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lFgar&Fj]lFearF]`tF$F$6(F0F^_qFhoF[^rF+Ffs6#Fdr6$7'FcuFhsF_tF_dpFfuFict>&F[xF\r -F[`l6#-F^x6$-F^x6$-FfblF[atF^atF`at>&F[xFcr-F[`l6#-F^x6$-F^x6$-FjdlF[btF\btF`a t>&F[xFat-Fabt6#-F^x6$-F^x6$-Fi_lFibtF\btF^at>Fcy-F[`l6#7%FeetF^ftFgft>&Fe^l6$F ]rF]r-Fhal6(-F[^r6%FgsF\qF^_qF]blFifsF\^l/F`bl7$,$FjsFaq,$F]`rFaqF_al>&Fe^l6$F] rFdr-Fhal6(-F[^r6%FesF\qF^_qF]blFifsF\^lFjgtF_al>&Fe^l6$FdrF]r-Fhal6(-F[^r6%F[q F`tF^_qF[blFifsF`_l/F`bl7$,$FasFaqF]htF_al>&Fe^l6$FdrFdr-Fhal6(-F[^r6%F[qF]tF^_ qF[blFifsF`_lF\itF_al>&Fe^l6$FaqF]r-Fhal6(-F[^r6%F[qF\qFfarF[blF]bl/F]^l%'yello wG/F`bl7$F^itF\htF_al>&Fe^l6$FaqFdr-Fhal6(-F[^r6%F[qF\qFdarF[blF]blF]jtF_jtF_al >F]]l-F[`l6%-F^x6$-F^x6$&Fe^l6$Fbfr8\q/FbfrFg\l/Fa[u;F]rFdrFbal/%%axesG%%noneG@ %-Fago6$F]rF]_rC$-Fdel6#Q9The~parametrized~domain:F$-Fdel6#-F[`l6'7$F]]l-%'base 3dGFi^rF_alFbal/%+lightmodelG%'light4GFe[uF$@%-Fago6$FdrF]_rC$-Fdel6#Q1The~vect or~fieldF$-Fdel6#-F[`l6'7%F^[tFd\uFcyF_alFbalFf\uFe[uF$@%-Fago6$FaqF]_rC$-Fdel6 #QUThe~rotation~vector~field~of~the~given~vector~field:F$-Fdel6#-F[`l6'7%F`]tFd \uFcyF_alFbalFf\uFe[uF$F$@%-Fago6$F_tF]_rC+-Fa[m6$QAParametrization:~r(u,v,w)~= ~%a~|+F$F]jr-Fa[m6$Q;Domain~of~definition:~%a~|+F$Ffs-Fa[m6$Q>Vector~field:~V(x ,y,z)~=~%a~|+F$Fa^s-Fa[m6#Q/Jacobi(u,v,w):F$-Fdel6#/.-F^\sFa^rFe_u-Fa[m6#Q;The~ rotation~vector~field:F$-Fdel6#/.-F`hs6#Fc^rFc]t-Fa[m6#QhnThe~total~rotation~of ~the~vector~field~over~the~given~domain:F$-Fdel6#/.-%$IntG6$F]`u/%#muG;&%&Omega G6#F[^r%!G-F^eq6#F`hsF$6#QdpFinder~og~viser~totale~rotation~(eng.:~total~vortic ity)~af~vektorfelt~paa|+rumligt~omraadeF$F$F$6#F\s6\q7'FcuFhsF]wFdrFfuFbhlFbhlF ifpFgfl7'FcuFhsF^glF\dmF`fl7'FcuFhsFhsFhgpFbflFcfl7'FcuFhsFhsF^glFbflFgflFiflFc flF\gl7'FcuFhsF^glFjflF`flF`hlF`hlF`hl7'FcuFhsFhsF\dmFbflFgflFgflFagl7'FcuFhsF \sF^glF`fl7'FcuFhsFhsF^hrF`fl7'FcuFhsFhs"#fF`flFhjqFhjqFhjq7'FcuFhsFhsF[\sF`fl7 'FcuFhsFhs"$r"F`fl7'FcuFhsFhsF_clF`fl7'FcuFhsFhs"$a"F`fl7'FcuFhsFhsF_gpF`flFfbu 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sF_alF$F$6$F`euF^`s6#Fdr6$7'FcuFhsF`[l"#=Ffu7'FcuFhsF`[lFhflFfu>Fi]sw,FjeoF$6$F htFitF$-Fibw6&Fe`v/F]^l-F^cw6&%$RGBG,&$F_dpFhsF]r-Fcx6#*&,&-Fj]l6#-Fi[p6#Fg`vF] rFhevFhsF]r,&T+F]rFhevFhsFhsFhs,&FbdwF]rFcdwFhs,&FbdwF]rFcdwFhs/%-transparencyG $F_yF_yF_alF$F$6,F`euF^`sFhoF\fuF\bqFeqF^cqF`hsF]cqFchs6#FdrFdcw>Ffr-F[`l6$7#-F ^x6$-F^x6$-Fi]sF_iu/Fahr;F_y,&FbguF]rFhsF]r/F_`r;F_y,&FeguF]rFhsF]rF_al@%-Fago6 $F]rFhcrC$-Fdel6#Q2Parameter~domain:F$-Fdel6#-F[`l6%7%FasFjs-%'base2dGFjpF_alFe [uF$@%-Fago6$FdrFhcrC$-Fdel6#QUThe~parametrized~surface~and~the~axis~of~revolut ion:F$-Fdel6#-F[`l6&7&F^`wFe`wF_jrFd\uF_alFe[uFbalF$@%-Fago6$FaqFhcrC$-Fdel6#Qi pApproximation~of~the~surface,~the~axis~of~revolution~and~an~indication~of~the~ weight~function:F$-Fdel6#-F[`l6'7%FfrF_jrFd\uF_alFe[uFf\uFbalF$F$@%-Fago6$F_tFh crC;-Fa[m6$Q?Parametrization:~r(u,v)~=~%a~|+F$Ff_vFh^u@%Fdt-Fa[m6$Q=Number~of~n et~elements:~%a~|+F$FfgsF$-Fa[m6$Q=The~density:~f(x,y,z)~=~%a~|+F$Fh^r-Fa[m6#Q- Jacobi(u,v):F$-Fdel6#/.-F^avFjpF^jw-Fa[m6$QJPoint~on~the~axis~of~revolution:~p~ =~%a~|+F$F^fu-Fa[m6$QHDirection~vector~for~the~axis:~e~=~%a~|+F$Fder-Fa[m6#QKTh e~distance~form~the~surface~to~the~axis:F$-Fdel6#/.-*$&%%distG6#%"lGFdrFjp-F^\s Fjp-Fa[m6#QXMoment~of~inertia~of~the~surface~around~the~given~axis:F$-Fdel6#/.- Fg`u6$-%".G6$FfgoF][x/Fj`u;&%"FG6#F\fuF_au-F^eq6#Ffir-Fdel6#/Fi[x-FcxFe\x-Fa[m6 #QFCalculation~of~the~moment~of~inertia:F$-Fdel6#/Fi[x-Fg`u6$-Fg`u6$*(-FfgoFi^r F]rFb[xF]rF]jwF]rF[blF]bl-Fdel6#/Fi[x-Fg`u6$-Fg`u6$*(-Feq6#-Fi[p6#Ff_vF]rFb[xF] rF^jwF]rF[blF]bl-Fa[m6#Q9-------INTERMEZZO:::::::F$-Fdel6#/-Fg`u6$-F_fsFjpF[bl& 7#-F\hq6$Fj^xF[q6#F[bl-Fdel6#/Fb^q-F\hqFi^x-Fdel6#/-Fg`u6$Fh^xF]bl-Fg`u6$-Fi\lF jclF]bl-Fdel6#/Fi_x&7#-F\hq6$F[`xF\q6#F]bl-Fdel6#/Fb^q-F\hqFj_x-Fa[m6#Q9------- CONCLUSION:::::::F$Ff[x-Fa[m6#Q<===========================F$F$6#QaoFinder~og~v iser~inertimomentet~af~fladestykke~med~givet~'massetaethed'F$F$F$6#Fdu6hq7'FcuF 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r/F[q;F_y%#PiG/F\q;,$FdfyFhsFdfy/F`bl7$FduF_]mF_al>Fcyw,F\vF$6$FhtFitF$Fa]wF$F$ F_^t6#F]rFf]w>Fe[lw,FftF$6$FhtFitF$F[^wF$F$F__t6#F]rF_^w>Fas-Fabt6#-F^x6$-Fcy6# F]gx/Fibv;F_yF_ex>Fjs-Fabt6#-F^x6$-Fe[l6#Fjfx/F\gx;F_yF\ex>Fjdlw,F\vF$6$FhtFitF $F`]lF$F$6(F.F[qFhoFfirF+Ffs6#F]rFa^l>F[xw,FftF$6$FhtFitF$Fh^lF$F$6(F/F\qFhoFfi rF+Ffs6#F]rFd_l>F^`w-F[`l6#-F^x6$-FjdlFhgyFigy>Fe`w-F[`l6#-F^x6$-F[xFahyFbhy>F] ]lF\aw>Fhdsw,FjeoF$6$FhtFitF$FbawF$F$6*FhoFfirFKFi`sF@FfblFLF^`s6#FdrF^bw@%2,&F _fsF]rFi\lFhsFcbw>F`asw,FjeoF$6$FhtFitF$FhbwF$F$6$F`euFhds6#FdrFdcw>F`asw,FjeoF $6$FhtFitF$-Fibw6&Fe`v/F]^l-F^cw6&F`dw,&FbdwF]rFcdwFhs,&FbdwF]rFcdwFhs,&FbdwF]r FcdwFhsF_ewF_alF$F$6,F`euFhdsFhoFfirF\bqFeqF^cqFi\lF]cqF_fs6#FdrFdcw>F[ds-F[`l6 $7#-F^x6$-F^x6$-F`asFifx/F\gx;F_y,&F\exF]rFhsF]r/Fibv;F_y,&F_exF]rFhsF]rF_al@%- Fago6$F]rF_jrC$FgfwFjfwF$@%-Fago6$FdrF_jrC$-Fdel6#QaoThe~parametrized~surface~w ith~center~of~mass~(if~it~is~welll~defined):F$-Fdel6#-F[`l6'7&F^`wFe`wF`]tFd\uF _alFe[uFf\uFbalF$@%-Fago6$FaqF_jrC$-Fdel6#QIApproximation~with~given~weight~and ~net:F$-Fdel6#-F[`l6'7&F[dsF]]lF`]tFd\uF_alFe[uFf\uFbalF$F$@%-Fago6$F_tF_jrC,-F a[m6$F_iwFfjxFh^u-Fa[m6$FciwFchs-Fa[m6$QHGiven~density~function:~f(x,y,z)~=~%a~ |+F$Fh^rFgiw-Fdel6#/.Fj`yFj`y-Fa[m6#Q,Total~mass:F$-Fdel6#/.-Fg`u6$Ffgo/Fj`u;&F b\xFe\xF_au-F^eqF`_y-Fdel6#/Fe_zF__y@%F]_yC$-Fa[m6#Q1Center~of~mass:~F$-Fdel6#/ -F^eqFjdy-FcxFjdy@%Fc_yC$Fa`z-Fdel6#Fg`z-Fa[m6#QGThe~center~of~mass~is~not~well ~definedF$F$6#QeoFinder~og~viser~fladeintegralet~af~'massetaetheden'~over~givet ~fladestykkeF$F$F$6#F\s6gpFbaxFcaxFifpFgflFgauFaflFcflFeflFgflFiflF[glF\glF]glF eglF`hlF`hlFeaxFaglFffpF^dpFgaxFgaxF[epFhaxFhaxFiaxFjaxF`hlFdfpFhgl7'FcuFhsFhs" $J"F`flF^dpFbdpF[epF^dpF^hlFhglFbbxFbbxFcdpFjgp7'FcuFhsFhsFfflF`flF_cpF`dpFcfl7 'FcuFhsFhs"$R"F`flFebxFfbxFeglFeglFiglFjglF[hlF[hlF]hlFccpFjgpFgbxFgbxFgbxF^dpF hbxF\duF[epF^dp7'FcuFhsFhs"#xFeu7'FcuFhsFhs"$<"FeuF[epF^dpF]cxF]duF[epF[epF^dpF 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r-F[alFcrF]rF\gsF]r;,&F\gsFhs-Fi[l6$F_y-FfalFatF]r,&-Fhy6$F]r-F[alFatF]rF\gsF]r >F_fs-F`[t6)Fg^]n/FfbmFa[^n/F_\oFj[^n/F`_oFc\^n/F`blF__rF`][lF\]t>F^av7#-F^x6$- F[`l6$7&Fd\uF_jrF]]lF[[^nFe[u/Fccq;F]rFf_l>F^_pF_jr@%-Fago6$F]rFgasC$-Fdel6#Q8T he~parametrized~curve:F$-Fdel6#-F[`l6(7$FibvFd\uF_alFbalFf\uFe[u/F_a[lFhfuF$@%- Fago6$FdrFgasC$Fggel-Fdel6#-F[`l6(7%F_fsF_jrFd\uF_alFbalFf\uFe[uFi^^nF$@%-Fago6 $FaqFgasC$-Fdel6#QenThe~corresponding~integral~curves~through~the~given~curve:F $-Fdel6#-F[`l6(7%Fd\uF_jrF]]lF_alFbalFf\uFe[uFi^^nF$@%-Fago6$F_tFgasC$-Fdel6#QZ The~forward~directed~flow~deformation~of~the~given~curve:F$-Fdel6#-F[`l6(7&Fd\u F]]lF_jr&F]gr6#Ff_lF_alFbalFf\uFe[uFi^^nF$@%-Fago6$F\sFgasC$-Fdel6#QdoAnimation ~of~the~deformation~via~the~integral~curves~of~the~vector~field:F$-Fdel6#-F[`l6 )F^av/%+insequenceGFgrF_alFbalFf\uFe[uFi^^nF$F$@%-Fago6$F`[lFgasC+-Fa[m6$QJPara metrization~of~the~curve:~r(u)~=~%a~|+F$F\a]nFh^u@%Fdt-Fa[m6$FciwFi\lF$-Fa[m6$F 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FbdwF]rFf`gnFhsF_alFbalF$F$62FhoFccqF1F`^mFKFjdlF@FfalF?F[dsF\bqFeqF^cqFi]sF]cq Ffr6#F]r6$7'FcuFhsF`[lFcg\lFfuFb[l>Ffbl-Fa]l6&Fa^gnF[bl/%*numpointsG"$+&F][`m>F cy-F^x6$-F[xF^gfn/F]grF\jq@%-Fago6$F]rFgjrC$F\dam-Fdel6#-F[`l6%F]c^lF_alFbalF$F $@%33F`[gnF^_pFi_lC&>Fi\l-Fhy6#-F^x6$-Fbeam6#-Fcx6#-Feq6#-Fi[p6#-Fccq6#,&FgsF]r F`gfnF]fam/F]gr;F_yFg[gn>F\gs-Fhbl6*FgdqFfeglFe[gn/Facl,$*(F[dsF]r-Fhy6$F_y-Feq 6#-Fi[p6#FgdqF]rFi\lFhsF]famF[i[lF][`mF_alFbal>F_fs-Fhbl6*FgdqFfeglFe[gn/Facl,$ *(F[dsF]r-Fi[lFcdgnF]rFi\lFhsFcgamF[i[lF]jtF_alFbal@%-Fago6$FdrFgjrC$Fdgam-Fdel 6#-F[`l6%7&FfgsF_fsF\gsFd\uF_alFbalF$F$@%-Fago6$FaqFgjrC$-Fdel6#QSApproximation ,~net,~and~indicated~weight~function:F$-Fdel6#-F[`l6'7%FcyFfblFd\uF_alFe[uFf\uF balF$@%-Fago6$F_tFgjrC,-Fa[m6$FbhamFa^gnFh^u@%Fa_r-Fa[m6$FciwFe^lF$FghamFjhamF] iam-Fa[m6#Q=Approximated~curve~integral:F$-Fdel6#/Fdt-Fcx6#Fdt-Fa[m6#Q6Exact~cu rve~integral:F$-Fdel6#/Fdgq-Fcx6#FdgqF$F^jqF$F$6#F\s6doF\g[lFcjqFifpFgflFgauFa] bmFcflFb]bmFgflFiflF`dpF\glFc]bmFcj`mF`hlFhjqFaglFfgpF^dpFhjqFhjqF[epFdjqFejqFc dpFfjqFhjqF^dpFcg^lFbglFeglFeglFfglF`hlFbcuFjgpF^bu7'FcuFhsFhsFcg\lF`flFejqFebx F^dpFebclFgcxF[epF[epFjgpFcdpFbbxFidpF^dpFbduFgcxF[epF[epF^dpF^gpF]duF[epF_i[mF fhpFggpFh]bmFj]bmF[epFfcxFebclF[^bmFhg^lFgg[lF\gpFbhpF[ep%(rumFlowGw,6)F9F+F][r F\[pF,>F_[r7+F]rFdrFaqF_tF\sF`[lFduFc[lFhu>FfbzFhbz6doF.F/F0F1F2F3F4FTFSFebqFUF ^f\nF_f\nF`f\nFaf\nFbf\nFcf\nFdf\nFef\nFff\nFgf\nFhf\nFif\nFjf\nF[g\nF\g\nF]oF< F=F>F?FWFGFdbqFcbqF]g\nF^g\nF_g\nFg\\lFfg\nFgg\n%&siderGF\pFag\nFbg\nFcg\nFdg\n FHFVF@Fb[rFAFBFc[rFh[rFg[rFf[rFCFe[rFa[rFd[rFDFg\rFhoFjhoFe]rFioFjoFf]r%"dGFhg \nF[p6#Fjg\nF$C9>FedflF\^r>FapFd^r@%F_h\n>F\yFdhgn>F\yF]gu@%Fch\nC$>F_fsF__r@%3 3-Fiq6$F`^ilF^r-Fiq6$F`_ilF^r-Fiq6$F\`ilF^r>FdgqFgrC$>FdgqF%>F_fsFc`rC$>FdgqF%> F_fsFc`r>&F\gxF\r-Fcx6#*&FdsF]rF`^ilFhs>&F\gxFcr-Fcx6#*&F\tF]rF`_ilFhs>&F\gxFat -Fcx6#*&FcarF]rF\`ilFhs@%Fiar>Ff\pFgr>Ff\pF%>F`]t7#-F^x6$-F^x6$-F^x6$-Fcx6#-Fed 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FitF$Fej\lF$F$F_^t6#FdrFhj\l>F\fuw,Ff^tF$6$FhtFitF$F][]lF$F$F__t6#FdrF`[]l>F^fu w,FjeoF$6$FhtFitF$Fe[]lF$F$F_`t6#FdrFh[]l>&FhcrF\r-F[`l6#-F^x6$-F^x6$-F_c\lFfeh nFi\hnFiehn>&FhcrFcr-F[`l6#-F^x6$-F^x6$-F\fuFbfhnFg\hnFiehn>&FhcrFat-Fabt6#-F^x 6$-F^x6$-F^fuF[ghnFg\hnFi\hn>Fahr-F[`l6%7%F\cinFecinF^dinF_alFbal>Fdao-Fccq6%F] ]lFfsF_fs>Fg_y7#-F^x6$-F[`l6$7&Fd\uF^[tFbsFb]inFe[u/Fgjr;F_yFeu>FdqF^[t@%Fa^`mC $F\\u-Fdel6#-F[`l6(7$F_`rFd\uF_alFbalFf\uFe[uFi^^nF$@%F]_`mC$F\\u-Fdel6#-F[`l6( 7$F^[tFd\uF_alFbalFf\uFe[uFi^^nF$@%Ff_`mC$Fggel-Fdel6#-F[`l6(7%F\gsFdqFd\uF_alF balFf\uFe[uFi^^nF$@%Fa``mC$Fggel-Fdel6#-F[`l6(7%F\gsF_`rFd\uF_alFbalFf\uFe[uFi^ ^nF$@%F]a`mC$-Fdel6#QfnThe~corresponding~integral~curves~through~the~given~doma in:F$-Fdel6#-F[`l6(7%Fd\uF^[tFbsF_alFbalFf\uFe[uFi^^nF$@%Fia`mC$-Fdel6#QZThe~fo rward~directed~flow-deformation~of~the~given~domainF$-Fdel6#-F[`l6(7&Fd\uFbsF_` rFdaoF_alFbalFf\uFe[uFi^^nF$@%-Fago6$FduF\yC$-Fdel6#QZStart~and~end~of~the~defo 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a\n>Fjj[lw,FjeoF$6$FhtFitF$F_dhlF$F$F\^\n6#FdrFe_hl>Fha\nFia\n>Fhdsw,F\ehlF_b\n 6#Fab\nF$C'>F[qFcehl>F\qF\fhl>F^_qFfb\n>F`^mw,FjeoF$6$FhtFitF$F_c\nF$F$Fbhhl6#F drFec\nFhc\nFcd\nF$Fa[il6#F_tFfd\n>Fjdlw,Fj[ilF$6$FhtFitF$F\\ilF$F$F]e\n6#FdrF_ e\n>F[xFf\il@%F^]ilC8>F`hsw,Ff]tF$6$FhtFitF$FactF$F$6(F.F[qFhoFg_yF+Ffs6#FdrFgc t>Ffgsw,Ff^tF$6$FhtFitF$F]dtF$F$6(F/F\qFhoFg_yF+Ffs6#FdrFcdt>F\gsw,FjeoF$6$FhtF itF$FhdtF$F$6(F0F^_qFhoFg_yF+Ffs6#FdrFbet>F`^il-F[`l6#-F^x6$-F^x6$-F`hs6$Fa_boF i^il/F`fvF\_ilF]_il>F`_il-F[`l6#-F^x6$-F^x6$-Ffgs6$F__boFi^il/F[gzFj_ilF]_il>F \`il-Fabt6#-F^x6$-F^x6$-F\gs6$F__boFa_boF_eboFedbo>F[dsFf`il>Fj`il-Fhal6(-Fg_yF igtF]blFifsF\^lF^ailF_al>Fcail-Fhal6(-Fg_yFdhtF]blFifsF\^lF^ailF_al>Fhail-Fhal6 (-Fg_yF[itF[blFifsF`_lF\bilF_al>F`bil-Fhal6(-Fg_yFeitF[blFifsF`_lF\bilF_al>Febi l-Fhal6(-Fg_yF\jtF[blF]blF]jtFibilF_al>F\cil-Fhal6(-Fg_yFgjtF[blF]blF]jtFibilF_ al>Fdt-F[`l6%-F^x6$-F^x6$&Fafq6$F_c\lFjez/F_c\lFg\l/FjezFd[uFbalF[e]l>Fbsw,FeyF $6$FhtFitF$FcisF$F$6(FF]]lw,FeyF$6$FhtFitF$FbjsF$F $F`hbo6#F]rF]js>Fe^l-Fhy6#-F^x6$,&-FbsFc\xF]r-F]]lFc\xFhsFaj\l>F]gr*&FacwF]rFe^ lF]r>Fi_l-F`[t6(-FfirFi^rF`eilFheilF`filF[]tF\]t@%Fe\zC$F\\uF\gilF$@%Fi\zC$F[ge l-Fdel6#-F[`l6'F]dimFe[uFf\uF_alFbalF$@%Fe]zC$F^hilFahilF$F$@%Fa^zC2-Fa[m6$Fg^u Ff^\nFh^u-Fa[m6$F^iilFciboF_iil-Fdel6#Q]oCalculation~of~the~fluxes~OUT~of~the~r espective~boundary~surfaces:F$F_iil-Fa[m6%Q>Partial~flux~[1,1]:~%a~=~%a~|+F$Fhi ilFiiil-Fa[m6%Q>Partial~flux~[1,2]:~%a~=~%a~|+F$Fh\ilF`jil-Fa[m6%Q>Partial~flux ~[2,1]:~%a~=~%a~|+F$FgjilFhjil-Fa[m6%Q>Partial~flux~[2,2]:~%a~=~%a~|+F$Fj\ilF_[ jl-Fa[m6%Q>Partial~flux~[3,1]:~%a~=~%a~|+F$Ff[jlFg[jl-Fa[m6%Q>Partial~flux~[3,2 ]:~%a~=~%a~|+F$F\]ilF^\jlF_iil-Fdel6#QZThe~total~flux~OUT~through~the~boundary~ surfaces~is~thus:F$F_iil-Fdel6#/Fh\jl-F^eq6#-FcxF[]jlF$Fde\nF$F$6#F\s6\qF]_^mFb hlFbhlFifpFgflFgauFhauFcflFiauFgflFiflFcflF\glFjauF`hlF`hlF`hlF[buFgflF\buF[b`l 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e~parametrized~image~domain~with~center~of~mass~(if~it~is~welll~defined):F$-Fde l6#-F[`l6%7&Fge_lF^f_lF][`lFfalF_alFe[uF$@%-Fago6$FaqFccqC$-Fdel6#QRApproximati on~with~given~net~and~weight~function:F$-Fdel6#-F[`l6%7%F_fsF\gsF][`lF_alFe[uF$ F$@%-Fago6$F_tFccqC0-Fa[m6$F_iwFj`yFh^uFf^z-Fa[m6$QFGiven~density~function:~f(x ,y)~=~%a~|+F$Fd\`lFgiw-Fdel6#/.Fb`hoFb`hoFdacm-Fdel6#/F^`sFg^ho-Fa[m6#Q9Calcula tion~of~the~mass:F$-Fdel6#/.-Fg`u6$Ffgo/Fj`u;&Fb\xFg]`lF_au-Fg`u6$-Fg`u6$*&F^`h oF]rFe[ioF]rF[blF]bl-Fdel6#/F_\io-Fg`u6$-Fg`u6$*&F^`hoF]rFb`hoF]rF[blF]bl-Fdel6 #/F_\io-Fg`uF^a_l-Fdel6#/F_\ioF^`s@%Fe^hoC$Fa`z-Fdel6#/-F^eqFhaho-FcxFhaho@%Fi^ hoC$-Fa[m6#Q2Masse-midtpunkt:~F$-Fdel6#F^^ioF]azF$6#QfrFinder~og~viser~(hvor~mu ligt)~planintegralet~af~|+'massetaetheden'~ff(x,y)~over~givet~plant~omraade,~ek sakt~og~approksimeretF$F$F$6#F\s6fpFibemFcaxFifpFgflFgauFaflFcflFeflFgflFiflF[g lF\glF]glFeglF`hlF`hlFeaxFaglFffpF^dpFgaxFgaxF[epF[b`lF[b`lF\b`lF[dpF_cpFdazF^d 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'mlet23G%'mlet33GF]jlFb^jl6#Fg^jlF$C'@%3Fj^jl/FfsFaqC&>F[q-Fb^m6#-Fg^m6$/F[_mQ8 Parametrization~requestF$7#7#7%-F`_m6$-Fd_m6#QMPlease~enter~a~parametrization~o f~the~curve:F$Fh_m7%7$-F`_m6$-Fd_m6#Q>1.~coordinate~function~of~u:~F$Fb[qFe[q7$ -F`_m6$-Fd_m6#Q>2.~coordinate~function~of~u:~F$Fb[qFc\q7$-F`_m6$-Fd_m6#Q>3.~coo rdinate~function~of~u:~F$Fb[qF^]qFc]q>FfbmFi]q@%55/7#&FfbmF\rFhbjl/7#&FfbmFcrFh bjl/7#&FfbmFatFhbjl>F_\oF]r>F_\oF_y@%Fgf[mO-Fcp6$-Fbap6#7%-Fhjl6#F`eio-Fhjl6#Fc eio-Fhjl6#FfeioF.F$F$@%3F[ejl/FfsFdrC&>F\q-Fb^m6#-Fg^m6$F]cio7#7#7%-F`_m6$-Fd_m 6#QUPlease~enter~a~parametrization~of~the~planar~domain:F$Fh_m7$7$-F`_m6$-Fd_m6 #QD1.~coordinate~function~of~u~and~v:~F$Fb[qFe[q7$-F`_m6$-Fd_m6#QD2.~coordinate ~function~of~u~and~v:~F$Fb[qFc\q-Fb`m6%Fbdm-Fg`m6#7$Fj[qFf\qFi`m>FfbmFdijl@%F]e io>F_\oF]r>F_\oF_y@%Fgf[mO-Fcp6%-Fbap6#7$F`fioFbfioF.F/F$F$@%3F[ejlFfbioC&>F^_q -Fb^m6#-Fg^m6$F]cio7#7#7%-F`_m6$-Fd_m6#QOPlease~enter~a~parametrization~of~the~ surface:F$Fh_m7%FhgioF^hio7$-F`_m6$-Fd_m6#QD3.~coordinate~function~of~u~and~v:~ F$Fb[qF^]qFc]q>FfbmF^b[m@%F\eio>F_\oF]r>F_\oF_y@%Fgf[mO-Fcp6%F]fioF.F/F$F$@%3Fb \[mFfbioC&>F`^m-Fb^m6#-Fg^m6$F]cio7#7#7%-F`_m6$-Fd_m6#QQPlease~enter~a~parametr ization~of~the~3D~domain:F$Fh_m7%7$-F`_m6$-Fd_m6#QH1.~coordinate~function~of~u, ~v,~and~w:~F$Fb[qFe[q7$-F`_m6$-Fd_m6#QH2.~coordinate~function~of~u,~v,~and~w:~F $Fb[qFc\q7$-F`_m6$-Fd_m6#QH3.~coordinate~function~of~u,~v,~and~w:~F$Fb[qF^]qFc] q>FfbmF^am@%F\eio>F_\oF]r>F_\oF_y@%Fgf[mO-Fcp6&F]fioF.F/F0F$F$@%/F_\oF]rC$Fc_q- FjdoFf`vF$6#QBCall~for~input~to~r~and~check~it.F$F$F$6#Fdr6OF\aqF^dpF]aqFdjq7'F cuFhsFhsFe]bmFbfl7'FcuFhsF[\sFegpF`fl7'FcuFhsFf]jlFg]bmF`flF^dp7'FcuFhsFhsFegpF _ipF[epF[epF^dpF]aqFdjqF\h[mF]h[mF^h[mF^dp7'FcuFhsFhsF\hlF_ipF[epF[epF^dpF]aqFd jqFf^joFg^joFh^joF^dp7'FcuFhsFhsFagpF_ipF[epF[epF^dpF]aqFdjqFf^joFg^joFh^joF^dp 7'FcuFhsFhsFf]jlF_ipF[epF[epFjgpFfaqFj`\mF[ep%(kurveImGw,6)F9F+F\bqFiduFjduF,>F 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