)GF$C?
@%F^`clC$>Fc[lFjao@%3-Fjz6$&Fc[lF9F][l-Fjz6$&Fc[lFBF][l>FcqFd[lC$>FcqF%>Fc[lFh[
lC$>FcqF%>Fc[lFh[l>F`dhl-F<6#*&F?F:F[`[mFE>F]ehl-F<6#*&FMF:F^`[mFE>Fj_m7#-Fer6$
-Fer6$-F<6#-F\p6$,&FDF:*&FbtF:F`dhlF:F:,&FQF:*&F]sF:F]ehlF:F:/Fbt;FcsF[`[m/F]s;
FcsF^`[m@%F[iqC$>F8Fd[l>F]o-Fjz6$-F[al6$-F[al6$-F<6#&&Fj_m6#%#iiGF\^l/.F_c[m;F:
*&F[`[mF:F^`[mF:FgalF][lC$>F8F%>F]oF%>Fbeof*FeqF$FboF$FfblF$F$Ffcl@%F\av>F`cm-F
jz6$-F[al6$-F[al6$-F<6#-FbeoFga[m/Fc[m;F:F[`[m/Fe[m;F:F^`[mF][l>F`cmF%>Fhgof*F\
^lF$FboF$-F_^l6#-Fer6$-F<6#Ff[dl/F__l;F:*&FefwF:F_dwF:F$F$6(FdvFj_mFi^lF^fpF,Fc
[l>Fghof*F\^lF$FboF$-Ff_lF\e[mF$F$Fde[m>FifpFf^dl>F`rFf_v>FeimF^dcl@%4Feim-Fc\n
6#QcpNB:~Tidsintervallet~skal~indeholde~t~=~0~s|`y~integralkurverne~starter~p|`
y~den~givne~flade!F$F$>F\z-F]bp6&/Fdjt-FdjtF[fl/Ffy-FfyF[flF[ecl-FG6%FdjtFfyFgy
>Fgbm6%/-Fgel6$Fdf[mF`sFj`m/-Fgel6$Fff[mF`sFham/FfeclF]bm>Fg\l-F`fcl6$7&Fgbm/-F
djtFefclFbiu/-FfyFefclFhfcl/FdfclF[gcl7%Fdf[mFff[mF\ecl>Fj]l-Fdel6'-F]bp6$Fg\lF
[h[mFbiuFhfclF[gclF`s>Fc_l-Fdel6&-Fj]l6$FjhnF`sF[pF`oF`s>Fh_l7#-Fer6$7%/Fgg[mF]
hcl/Fig[mF`hcl/FdfclFbhclFchcl@%3333F\avF`cmFcqF`rFeimCC>Fey-Fbr6#-Fer6$-Fer6$-
Ffo6%-Fc_l6%Fha[mFja[mF`sFgiclFgdmF\b[mF^b[m>F\\m-Fdel6%-Fc_l6%F[pF`oF``vF[pF`o
>Fe\m7#-Fer6$-Fer6$-Fer6$-F<6#-Fc_l6%Fha[m,&FQF:*&F]sF:F`dhlF:F:FjjclF\b[mF^b[m
F^[dl>F_]mf*F\^lF$FboF$Fb[dlF$F$6&F[_clFe\mFi^lF^fp>Fc^mf*F\^lF$FboF$F`\dlF$F$F
d[\m>FacoFb\dl>Fhgof*F\^lF$FboF$F[e[mF$F$Fde[m>Fghof*F\^lF$FboF$Fge[mF$F$Fde[m>
Fh^o-F`fl6#-F_^l6#-Fer6$-Fer6$Fcd[m/FbtFed[m/F]sFgd[m>Fgbof*F`dlF$FboF$-Ffo6&F`
\mF]pFgdmFechlF$F$Fep>Fdevf*FiclF$FboF$-Ffo6&Fi\mF^qFgdmFechlF$F$Fep>&FdfrF9-Fb
r6#-Fer6$-Fgbo6#Fja[mF^b[m>&FdfrFB-Fbr6#-Fer6$-Fdev6#Fha[mF\b[m>Fc`lFaco>F[en-F
_^l6$Fg`mFh^o>8[o-Fbr6%7$F`]\mFh]\mFg^mFj^m>Fhen-Fa_m6'Fc_mFbhlF\il/Fe_m7$,$F[`
[mFjn,$F^`[mFjnFg^m?(F^fpF:F:Fh_dlFd[lC$>&Fd`mFhf\lf*FeqF$FboF$-Fggl6%F\pFAF_`d
lF$F$6(F`^clFfzFi^lF^fpFg^clFc_l>&FfaoFhf\l-Fa_m6(-Fb_\mFeqFbhlF\ilF[_\mFg^mFj^
m>F__mFj`dl>Fgdo-F^jm6(Fgf[m/FdjtF[adl/FfyFdadl/FgyF]bdlF]\nF_\n>Fbel7#-Fer6$-F
br6%7&-F\]nFhf[mFd^\mFeyFh_\mFdcdlF]]n/F^fpFgcdl>FezFhen-Fc\n6#Q;Den~parametris
erede~flade:F$-Fc\n6#-Fbr6'7$FhenFj`\mFg^mFj^mF`]nF]]nFc]n-Fc\n6#-Fbr6'7%FgdoFh
enFj`\mFg^mFj^mF`]nF]]n-Fc\n6#-Fbr6'7%FgdoFd^\mFj`\mFg^mFj^mF`]nF]]n-Fc\n6#QUDe
~tilh|czrende~integralkurver~gennem~den~givne~flade:F$-Fc\n6#-Fbr6(7%Fj`\mFd^\m
FeyFg^mFj^mF`]nF]]nFdcdl-Fc\n6#QWDen~fremadrettede~flow-deformation~af~den~givn
e~flade:F$-Fc\n6#-Fbr6(7&Fj`\mFeyFhen&FfaoFgedlFg^mFj^mF`]nF]]nFdcdlFhedl-Fc\n6
#-Fbr6(FbelF_fdlFg^mFj^mF`]nF]]nF$-F\_n6$QPParameterfremstilling~for~fladen:~r(
u,v)~=~%a~|+F$Fc_m-F\_n6$Q8Definitionsm|ayngde:~%a~|+F$FA@%Fcq-F\_n6$F_aqFc[lF$
-F\_n6$Fd_nFgf[mFifdl-F\_n6#QgnIntegralkurven~igennem~punktet~(a,~b,~c)~for~vek
torfeltet:~|+F$-Fc\n6#/.Fdf[mFcgdl-Fc\n6#/.Fff[mFjgdl-Fc\n6#/FbgdlF_hdl6#Q^tFin
der~og~viser~V~omkring~~given~flade~samt~de~tilh|czrende~integralkurver~i~tidsi
ntervallet~T~og~den~tilh|czrende~deformation~af~den~givne~grundflade.F$F$F$%+ku
rveIntGoGf*Fdj[lF_cs6#%HRev.~20.01.04~(S.Markvorsen@mat.dtu.dk)GF$C&>F8F[^p>F]o
f*F[flF$FboF$Fb^pF$F$F_gs>Fgpf*F[flF$FboF$FbgsF$F$FegsFggs6#QYFinder~|+kurveint
egralet~af~funktion|+langs~given~rumkurveF$F$F$%-rumIntApproxGf*F[cn6MFgwFfwFhy
FhwFfxFdvFevFfvFgvFdcnF,FccnFc]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFg]oF]wF^wF_w
Fh]oFi]oFhvFgcnF^vFecnFfcnFj]oFgizFjyF[zFhizFciuFdiuFbiuFayF$C<@%FazC$>Fh_lFfz@
%33-Fjz6$FadsF][l-Fjz6$F[i]lF][l-Fjz6$&Fh_lFRF][l>FcqFd[lC$>FcqF%>Fh_lF^\nC$>Fc
qF%>Fh_lF^\n>Fbjz-F<6#*&F?F:FadsFE>F\ael-F<6#*&FMF:F[i]lFE>&Fe\mFR-F<6#*&FZF:F[
g\mFE@%Fe__lC%>F8Fd[l>F\z7#-Fer6$-Fer6$-Fer6$-F<6#-F\p6%,&FDF:*&FbtF:FbjzF:F:,&
FQF:*&F]sF:F\aelF:F:,&FhnF:*&FaoF:F\h\mF:F:/Fbt;F:Fads/F]s;F:F[i]l/Fao;F:F[g\m>
F]o-Fjz6$-F[al6$-F[al6$F\_v/Fdal;F:*(FadsF:F[i]lF:F[g\mF:FgalF][l>F8F%@%F\av>Fg
p-Fjz6$-F[al6$Fbav/F`]pFcj\mF][l>FgpF%@%FgpC$>F^fp-F_^l6#-Fer6$Fcav/F^]pFcj\m>F
d`m-Ff_lFb[]mF$>Fc`lFi^o>F[enFb_o>FhenFi_o>Fgyf*F`_oF$FboF$F_`oF$F$F\\[m>Fc[lf*
F`_oF$FboF$Fj`oF$F$F_\[m>Fgdof*F_oF$FboF$Fb\[mF$F$Fc_el>Fh^of*FidoF$FboF$FjdoF$
F$F[][m>F[jrF]][m@%F]fvCL>Fgbmf*F\^lF$FboF$FbfvF$F$6&FdvF\zF,Fh_l>Fg\lf*F\^lF$F
boF$F]gvF$F$Fh\]m>Fj]lF_gv>Fezf*F_oF$FboF$FjhvF$F$F_iv>Fdblf*FipF$FboF$FbivF$F$
F_iv>Fj_mf*FeqF$FboF$FiivF$F$F_iv>Fbj_l-Fbr6#-Fer6$-Fer6$-Fez6$Fai\m,&FhnF:*&F`
sF:F\h\mF:F:/F]s;FcsF[i]l/F`s;FcsF[g\m>Fdj_l-Fbr6#-Fer6$-Fer6$-Fdbl6$F_i\mF[^]m
/Fbt;FcsFadsF_^]m>Ffj_l-Fht6#-Fer6$-Fer6$-Fj_m6$F_i\mFai\mFj^]mF]^]m>Fdjt-Fbr6%
7%Fbj_lFdj_lFfj_lFg^mFj^m>Fc^mf*F_oF$FboF$F\]wF$F$Fep>F__mf*FipF$FboF$F`]wF$F$F
ep>Fffmf*FeqF$FboF$Fd]wF$F$Fep>Fb[y-Fbr6#-Fer6$-Fer6$-Fc^mFj]]mF]^]mF_^]m>Fg[y-
Fbr6#-Fer6$-Fer6$-F__mFi^]mFj^]mF_^]m>F\\y-Fht6#-Fer6$-Fer6$-FffmFd_]mFj^]mF]^]
m>Ffy-Fbr6#7%Fb[yFg[yF\\y>&F_]mFg_w-Fa_m6(Fj_wF\ilFheoFb\m/Fe_m7$,$F[i]lFS,$F[g
\mFSFg^m>&F_]mFb`w-Fa_m6(Fe`wF\ilFheoFb\mF_b]mFg^m>&F_]mFi`w-Fa_m6(F\awFbhlFheo
F[]m/Fe_m7$,$FadsFSFbb]mFg^m>&F_]mFcaw-Fa_m6(FfawFbhlFheoF[]mF[c]mFg^m>&F_]mFja
w-Fa_m6(F]bwFbhlF\ilFegp/Fe_m7$F]c]mFab]mFg^m>&F_]mFcbw-Fa_m6(FfbwFbhlF\ilFegpF
fc]mFg^m>Faco-Fbr6%-Fer6$-Fer6$&F_]mF`cwFacwFbcwFj^mFdcw>FfaoFC>Fhcwf*F`_oF$Fbo
F$FjcwF$F$6,F*F\pF_wF[enF.Fe\mFbiuFfaoFh]oFhen>Fjdwf*F`_oF$FboF$-F]ew6$f*F`ewF$
FboF$FaewF$F$FiewFjewF$F$6*Fj]oF^irF^wFc`lF.Fe\mFbiuFfao>F^fwf*F`_oF$FboF$F`fwF
$F$6,F*F\pF^wFc`lF.Fe\mFbiuFfaoF_wF[en>F]gwf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$Fbg
wF$F$FggwFhgwF$F$6*Fj]oF^irFh]oFhenF.Fe\mFbiuFfao>F\hwf*F`_oF$FboF$F^hwF$F$6,F*
F\pF^wFc`lF.Fe\mFbiuFfaoFh]oFhen>Fbhwf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$FghwF$F$F
\iwF]iwF$F$6*Fj]oF^irF_wF[enF.Fe\mFbiuFfao@%2,&F^fpF:Fd`mFEFciw>Fhjlf*F`_oF$Fbo
F$FfiwF$F$Fejw>Fhjlf*F`_oF$FboF$FhjwF$F$6,Fj]oF^irF*F\pF\cnFGFdiuFd`mFciuF^fp>F
\jm-Fbr6%7#-Fer6$-Fer6$-Fer6$-Fhjl6%F_i\mFai\mF[^]m/Fbt;Fcs,&FadsF:FEF:/F]s;Fcs
,&F[i]lF:FEF:/F`s;Fcs,&F[g\mF:FEF:Fg^mFj^m>Fifp-F^_[l6$-F^_[l6$-F^_[l6$*,-FG6#-
F`cl6#-F\pFcg]mF:-Fc[lFcg]mF:FbjzF:F\aelF:F\h\mF:/Fc[mFeg]m/Fe[mFhg]m/.F`sF[h]m
F]]x-Fc\n6#-Fbr6'7&Fbj_lFdj_lFfj_lFe]xFg^mF`]nF]]nFj^mFf]x-Fc\n6#-Fbr6'Fge[lFg^
mF]]nF`]nFj^mFb]^l-Fc\n6#-Fbr6'F^guFj^mFg^mF`]nF]]nF_\al-Fc\n6#-Fbr6'Fj]nFg^mF`
]nF]]nFj^mF$F^_xF__n-F\_n6$F_aqFh_lFi^^lFf_x-Fc\n6#/.-Fc[lF`_oFfj]m-F\_n6#Q@App
roksimeret~rumintegral~af~f:F$-Fc\n6#/Fifp-F<6#Fifp-F\_n6#Q4Eksakt~rumintegral:
F$Fi\al6#QhpFinder~og~viser~approksimation~til~rumintegralet~af~given~funktion~
over~givet~rumligt~omr|`yde.F$F$F$%*kurveImGoGf*F_]oF\^blFayF$C)>F8F[^p>F]of*F[
flF$FboF$FeablF$F$F_gs>FdblFgao>Fezf*F[flF$FboF$F^bblF$F$Febo>Fh^of*F[flF$FboF$
FabblF$F$Febbl>Fgpf*F[flF$FboF$-F[gl6#FhbblF$F$Fibbl>FhgoFigs6#QasFinder~inerti
momentet~af~rumkurve~med~v|aygtfunktion~|+omkring~given~linje~(ved~pkt~med~retn
ingsvektor~|+ikke~n|czdvendigvis~enhedsvektor).~F$F$F$%(fladeImGf*F_]o6RFfxFfwF
gwFhyFhwF.F/F0F_cnF`cnF2F^cnFhvFivFgcnFevFfvFgvFccnFdcnFc]oFciuFdiuF,FdvF_vF[fq
F]^oF^^oF_^oF`^oFacnFfcnFecnF^wF_wF`wF^vFicnFa^oFb^oFc^oFd^oFe^oF]wFjcnFjyF[zFa
yF$CR@%F_goC$>FacoFbgo@%Ffidl>F`rFd[lC$>F`rF%>FacoFh[lC$>F`rF%>FacoFh[l>FgpF[iq
>Fj`mFejdl>FhamFjjdl>Fgdo7#-Fer6$-Fer6$F_`]lF[\elF]\el>F]o-Fjz6$-F[al6$Fd\elFga
lF][l>FcqF`]el@%FcqC$>FhgoF`^el>FghoFf^elF$@%Fe_v>F[enFd[l>F[enF%@%F][p>FhenFd[
l>FhenF%>F\jmF]dn>F^irFddn>F[jrf*FeqF$FboF$FgjnF$F$6&F^wF\jmF_wF^ir>F_jrf*FeqF$
FboF$F^[\lF$F$6$F`wF[jr>FfyFgao>Fd`mf*F[flF$FboF$FbcvF$F$6&F`]oFfzF`^oFfy>Fdjtf
*FeqF$FboF$-Fecm6#,&FbclF:F`pFEF$F$Febo>Fbelf*FeqF$FboF$,&Fb[oF:*$-Fibl6%FafmF_
_lFdclFCFEF$F$6&F_^oFdjtF`^oFfy>Fh^of*FeqF$FboF$-F[gl6#*(F]clF:FfglF:Fd]rF:F$F$
6*F*F\pF\cnFGFe^oFbelF^vF_jr>Feimf*F`dlF$FboF$F`a\lF$F$6&F_vFh^oF+FA>Fc`lf*F`dl
F$FboF$Fh^pF$F$6$F]wFeim>Fifp-F[gl6#-F^hl6$-FeimF`dlF\il@%33F`d]lFhenF[enC<>Fe\
m-Fa_m6'Fc_mFbhlF\il/Fe_m7$,$Fg]wFjn,$F`^wFjnFg^m>F_]mFjd]l>Fc^mf*F\^lF$FboF$F`
e]lF$F$6&FgcnF_]mF,Faco>F__mf*F\^lF$FboF$Fje]lF$F$Ffb^m>FffmF\f]l>F^fp-F\am6*-F
d`mF[fl/F`s;,$-F_^l6$F_hvFffmFEFac^mF`hv/Fcam-F_^l6$Fehv,$FffmFghvFiemFb\mFg^mF
j^m>Fj`mFgjdl>FhamF\[el>Fj]lf*F`dlF$FboF$Fef]lF$F$F_iv>Fc_lf*FiclF$FboF$F[g]lF$
F$F_iv>F]\l-Fht6#-Fer6$Fcg]lFi`el>Fb\l-Fht6#-Fer6$F\h]lFcael>Fgbmf*F`dlF$FboF$F
^\mF$F$Fep>Fg\lf*FiclF$FboF$Fg\mF$F$Fep>Fads-Fbr6#-Fer6$Fih]lFi`el>F[i]l-Fbr6#-
Fer6$F`i]lFcael>F\\mFbi]l>Fgyf*FeqF$FboF$Fgi]lF$F$6*F*F\pF^wF\jmF.F\zF_wF^ir@%2
,&FhgoF:FghoFEFciw>Fhjlf*FeqF$FboF$Fgj]lF$F$6$FacnFgy>Fhjlf*FeqF$FboF$F\[^lF$F$
6,FacnFgyF*F\pF\cnFGFdiuFghoFciuFhgo>Fc[l-Fbr6$7#-Fer6$-Fer6$F`g`lF^eelFaeelFg^
mF]]xF\]^l-Fc\n6#QWDet~parametriserede~billed-omr|`yde~og~omdrejningsaksen:F$-F
c\n6#-Fbr6&7&FadsF[i]lF^fpF[]nFg^mF]]nFj^m-Fc\n6#Q[oApproksimeret~flade,~omdrej
ningsaksen~og~antydedet~v|aygtfunktion:F$-Fc\n6#-Fbr6'7%Fc[lF^fpF[]nFg^mF]]nF`]
nFj^mF$F[_nF__n@%F`rFhfelF$-F\_n6$QFAngivet~masse-t|aythed:~f(x,y,z)~=~%a~|+F$F
`jmFe_s-Fc\n6#/.-F_jrFeqFbh^mF^`xFa`x-F\_n6#QMAfstanden~fra~fladen~til~aksen~er
~givet~ved:F$-Fc\n6#/.-F\axFeq-FbelFeq-F\_n6#Q<Inertimoment~om~givne~akse:F$-Fc
\n6#/.-F\bs6$F[bxF^bs-F[glF^[^m-Fc\n6#/Fbi^mF][^mFcjbl-Fc\n6#/Fbi^m-F\bs6$-F\bs
6$*(FfcxF:F[i^mF:Fah^mF:FbhlF\il-Fc\n6#/Fbi^m-F\bs6$-F\bs6$*(FhhnF:F[i^mF:Fbh^m
F:FbhlF\ilF_exF`ielFiielF]jelFejelF\[flFjhxF_i^mF]ix6#Q`oFinder~og~viser~inerti
momentet~af~fladestykke~med~givet~'masset|aythed'F$F$F$%,GaussFluxGoGf*FiuFdixF
[jxF$C:>Fb[yFc[y>Fg[yFh[y>F\\yF]\y>F_ayF`ay>FbayF`ay>FdayFeay>FgayFeay>FiayFjay
>F\byFjay>F^byf*F_oF$FboF$F`byF$F$F]dl>FcbyFdby>Fjbyf*F_oF$FboF$F\cyF$F$F]dl>F_
cyF`cy>Ffcyf*FipF$FboF$FhcyF$F$F]dl>F[dyF\dy>Fbdyf*FipF$FboF$FddyF$F$F]dl>FgdyF
hdy>F^eyf*FeqF$FboF$F`eyF$F$F]dl>FceyFdey>Fjeyf*FeqF$FboF$F\fyF$F$F]dl>F_fyF`fy
>F[enf*FgfyF[gyF\gyF$C'>F8F]dn>F]oFddn>FgpFbgy>Fcqf*FeqF$FboF$FjgyF$F$FhhyFihyF
giyF$F$>Fgbmf*F`cwF$FboF$F[jyF$F$F`jy>Fg\l-F[gl6#FbjyFhgzF$F$%-tangKurveIntGf*F
iu6GFfxFfwFgwFiwF`foFd^oFgvF.F[wF\wFevFfvFbvFixFjxF[yFjvF]vF^vF_vF,FhwFdvFexFjw
FcwFdwFewF[xF\xF]xF^xF_xF`xFgxFaxFbwF]^blF$CE@%FazC$>Fj_mFfz@%Fdgo>FhgoFd[lC$>F
hgoF%>Fj_mF\hoC$>FhgoF%>Fj_mF\ho>FgpFjal>FbhoFcho>FghoFhho>F]oFdio>Fj]lf*FiclF$
FboF$FbjoF$F$Ffcl>FcqFb]p@%F_cpC5>F`rFdcp>F\zF^dp>Fh_lf*F\^lF$FboF$FddpF$F$Faep
>Feyf*F\^lF$FboF$FdepF$F$Faep>FgbmFfep>F[jr*&F\dmF:FgbmF:>F\\mFafp>Fc_lF[jp>Fe\
mf*F[flF$FboF$-F\am6)Fgjp/F[p;Fcs*&F\dmF:F]dmF:/Fcam*&F`[qF:F]dmF:FbdmFddmFg^mF
a[qF$F$6*F*F\pFjuFGF\wFc_lFbwF[jr>F_]mf*F[flF$FboF$-F\am6'Fgjp/F[p;*&F\dmF:F]dm
F:*&FcemF:F]dmF:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F[pFEF:FbdmFiemF$F$Fb`_m>Fc^mf*F
[flF$FboF$Fb\qF$F$Ff\q>F__mFh\q>Fgdo-F^jm6(F`jmFh]qF`^qFh^qF]\nF_\nF`_q-Fc\n6#-
Fbr6&Fg_qFg^mF`]nFj^mFc]n-Fc\n6#-Fbr6&7$FgdoF[]nFg^mF`]nFj^m-Fc\n6#QEKurven~og~
vektorfeltet~langs~kurven:F$-Fc\n6#-Fbr6&7%F\\mF__mF[]nFg^mF`]nFj^mF$>FffmF[^p>
Fezf*F[flF$FboF$Fb^pF$F$Fi^p>Fdblf*F[flF$FboF$-F[gl6#-Fibl6%FfjoF^`pFdclF$F$6(F
*F\pFjuFGF]vFffm>Faco-F[gl6#-F^hl6$-FdblF[flFfcpFh`qF__nF]aqF]_sFiaq-F\_n6#Q;Ve
ktorfeltet~langs~kurven:F$-Fc\nFdbqFb_n-F\_n6#Q+Jacobi(u):F$Fjcq-F\_n6#Q<Tangen
tielle~kurveintegral:F$-Fc\n6#/.-F\bs6$-F\bx6$FGFa]oFag^lFaco-Fc\n6#/Fcd_m-F<6#
Faco-F\_n6#QNBeregning~af~det~tangentielle~kurveintegral~:F$-Fc\n6#/Fcd_m-F\bs6
$-F\bx6$FG.F^b[lFbhl-Fc\n6#/Fcd_m-F\bs6$-F\bx6$FebqF^b[lFbhl-Fc\n6#/Fcd_m-F\bs6
$-FdblFiclFbhlF_ex-Fc\n6#/Fbf_m&7#-F^hl6$Fdf_mF[pF\fx-Fc\n6#/F`fx-F^hlFcf_mFjhx
F`d_mF]ix6#QdsFinder~og~viser~(hvor~muligt)~det~tangentielle~kurveintegral~af~v
ektorfeltet~V(x,y,z)~langs~en~given~rumkurve~med~parameterfremstilling~rF$F$F$%
.traadModelRumGf*F)F-F3F$C,>F7F;>FIFJ>FVFW>F]of*F_oF$FboF$F\]wF$F$Fep>Fgpf*FipF
$FboF$F`]wF$F$Fep>Fcqf*FeqF$FboF$Fd]wF$F$Fep>F_rFar>FgsFhs>FftFgtFbuFeuF$F$%4tr
aadModelFladeGrayGf*F)F_etFayF$C)>F7F;>FIFJ>F]of*F`dlF$FboF$-Ffo6%F`\mF]pFbpF$F
$Fep>Fgpf*FiclF$FboF$-Ffo6%Fi\mF^qFbpF$F$Fep>FjglFhet>FehlF_ftFeftFhftF$F$%/tan
gKurveIntGoGf*F^jnFh]_mFae\mF$C&>FffmF[^p>Fezf*F[flF$FboF$Fb^pF$F$Fi^p>Fdblf*F[
flF$FboF$F[c_mF$F$F_c_m>FacoFac_m6#QdoFinder~det~tangentielle~kurveintegral~af~
vektorfelt~langs~given~rumkurve.F$F$F$%(rumCmGoGf*Fdj[lFe[[mFayF$C,>Fc`lFi^o>F[
enFb_o>FhenFi_o>Fgyf*F`_oF$FboF$F_`oF$F$F\\[m>Fc[lf*F`_oF$FboF$Fj`oF$F$F_\[m>F[
jr-F[gl6#-F^hl6$-F[gl6#-F^hl6$-F[gl6#-F^hl6$-F[gl6#Ffj]mFbhlF\ilFheo>Fgdof*F_oF
$FboF$Fb\[mF$F$Fc_el>Fh^of*FidoF$FboF$FjdoF$F$F[][m>F_jrF]][m>F]]r7%-F[gl6#*&-F
^hl6$-F[gl6#-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(&F__oF9F:FadxF:Ffj]mF:FbhlF\ilFheoF:F
_jrFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(&F__oFBF:FadxF:Ffj]mF
:FbhlF\ilFheoF:F_jrFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[gl6#-F^hl6$-F[gl6#*(&F__o
FRF:FadxF:Ffj]mF:FbhlF\ilFheoF:F_jrFE6#QdoFinder~massemidtpunktet~for~givet~rum
ligt~omr|`yde~~med~givet~'masset|aythed'F$F$F$%'rumIntGf*F[cn6PFgwFfwFhyFhwFfxF
biuFdvFevFfvFgvFdcnF,F_vF[fqFeiuFciuFdiuFccnFc]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnF
f]oFg]oF]wF^wF_wFh]oFi]oFhvFgcnF^vFecnFfcnFj]oF[^oFjyF[zF\^o6#%HRev.~11.07.04~(
S.Markvorsen@mat.dtu.dk)GF$CM@%FazC$>FeyFfz@%Fc^_l>FcqFd[lC$>FcqF%>FeyF^\nC$>Fc
qF%>FeyF^\n>&FdblF9Fagq>&FdblFBFegq>&FdblFRFa__l@%Fe__lC%>F8Fd[l>Fgbm7#-Fer6$-F
er6$-Fer6$-F<6#-F\p6%,&FDF:*&FbtF:Fb``mF:F:,&FQF:*&F]sF:Fd``mF:F:,&FhnF:*&FaoF:
Ff``mF:F:FfhqFhhqFg`_l>F]o-Fjz6$-F[al6$-F[al6$.-F<6#&&FgbmFh^lF\^lFaa_lFgalF][l
>F8F%@%F\av>Fgp-Fjz6$-F[al6$.-F<6#-FG6#-F`cl6#&FgbmF]]pFe^jlF][l>FgpF%@%FgpC$>F
c^m-F_^l6#-Fer6$F`c`m/F^]pFba_l>F__m-Ff_lF\d`mF$>Fc[lFi^o>FhjlFb_o>F\jmFi_o>F_j
rf*F`_oF$FboF$F_`oF$F$6(F^wFc[lF_wFhjlFh]oF\jm>F]]rf*F`_oF$FboF$Fj`oF$F$6$FgcnF
_jr>F\\mf*F`_oF$FboF$-F[gl6#*&FadvF:Ff`oF:F$F$6(F*F\pF\cnFGF^vF]]r>Ffyf*F_oF$Fb
oF$Fb\[mF$F$6*F*F\pF\cnFGF^vF]]rF+FA>Fe\mf*F_oF$FboF$FafmF$F$6$Fg]oFfy>Fgyf*Fid
oF$FboF$FjdoF$F$6&Fg]oFfyF+FA>F_]mf*FidoF$FboF$Fh^pF$F$6$F]wFgy>Fbel-F[gl6#-F^h
l6$-FgyFidoFheo@%F]fvCK>Fg\lf*F\^lF$FboF$FbfvF$F$6&FdvFgbmF,Fey>Fj]lf*F\^lF$Fbo
F$F]gvF$F$Fgf`m>Fc_l-F_^l6#-Fer6$,&FegvF:F^`lFEF``l>Fgdof*F_oF$FboF$FjhvF$F$F_i
v>Fh^of*FipF$FboF$FbivF$F$F_iv>Fc`lf*FeqF$FboF$FiivF$F$F_iv>Fa^gl-Fbr6#-Fer6$-F
er6$-Fgdo6$Fha`m,&FhnF:*&F`sF:Ff``mF:F:FfcrFg\`l>Fj^gl-Fbr6#-Fer6$-Fer6$-Fh^o6$
Ffa`mF_h`mF`drFg\`l>Fc_gl-Fht6#-Fer6$-Fer6$-Fc`l6$Ffa`mFha`mF`drFfcr>F^ir-Fbr6%
F^`glFg^mFj^m>Fhgof*F_oF$FboF$F\]wF$F$Fep>Fghof*FipF$FboF$F`]wF$F$Fep>Facof*Feq
F$FboF$Fd]wF$F$Fep>&F[enF9-Fbr6#-Fer6$-Fer6$-FhgoF^h`mFfcrFg\`l>&F[enFB-Fbr6#-F
er6$-Fer6$-FghoFih`mF`drFg\`l>&F[enFR-Fht6#-Fer6$-Fer6$-FacoFbi`mF`drFfcr>F[jr-
Fbr6#7%F]j`mFfj`mF_[am>&Fj_mFg_wFe``l>&Fj_mFb`wF]a`l>&Fj_mFi`wFaa`l>&Fj_mFcawFh
a`l>&Fj_mFjawF\b`l>&Fj_mFcbwFbb`l>Fdjt-Fbr6%-Fer6$-Fer6$&Fj_mF`cwFacwFbcwFj^mFd
cw>F\zF:>&Fd`mFg_wf*F`_oF$FboF$FjcwF$F$6,F*F\pF_wFhjlF.FdblFbiuF\zFh]oF\jm>&Fd`
mFb`wf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$FaewF$F$FiewFjewF$F$6*Fj]oFd`mF^wFc[lF.Fd
blFbiuF\z>&Fd`mFjawf*F`_oF$FboF$F`fwF$F$6,F*F\pF^wFc[lF.FdblFbiuF\zF_wFhjl>&Fd`
mFcbwf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$FbgwF$F$FggwFhgwF$F$6*Fj]oFd`mFh]oF\jmF.F
dblFbiuF\z>&Fd`mFi`wf*F`_oF$FboF$F^hwF$F$6,F*F\pF^wFc[lF.FdblFbiuF\zFh]oF\jm>&F
d`mFcawf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$FghwF$F$F\iwF]iwF$F$6*Fj]oFd`mF_wFhjlF.
FdblFbiuF\z@%2,&Fc^mF:F__mFEFciw>Fifpf*F`_oF$FboF$FfiwF$F$6$Fj]oFd`m>Fifpf*F`_o
F$FboF$FhjwF$F$6,Fj]oFd`mF*F\pF\cnFGFdiuF__mFciuFc^m>F^fp-Fbr6%7#-Fer6$-Fer6$-F
er6$-Fifp6%Ffa`mFha`mF_h`m/Fbt;Fcs,&F]\lF:FEF:/F]s;Fcs,&Fb\lF:FEF:/F`s;Fcs,&Ff^
_lF:FEF:Fg^mFj^mF]]x-Fc\n6#-Fbr6'7&Fa^glFj^glFc_glFe]xFg^mF]]nF`]nFj^mFf]x-Fc\n
6#-Fbr6'F\]blFg^mF]]nF`]nFj^mFb]^l-Fc\n6#-Fbr6'7$F[jrF[]nFj^mFg^mF`]nF]]n-Fc\n6
#QOApproksimation,~net-finhed,~funktions-v|ayrdier:F$-Fc\n6#-Fbr6'7$F^fpF[]nFg^
mF]]nF`]nFj^mF$F^_xF__nF[_sFi^^lFf_x-Fc\n6#/.-F]]rF`_oFibam-F\_n6#Q@Rum-integra
let~af~funktionen~f:F$-Fc\n6#/.-F\bs6$FbbqF^bx-F[gl6#Fbel-Fc\n6#/F`cam-F<Fdcam-
F\_n6#Q=Beregning~af~rum-integralet:F$-Fc\n6#/F`cam-F\bs6$-F\bs6$-F\bs6$*&FadxF
:FhbamF:FbhlF\ilFheo-Fc\n6#/F`cam-F\bs6$-F\bs6$-F\bs6$*&FadxF:FibamF:FbhlF\ilFh
eoF_ex-Fc\n6#/-F\bs6$-F\\mF`_oFbhl&7#-F^hl6$FeeamF[pF\fx-Fc\n6#/F`fx-F^hlFdeam-
Fc\n6#/-F\bs6$FceamF\il-F\bs6$-Fe\mF_oF\il-Fc\n6#/Fcfam&7#-F^hl6$FefamF`oFagx-F
c\n6#/F`fx-F^hlFdfam-Fc\n6#/-F\bs6$FafamFheo-F\bs6$-F_]mFidoFheo-Fc\n6#/Ffgam&7
#-F^hl6$FhgamFaoFehx-Fc\n6#/F`fx-F^hlFggamFjhxF]camF]ix6#Q^oFinder~og~viser~rum
integralet~af~funktion~over~givet~rumligt~omr|`ydeF$F$F$%(fladeCmGf*F[cn6IFfxFf
wFgwFhyFhwF.F/F0F_cnF`cnF2F^cnFhvFivFgcnFevFfvFgvFccnFdcnFc]oF,FdvF_vF[fqFciuFd
iuFacnFfcnFecnF^wF_wF`wF^vFicnF]wFjcnFjyF[zFayF$C>@%FazC$>FhgoFfz@%Fd^]l>F`rFd[
lC$>F`rF%>FhgoFh[lC$>F`rF%>FhgoFh[l>FgpF[iq>Fj`mFb_]l>FhamFf_]l>FghoFj_]l>F]oF\
a]l>FcqFfa]l@%FcqC$>Fh^oF^b]l>Fc`lFdb]lF$>FfyF]dn>FgyFddn>Fc[lf*FeqF$FboF$FgjnF
$F$Fib]l>Fhjlf*FeqF$FboF$F^[\lF$F$F\c]l>Facof*FeqF$FboF$F_c]lF$F$Fbc]l>F^irf*F`
dlF$FboF$FafnF$F$Fec]l>Fgdof*F`dlF$FboF$Fh^pF$F$6$F]wF^ir>F_jr-F[gl6#-F^hl6$-F^
irF`dlF\il@%5-Fjz6$-F<6#F_jrFbhs4-Fjz6$Ff[bmF][l>F]]r7%-F[gl6#*&-F^hl6$-F[gl6#-
F^hl6$-F[gl6#*(FghnF:FhhnF:F`_^lF:FbhlF\ilF:F_jrFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6
$-F[gl6#*(FiinF:FhhnF:F`_^lF:FbhlF\ilF:F_jrFE-F[gl6#*&-F^hl6$-F[gl6#-F^hl6$-F[g
l6#*(Fa^\lF:FhhnF:F`_^lF:FbhlF\ilF:F_jrFEF$@%F`d]lC;>Fe\mFcd]l>F_]mFjd]l>Fc^mf*
F\^lF$FboF$F`e]lF$F$Fge]l>F__mf*F\^lF$FboF$Fje]lF$F$Fge]l>FffmF\f]l@%Fd[bmC$>Fe
z,$FffmFj\t>Fdbl-F<6#-FecmF[`^lC$>FezFa^t>FdblFc^t>Fj_m-Fa_m6'7%,&Fb``mF:*(FezF
:FehpF:FghpF:F:,&Fd``mF:*(FezF:FehpF:F\ipF:F:,&Ff``mF:*&FezF:F`ipF:F:FaipFdipFg
ipFg^m>Fj]lf*F`dlF$FboF$Fef]lF$F$F_iv>Fc_lf*FiclF$FboF$F[g]lF$F$F_iv>F]\lF_g]l>
Fb\lFhg]l>Fgbmf*F`dlF$FboF$F^\mF$F$Fep>Fg\lf*FiclF$FboF$Fg\mF$F$Fep>FadsFeh]l>F
[i]lF\i]l>F\\mFbi]l>F[enf*FeqF$FboF$Fgi]lF$F$Faj]l@%Fcj]l>Fdjtf*FeqF$FboF$Fgj]l
F$F$Fij]l>Fdjtf*FeqF$FboF$F\[^lF$F$Fe[^l>FhenFg[^lF]]xF\]^l-Fc\n6#Q[pDet~parame
triserede~billed-omr|`yde~med~massemidtpunkt~(hvis~det~er~veldefineret):F$-Fc\n
6#-Fbr6'7&FadsF[i]lFj_mF[]nFg^mF]]nF`]nFj^m-Fc\n6#QYApproksimation~med~den~givn
e~v|aygtfunktion~og~net-finhed:F$-Fc\n6#-Fbr6'7&FhenF\\mFj_mF[]nFg^mF]]nF`]nFj^
mF$F[_nF__nFg^^lF[h^mFe_sF\_^l-F\_n6#Q.Totale~masse:F$-Fc\n6#/Fg_^l-F[glFg[bm-F
c\n6#/Fg_^lFf[bm@%Fd[bmC$F[dt-Fc\n6#/Fj_^lF_`^l@%Fh[bmC$F[dt-Fc\n6#Fj_^lFhdt6#Q
doFinder~og~viser~fladeintegralet~af~'masset|aytheden'~over~givet~fladestykkeF$
F$F$%/traadProjFladeGf*6%F*F+F[v6aoF]vF^vF_vF`vFavFbvFcvFdvFevFfvFgvF.F/F0F2Fhv
FivFjvF,F[wF\wF]wF^wF_wF`wFawFbwFcwFdwFewFfwFgwFhwFiwFjwF[xF\xF]xF^xF_xF`xFaxFb
xFcxFdxFexFfxFgxFhxFixFjxF[yF\yF]yF^yF_yF`yFhyFjyF[zF]zF^zF_z%$rxyG%$ryzG%$rxzG
%#xyG%#xzG%#yzG6#%HRev.~28.07.04~(S.Markvorsen@mat.dtu.dk)GF$C3@%/FbzFSC$>FezFG
@%Fhz>Fc[lFd[lC$>Fc[lF%>FezFh[lC$>Fc[lF%>FezFh[l>F]\lF^\l>Fb\lFc\l>Fg\l7#-Fer6$
-Fer6$F]]lF`^mFf]m>Fj]lf*F\^lF$FboF$-F_^l6#-Fer6$Fc^l/Fi^l;F:*&,&F^_lF:F:F:F:,&
F`_lF:F:F:F:F$F$Fa_l>Fc_lf*F\^lF$FboF$-Ff_lF]fbmF$F$Fa_l>Fh_lf*F\^lF$FboF$,&Fh^
pF:Fe\qFEF$F$6&FevFj]lFfvFc_l>F\f_lf*FeqF$FboF$7%Fcj^lFgj^lFcsF$F$Fdj^l>F^g_lf*
FeqF$FboF$7%Fcj^lFcs&FbclFRF$F$Fdj^l>Fef_lf*FeqF$FboF$7%FcsFgj^lFbgbmF$F$Fdj^l>
F__m-Fa_m6&Fc_mFbhlF\ilFg^m>F\\u-Fa_m6'-F\f_lFeqFbhlF\ilF]atFa[q>F_cu-Fa_m6'-Fe
f_lFeqFbhlF\ilF]atFb\m>F]\u-Fa_m6'-F^g_lFeqFbhlF\ilF]atF[]m-Fc\n6#-Fbr6&Fj\nFdc
wFg^mFj^m-Fc\n6#-Fbr6&7&F\\uF_cuF]\uF[]nFdcwFg^mFj^m-Fc\n6#-Fbr6&7'F\\uF_cuF]\u
F__mF[]nFdcwFg^mFj^m6#Q[oViser~tr|`ydmodel~projektioner~p|`y~koordinatplaner~og
~fladestykket.F$F$F$Fe[ilf*F)FffhlF`jnF$C)>F^byFifhl>FjbyF`ghl>FfcyFcghl>FbdyFi
ghl>F^eyF\hhl>FjeyFahhl-Fbr6%-Fer6$-Fer6$&F8F`cwFacwFbcwFj^mF]]nF\ihlF$F$%'plan
CmGf*F[cn6EFfxFfwFgwFhyFhwFbjmFcjmF^wF_wF.F2F/F0F^cnF_cnF]wF`cnFacnFbcnFccnFciu
FdiuFdcnF,FdvFecnFhvFfcnFgcnFhcnF^vFicnFjcnFjyF[zF3F$C@@%FazC$>FacoFfz@%Ffidl>F
`rFd[lC$>F`rF%>FacoFh[lC$>F`rF%>FacoFh[l>FgpF[iq>FdjdlFejdl>FijdlFjjdl>FgdoF^[e
l>F]oF`\el>FcqF`]el@%FcqC$>Fj_mF`^el>FhgoFf^elF$>Fg\lF]dn>Fj]lFddn>Fdjtf*FeqF$F
boF$F]enF$F$Ffen>Ffyf*FeqF$FboF$FjenF$F$Fa^[m>Fc^mf*F`dlF$FboF$FafnF$F$6*F*F\pF
\cnFGF^vFfyF+FA>FhjlFdgn@%5-Fjz6$-F<6#FhjlFbhs4-Fjz6$F[]cmF][l>F\jm7$-F[gl6#*&-
F^hl6$-F[gl6#-F^hl6$-F[gl6#*(FghnF:FhhnF:-FfyFeqF:FbhlF\ilF:FhjlFE-F[gl6#*&-F^h
l6$-F[gl6#-F^hl6$-F[gl6#*(FiinF:FhhnF:F^^cmF:FbhlF\ilF:FhjlFEF$@%F`d]lC8@%Fi\cm
C$>Fdbl$Ff`lF\dp>Fgho-Fecm6#F\jmC$>FdblFa^t>Fgho7$FcsFcs>F\zf*FeqF$FboF$Fcj^lF$
F$Fdj^l>Fgbmf*FeqF$FboF$Fgj^lF$F$Fdj^l>F_]mf*F`dlF$FboF$Fef]lF$F$F_iv>F__mf*Fic
lF$FboF$F[g]lF$F$F_iv>Fez-Fbr6#-F_del6%7#-Fer6$7$,&Fbj_lF:*&FdblF:-Fhhp6#,$*&Fb
tF:FcipF:#F:FPF:F:,&Fdj_lF:*&FdblF:-FfhpF\acmF:F:/Fbt;F:F]_mFbpFg^m>FbjzFc`el>F
\aelF]ael>Feyf*F`dlF$FboF$Fjj^lF$F$Fgael>F\\mf*FiclF$FboF$Fe[_lF$F$Fgael>FadsF[
bel>F[i]lFabel>Fc`lFgbel>Fffmf*FeqF$FboF$F[celF$F$Fhcel@%2,&Fj_mF:FhgoFEFciw>Fh
^of*FeqF$FboF$F^delF$F$F`del>Fh^of*FeqF$FboF$FcdelF$F$6,FacnFffmF*F\pF\cnFGFdiu
FhgoFciuFj_m>F[en-Fbr6$7#-Fer6$-Fer6$-Fh^oFf[elF^eelFaeelFg^mF]]xFdeelFdabm-Fc\
n6#-Fbr6%7&FadsF[i]lF^felFezFg^mF]]n-Fc\n6#QYApproksimation~med~den~givne~net-f
inhed~og~v|aygtfunktion:F$-Fc\n6#-Fbr6%7%F[enFc`lF^felFg^mF]]nF$F[_nF__nFhfel-F
\_n6$QDAngivet~masse-t|aythed:~f(x,y)~=~%a~|+F$F]gelFe_s-Fc\n6#/.F^^cmF^^cm-F\_
n6#Q'Masse:F$-Fc\n6#/FhjlF[]cm-F\_n6#Q1Masse-beregning:F$-Fc\n6#/Fg_^l-F\bs6$-F
\bs6$*&FhhnF:FddcmF:FbhlF\il-Fc\n6#/Fg_^l-F\bs6$-F\bs6$*&FhhnF:F^^cmF:FbhlF\il-
Fc\n6#/Fg_^l-F\bsFggn-Fc\n6#/Fg_^lFhjl@%Fi\cmC$F[dt-Fc\n6#/-F[glFc_cm-F<Fc_cm@%
F]]cmC$F[dt-Fc\n6#FjfcmFhdtFjinF$F$%*kurveKmGoGf*F^fo6UFfwFgwFiwF`foFd^oFgvF.F[
wF\wFevFfvFbvFixFjxF[yFjvF]vF^vF_vF,FhwFdvFafoFccnFdcnFc]oFexFjwFcwFdwFewF[xF\x
F]xF^xF_xF`xFgxFaxFbwFbfoFcfoFdfoFhyF]^oFefoFffoFgfoFhfoFifoFjfoF[goF$C,>F__mF[
^p>Fffmf*F[flF$FboF$Fb^pF$F$6$F]vF__m>Fiyf*FiclF$FboF$F]_pF$F$Fi_p>Fbeof*FiclF$
FboF$F\`pF$F$6*F*F\pF\cnFGFffoFiyF^vFffm>F`cmf*FiclF$FboF$Fc`pF$F$F\hcm>F[emf*F
iclF$FboF$Fg`pF$F$F\hcm>F^fp-F[gl6#-F^hl6$-FbeoFeqFbhl>Fd`mFj`p>FfaoF`ap>Fcjl-F
]bp6%F_bpF`bp7%F^fpFd`mFfao6#QfoFinder~kraftmomentet~(eng.:~torque)~om~punkt~af
~vektorfelt~paa|+v|aygtet~kurveF$F$F$%.planIntApproxGf*F[cn6HFfxFfwFgwFhyFhwFbj
mFcjmF^wF_wF.F2F/F0FciuFdiuF^cnF_cnF]wF`cnFacnFbcnFccnFbiuFdcnF,FdvFecnFhvFfcnF
gcnFhcnF^vFicnFjcnFjyF[zFhizFgizF3F$C=@%FazC$>FgdoFfz@%3-Fjz6$FajvF][l-Fjz6$Fa[
wF][l>F`rFd[lC$>F`rF%>FgdoFh[lC$>F`rF%>FgdoFh[l>FgpF[iq>Fdjdl-F<6#*&F?F:FajvFE>
Fijdl-F<6#*&FMF:Fa[wFE>Fh^o7#-Fer6$-Fer6$Fc[el/Fbt;F:Fajv/F]s;F:Fa[w>F]o-Fjz6$-
F[al6$-F[al6$.-F<6#&&Fh^oFh^lF\^l/Fdal;F:*&FajvF:Fa[wF:F^]elF][l>Fcq-Fjz6$-F[al
6$.-F<6#-FG6#-F`cl6#&Fh^oF]]p/F`]pFc\dmF][l@%FcqC$>Fe\m-F_^l6#-Fer6$F[]dm/F^]pF
c\dm>F_]m-Ff_lFg]dmF$>Fg\lF]dn>Fj]lFddn>Ffyf*FeqF$FboF$F]enF$F$Ffen>Fgyf*FeqF$F
boF$FjenF$F$6$FhcnFfy>Fffmf*F`dlF$FboF$FafnF$F$6*F*F\pF\cnFGF^vFgyF+FA>F_jrFa`r
>F[jr-F^_[l6$-F^_[l6$**Fh^[mF:-FgyFf[elF:FdjdlF:FijdlF:/Fc[m;Fcs,&FajvF:FEF:/Fe
[m;Fcs,&Fa[wF:FEF:@%F`d]lC7>F\zf*FeqF$FboF$Fcj^lF$F$Fdj^l>Fgbmf*FeqF$FboF$Fgj^l
F$F$Fdj^l>F__mf*F`dlF$FboF$Fef]lF$F$F_iv>Fezf*FiclF$FboF$F[g]lF$F$F_iv>F^[v-Fbr
6#-Fer6$-F__mFh`el/F]s;FcsFa[w>Fc[v-Fbr6#-Fer6$-FezFbael/Fbt;FcsFajv>Feyf*F`dlF
$FboF$Fjj^lF$F$Fgael>F\\mf*FiclF$FboF$Fe[_lF$F$Fgael>Fads-Fbr6#-Fer6$F_belFe`dm
>F[i]l-Fbr6#-Fer6$FebelF]adm>F[enFgbel>FghoFC>Fdblf*FeqF$FboF$7&F`fm,&F`fmF:*&F
]celF:FedwFEF:,(F`fmF:FcbdmF:*&-F<6#*&-FhewFcclF:FfcelF:F:FedwFEF:,&F`fmF:Febdm
F:F$F$6.FbjmF\zFcjmFgbmF^wFg\lF.Fc_lFbiuFghoF_wFj]l@%2,&Fe\mF:F_]mFEFciw>Fc`lf*
FeqF$FboF$F^delF$F$6$FacnFdbl>Fc`lf*FeqF$FboF$FcdelF$F$6,FacnFdblF*F\pF\cnFGFdi
uF_]mFciuFe\m>Fhen-Fbr6$7#-Fer6$-Fer6$-Fc`lFf[el/FbtF`_dm/F]sFc_dmFg^mF]]x-Fc\n
6#-Fbr6%7%F^[vFc[vFa]^lFg^mF]]nFb]^lFieelF_\al-Fc\n6#-Fbr6%7%FhenF[enF^felFg^mF
]]nF$@%3FgpF`r>Fc[l-F^_[l6$-F^_[l6$.*(F^_dmF:FdjdlF:FijdlF:F__dmFb_dmF$F[_nF__n
-F\_n6$F_aqFgdo-F\_n6$F[_^l-FG6%F\zFgbmFdjmFe_s-Fc\n6#/.-FgyFeqF]fdm-F\_n6#QBAp
proksimation~af~planintegralet:F$Fi\al-F\_n6#Q5Eksakt~planintegral:F$-Fc\n6#/F_
jrFf[bmFjinF$F$%-StokesFluxGoGf*F^jn6XFdvFevFfvFgvF.F/F0F2FhvFivFjvF,Fb]_l%'nor
maVGFceqF\wFbeqFdeqFeeqF]wF^wF_wF`wFawF^vFbwFcwFdwFewFfwFgwFhwFiwFjwF[xF\xF]xF^
xF_xF`xFaxFbxFfxFgxFhxFixFjxF[yF\fqF\yF]yF^yF_yF`yFayF$C*>Fj_mF]dn>FhgoFddn>Fgh
of*FeqF$FboF$FgjnF$F$6&F^wFj_mF_wFhgo>Fgdof*FeqF$FboF$F_[rF$F$6$F`wFgho>F__mFcc
_l>Feimf*FeqF$FboF$-F[gl6#-Fibl6%Fafm-FgglF]\oFdclF$F$6(F`wFghoF*F\pFbeqF__m>Fd
blf*F`dlF$FboF$F`a\lF$F$6&FhxFeimF+FA>Fbel-F[gl6#-F^hl6$-FdblF`dlF\il6#QZFinder
~fluxen~af~givet~vetorfelt~gennem~givet~fladestykkeF$F$F$%*fladeImGoGf*6'F*F+F\
cnF`]oFa]o6OFfxFfwFgwFhyFhwF.F/F0F_cnF`cnF2F^cnFhvFivFgcnFevFfvFgvFccnFdcnFc]oF
,FdvF_vF]^oF^^oF_^oF`^oFacnFfcnFecnF^wF_wF`wF^vFicnFa^oFb^oFc^oFd^oFe^oF]wFjcnF
jyF[zFayF$C,>FgyF]dn>Fc[lFddn>Fhjlf*FeqF$FboF$FgjnF$F$6&F^wFgyF_wFc[l>F\jmf*Feq
F$FboF$F^[\lF$F$6$F`wFhjl>F[enFgao>Fc`lf*FeqF$FboF$F\`^mF$F$Febo>F^fpf*FeqF$Fbo
F$Fa`^mF$F$6&F_^oFc`lF`^oF[en>Facof*FeqF$FboF$Fj`^mF$F$6*F*F\pF\cnFGFe^oF^fpF^v
F\jm>Fd`mf*F`dlF$FboF$F`a\lF$F$6&F_vFacoF+FA>F[jrFa_u6#QgnFinder~inertimomentet
~af~fladestykke~med~givet~'masset|aythed'F$F$F$%7traadModelFladeMedRandGf*F[dbm
F\vFayF$C;Fdy@%FgdbmC$>FezFG@%Fhz>Fc[lFd[lC$>Fc[lF%>FezFh[lC$>Fc[lF%>FezFh[l>F]
\lF`\l>Fb\lFe\l>F_rf*FiclF$FboF$FjclF$F$F]dl>Fgsf*F`dlF$FboF$FadlF$F$F]dl>Fftf*
FiclF$FboF$FfdlF$F$F]dl>F[elf*F`dlF$FboF$F]elF$F$F]dl>Fg\lFeebm>Fj]lf*F\^lF$Fbo
F$F\fbmF$F$Fa_l>Fc_lf*F\^lF$FboF$FgfbmF$F$Fa_l>Fh_lFi_l>Fc`l*&Fe`lF:Fh_lF:>F\\m
f*F`dlF$FboF$F^\mF$F$Fep>Fe\mf*FiclF$FboF$Fg\mF$F$Fep>F^]mF`]m>Fi]mFj]m>Fc^mFd^
m>F__mF`_m>Fj`mF[am>FhamFiam>F]bmF^bm>FbbmFcbm>FgbmFhbm-Fc\n6#-Fbr6%7$Fc^mFgbmF
g^mFj^m6#QGViser~totale~randkurve~og~fladestykke.F$F$F$%/rumIntApproxGoGf*F[cnF
]f\mFayF$C,>Fh_lFfz>FbjzFdg\m>F\aelFhg\m>F\h\mF]h\m>Fc`lFi^o>F[enFb_o>FhenFi_o>
Fgyf*F`_oF$FboF$F_`oF$F$F\\[m>Fc[lf*F`_oF$FboF$Fj`oF$F$F_\[m>FifpF^h]m6#Q_pFind
er~approksimation~til~rumintegralet~af~given~funktion~over~givet~rumligt~omr|`y
de.F$F$F$%*kurveCmGoGf*Fdj[lF_csFae\mF$C'>F8F[^p>F]of*F[flF$FboF$Fb^pF$F$F_gs>F
gpf*F[flF$FboF$FbgsF$F$Fegs>Fc^mFggs>F__mFhhs6#Q]pFinder~|+massemidtpunktet~af~
en~given~v|aygtet~|+rumkurve~~med~parameterfremstilling~rF$F$F$%+StokesFluxGf*F
iu6ZFdvFevFfvFgvF.F/F0F2FhvFivFjvF,Fb]_lFjfdmFceqF\wFbeqFdeqFeeqF]wF^wF_wF`wFaw
F^vFbwFcwFdwFewFfwFgwFhwFiwFjwF[xF\xF]xF^xF_xF`xFaxFbxF_vF[fqFfxFgxFhxFixFjxF[y
F\fqF\yF]yF^yF_yF`yF3F$CQ@%FazC$>FeyFfz@%Fdfq>FgyFd[lC$>FgyF%>FeyFh[lC$>FgyF%>F
eyFh[l>F_rFagq>FgsFegq>F8Figq@%F[iqC$>FfyFd[l>Fdjt-Fjz6$-F[al6$-F[al6$F`a_lFbjq
FgalF][lC$>FfyF%>FdjtF%>Fj_mF]dn>FhgoFddn>Fghof*FeqF$FboF$FgjnF$F$F`gdm>Fgdof*F
eqF$FboF$F_[rF$F$Fcgdm>F__mFcc_l>Fe\mf*FeqF$FboF$FfblF$F$Ffcl>F_]mf*FeqF$FboF$-
F<6#-Fibl6%-F__lF^clFcbemFdclF$F$6&F*F\pFbeqF__m@%3FdjtFfy>Fc[l-Fjz6$-F[al6$-F[
al6$-F<6#-Fe\mFahqFa\rFb\rF][l>Fc[lF%>Fiyf*FeqF$FboF$FggdmF$F$F\hdm>Ffaof*FeqF$
FboF$FafmF$F$6$FhxFiy>Fdblf*F`dlF$FboF$F`a\lF$F$6&FhxFiyF+FA>Fbelf*F`dlF$FboF$F
h^pF$F$6$F]wFdbl>FcjlFahdm@%33FfbemFgyFc[lCC>F]of*F\^lF$FboF$F^^lF$F$F]ar>Fgpf*
F\^lF$FboF$Fe_lF$F$F]ar>FcqFaar>Fh^oFcq>Fc^m-F`fl6#-F_^l6#-Fer6$-Fer6$F`cemFfhq
Fhhq>F\zf*F`dlF$FboF$F^\mF$F$Fep>Fgbmf*FiclF$FboF$Fg\mF$F$Fep>FbhoF`cr>FicrFjcr
>Fffm-F`fl6#-F_^l6#-Fer6$-Fer6$F^``lFfhqFhhq>F\zf*F`dlF$FboF$F^\mF$F$Fep>Fgbmf*
FiclF$FboF$Fg\mF$F$Fep>FbhoF`cr>FicrFjcr>F^fp-F_^l6$Fg`mFc^m>Fez-F_^l6$Fg`mFffm
>Fj]lFjdr>Fc_l-Fa_m6'Fc_mFbhlF\ilF^b`lFg^m>F[_pf*FeqF$FboF$-F\am6)-Fecm6#,&Fbcl
F:*(F`sF:FcbemF:FgglFEF:/F`s;Fcs*&F\dmF:F]dmF:/Fcam*&F`dmF:F]dmF:FbdmFddmFg^mFg
dmF$F$6*F*F\pFbeqF__mFeeqFezFbwFh^o>Fbeof*FeqF$FboF$-F\am6'Fdgem/F`s;*&F\dmF:F]
dmF:*&FcemF:F]dmF:/Fcam*&FfemF:,&*&F]dmF:FcemF:F:F`sFEF:FbdmFiemF$F$F]hem>F`cmf
*FeqF$FboF$F^fmF$F$6&FixF[_pFjxFbeo>Fh_l-Fbr6$7#-Fer6$-Fer6$-F`cm6%,&FDF:FchqF`
gm,&FQF:FehqF`gmF^hr/Fbt;FcsFea`l/F]s;FcsFi``lFg^m>FhjlFijr>F\\m-F^jm6(-F__mFaj
mF[[sFc[sF[\sF]\nF_\nFc\s-Fc\n6#-Fbr6'Fj\sFg^mFj^mF]]nF`]nFh_q-Fc\n6#-Fbr6'7$Fh
jlF[]nFg^mFj^mF]]nF`]n-Fc\n6#QCVektorfeltets~rotationsvektorfelt:F$-Fc\n6#-Fbr6
'Fg_qFg^mFj^mF]]nF`]n-Fc\n6#QDRotationsvektorfeltet~(ved~fladen):F$-Fc\n6#-Fbr6
'Fg]sFg^mF]]nF`]nFj^mFh]sF$F[_nF__n@%FgyF[_sF$Fb_nFe_s-Fc\n6#/.F^i\lF^i\l-F\_n6
#Q<Rotationen~af~V~(punktvis):F$-Fc\n6#/.-Fd]_lF\e`lFajem-F\_n6#Q:Rotationen~af
~V~i~r(u,v):F$-Fc\n6#/Fd\fm-F__mFihnF]`s-Fc\n6#/-.Fe`sFeq-FghoFeq-F\_n6#QFFluxe
n~af~rot(V)~gennem~fladestykket:F$-Fc\n6#/.-F\bs6$-F\bx6$Fe\fmFff\lF^bsF_egl-Fc
\n6#/Fi]fmFfbnF]g\l-Fc\n6#/Fi]fm-F\bs6$-F\bs6$*&.F\^fmF:F]\fmF:FbhlF\il-Fc\n6#/
Fi]fm-F\bs6$-F\bs6$.-F\bx6$Fe\fmFd`sFbhlF\il-Fc\n6#/Fi]fm-F\bs6$-F\bs6$-FiyFeqF
bhlF\ilF_ex-Fc\n6#/-F\bs6$-FfaoFeqFbhl&7#-F^hl6$Fa`fmF[pF\fx-Fc\n6#/F`fx-F^hlF`
`fm-Fc\n6#/-F\bs6$F_`fmF\il-F\bs6$-FbelF`dlF\il-Fc\n6#/F_afm&7#-F^hl6$FaafmF`oF
agx-Fc\n6#/F`fx-F^hlF`afmFjhxFf]fmF]ix6#Q[pFinder~og~viser~(hvis~muligt)~fluxen
~af~givet~vetorfelt~gennem~givet~fladestykkeF$F$F$%&rumCmGf*F[cn6MFgwFfwFhyFhwF
fxFdvFevFfvFgvFdcnF,FccnFc]oF.Fd]oF/F0F1F_cnF`cnFe]oF2F^cnFf]oFg]oF]wF^wF_wFh]o
Fi]oFhvFgcnF^vFecnFfcnFj]oFdiuFciuF[^oFjyF[zF\^oFbiuFayF$C>@%FazC$>Fh_lFfz@%Fcf
\m>FcqFd[lC$>FcqF%>Fh_lF^\nC$>FcqF%>Fh_lF^\n>FbjzFdg\m>F\aelFhg\m>F\h\mF]h\m@%F
e__lC%>F8Fd[l>F\zFdh\m>F]oF\j\m>F8F%@%F\av>FgpFhj\m>FgpF%@%FgpC$>F_jrFa[]m>F[jr
Fg[]mF$>Fc`lFi^o>F[enFb_o>FhenFi_o>Fgyf*F`_oF$FboF$F_`oF$F$F\\[m>Fc[lf*F`_oF$Fb
oF$Fj`oF$F$F_\[m>Fgdof*F_oF$FboF$Fb\[mF$F$Fc_el>Fh^of*FidoF$FboF$FjdoF$F$F[][m>
FifpF]][m@%5-Fjz6$F][^mFbhs4-Fjz6$F][^mF][l>F^fp7%-F[gl6#*&F^\`mF:FifpFE-F[gl6#
*&F_]`mF:FifpFE-F[gl6#*&F`^`mF:FifpFEF$@%F\efmC$>Fey,$Fj]lFj\t>Fc_l-F<6#-FecmFh
f\lC$>FeyFa^t>Fc_lFc^t@%F^fvCL>Fgbmf*F\^lF$FboF$FbfvF$F$Fh\]m>Fg\lf*F\^lF$FboF$
F]gvF$F$Fh\]m>Fj]lF_gv>F\\m-Fa_m6'7%,&FdjdlF:*(FeyF:FehpF:FghpF:F:,&FijdlF:*(Fe
yF:FehpF:F\ipF:F:,&&Fc_lFRF:*&FeyF:F`ipF:F:FaipFdipFgipFg^m>Fezf*F_oF$FboF$Fjhv
F$F$F_iv>Fdblf*FipF$FboF$FbivF$F$F_iv>Fj_mf*FeqF$FboF$FiivF$F$F_iv>Fbj_lFc]]m>F
dj_lFb^]m>Ffj_lF]_]m>FdjtFf_]m>Fc^mf*F_oF$FboF$F\]wF$F$Fep>F__mf*FipF$FboF$F`]w
F$F$Fep>Fffmf*FeqF$FboF$Fd]wF$F$Fep>Fb[yF``]m>Fg[yFh`]m>F\\yF`a]m>FfyFha]m>F\b]
mF]b]m>Fdb]mFeb]m>Fhb]mFib]m>F_c]mF`c]m>Fcc]mFdc]m>Fic]mFjc]m>FacoF]d]m>FfaoFC>
Fhcwf*F`_oF$FboF$FjcwF$F$Fgd]m>Fjdwf*F`_oF$FboF$-F]ew6$f*F`ewF$FboF$FaewF$F$Fie
wFjewF$F$F]e]m>F^fwf*F`_oF$FboF$F`fwF$F$F`e]m>F]gwf*F`_oF$FboF$-F]ew6$f*F`ewF$F
boF$FbgwF$F$FggwFhgwF$F$Ffe]m>F\hwf*F`_oF$FboF$F^hwF$F$Fie]m>Fbhwf*F`_oF$FboF$-
F]ew6$f*F`ewF$FboF$FghwF$F$F\iwF]iwF$F$F_f]m@%2,&F_jrF:F[jrFEFciw>Fhjlf*F`_oF$F
boF$FfiwF$F$Fejw>Fhjlf*F`_oF$FboF$FhjwF$F$6,Fj]oF^irF*F\pF\cnFGFdiuF[jrFciuF_jr
>F\jmFif]mF]]x-Fc\n6#-Fbr6&Fci]mFg^mF]]nFj^mFf]x-Fc\n6#-Fbr6&Fge[lFg^mF]]nFj^mF
dabm-Fc\n6#-Fbr6'7%FfyF\\mF[]nFj^mF`]nFg^mF]]n-Fc\n6#QSApproksimation,~net-finh
ed~og~antydet~masset|aythed:F$-Fc\n6#-Fbr6'7%F\jmF\\mF[]nFg^mF]]nF`]nFj^mF$F^_x
F__nF`j]mFc_xFf_xFbj]mFedcm-Fc\n6#/F`camFifp-Fc\n6#/F`camF][^m@%F\efmC$F[dt-Fc\
n6#/Fgf\lF\g\l@%F^efmC$F[dt-Fc\n6#Fgf\lFhdt6#Q]pFinder~og~viser~massemidtpunkte
t~for~givet~rumligt~omr|`yde~~med~givet~'masset|aythed'F$F$F$%3rumRandTotalVrid
GoGf*F^jnFadalF[jxF$C8>F_ayF`ay>FbayF`ay>FdayFeay>FgayFeay>FiayFjay>F\byFjay>F^
byf*F_oF$FboF$F`byF$F$F]dl>FcfalFdby>Fjbyf*F_oF$FboF$F\cyF$F$F]dl>FgfalF`cy>Ffc
yf*FipF$FboF$FhcyF$F$F]dl>F[galF\dy>Fbdyf*FipF$FboF$FddyF$F$F]dl>F_galFhdy>F^ey
f*FeqF$FboF$F`eyF$F$F]dl>FcgalFdey>Fjeyf*FeqF$FboF$F\fyF$F$F]dl>FggalF`fy>Fc`lf
*FgfyFjgalF]fqF$C.>F8F]dn>F]oFddn>Fgpf*FeqF$FboF$FgjnF$F$F[[o>F`rf*FeqF$FboF$F_
[rF$F$Fd[o>Fcqf*FeqF$FboF$FdhalF$F$Fhhy>F\zf*F`dlF$FboF$Fe^rF$F$F]ial>Fgbmf*F`d
lF$FboF$F^_rF$F$F]ial>Fg\lf*F`dlF$FboF$Fe_rF$F$F]ial>Fj]lFcial>Fc_lFiial>Fh_lF_
jalFdjalFgjalF$F$>Fgbmf*F`cwF$FboF$F[jyF$F$F[[bl>Fg\lFbjyFg\lFh]blF$F$F$
