SI.recall.geometgfsurfaessetli...

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SI.recall.geometgfsurfaessetli_gE.GR?regularsurfaeuc1R2-- imfeoning 1 0 coinatecx.ME : aiented.chwseaunitnormdvectwfeldvtfnstandse.co ud fundamentd fumsg = f.jdisdxwheneg.cn = < 姿 : , 姿Isual īnner pwductof RA hijdxsdi where hijlx ) = < , v > : 3 = - < , > = > 2 Gausscurvatureandmeancurvaturegisymmetnicandp.si tiuedefinitelinnerpwduc.tl Deuotetheinverseofoefficientmatnxby.fi ig gfk =_= Ìfgji l. , K = detg A = ( detgideth ) H = trg A = cfhjilsomebookhasafactwfksremi.IT heorema Egregium ) kdependsonymg.butntA.lGauss-BonnetJFcrchrsedE.SK = 2 X ( E ) Tgdidi

Transcript of SI.recall.geometgfsurfaessetli...

Page 1: SI.recall.geometgfsurfaessetli gE.GR?regularsurfaeuc1R2--homepage.ntu.edu.tw/~cjtsai/teaching/20g2/01.pdf · 㱺 類huhigik + 隋n.ee 器 + 孝 ⼆ 器 ⼀ 陆hej ⼀六年 hie ⼆

SI.recall.geometgfsurfaessetli_gE.GR?regularsurfaeuc1R2--imfeoning10 coinatecx.ME:

aiented.chwseaunitnormdvectwfeldvtfnstandse.coud fundamentdfumsg= 淳f.jdisdxwheneg.cn = <姿:

,

姿⼦ 〉

Isual īnner pwductofRA⼆

豪 hijdxsdi where hijlx) = <器, v >

△𦴢蕊蕊蕊鬣:嗡懸点 3= - < 器

,意 > = 器器 >

2

Gausscurvatureandmeancurvaturegisymmetnicandp.sitiuedefinitelinnerpwduc.tl

Deuotetheinverseofoefficientmatnxby.fiig的 前 gfk =_= Ìfgjil.識諁求 ,韔呦

K = detg A = (detgideth )H = trg A = 嘉cfhjilsomebookhasafactwfksremi.ITheorema Egregium )kdependsonymg.butntA.lGauss-BonnetJFcrchrsedE.SK些 = 2 元 X ( E )

Tgdidi

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SILeri-aitaconnectionandtheGdazz.iquatimThwughtheparametnization , 忘 is īdentifiedlvisudized as 姿i.ILevi-aitacmnect.in

T

烈樂州前 交迎㔇K

=䎚忘 培噶iiknn ⼆点 屋器⼆点噀 ⼀点 咸⼼ 忌

n-Deutthecoeficientbgdjich-R.ie三点 牡 (熱 + 發 ⼀ 愁

20 Thedualconnectiononthecotagentbundle.VE/R=sV*ElR.but ingeneral NOcanonicdisomwphismtfVcomeswithaninnerproduct.li⼀ 只R. (W ) = < w , 以

Inthecurrentcase.T.EETIV⼆点 ci我 的 Īlgajdi =Vbifl7#=副8的定 )忘 ⼀ É 7idxi = 7

Defne 7 n TT by7上 2 = (0 7

#jbmiNote that fòray fET(Mi R) ,

Jlfy ) = f 7 7 + dfz

Interms of component 套⾺ 写⼼ ) ⼆点 Zidx(巡 守 ; i

= 哭 ⼀点 百垓

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z

Inotherwords,

Jdxi = - E家 df就 ⼼ 0☆ Fnmnowon.usetheEinsteinsummatimconventimName.ly

, repeatedindieslusualyoneupper.me/ower)aresummed.3° Moregenerally ,

extend 7 to sectims of TMDTMbytheLeibniz rule :

否ldxbdxk ) = (套dxipdxktdxb (否!划= (⼀ ⾨主 dxyxodxktdixo ( ⼀ 陛 dxg

⇒ 套A - 7点出edisdxf =hkejidxkxodxewhevehkeii-217.ih.ge⼀ 限 hkj录imwefficieut.dxk.de

cht 各ig 三 0 ( g ìsisarallel"

)

ADecomposethedeniat.ve f 發j īn 录 īntotangentandnormdcomponents.andtkeonemwedenvat.ve剥器 = 尉丞 +hiiyf 筺熈訫.cn

2ㄨ 2xenwmal component :Gdazzi quatim

ㄑ 2,2> = I => 哭。上 2

录 = - hu䄩 器(Èi 硟識 + 器 ⼆處 啩…我的⼼如

= 1 .器,成 = ( 哭 + Thie ) 0

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⇒ 類 + 隋 n.ee 器 + 孝huhigik⼆ 器 ⼀ 陆 hej ⼀六年 hie⼆ 也幾.tt - ㄗ寺 hu) ⼀ 陆hejli= hkjji ⼆ hjkii

Gdazziequat.im hijik ⼆ hjkii

台 正 StokesldivegenetheuemmeansthechosendimM = n ← eetus dothegeneral caseoientatimg-gdxsdi.Iioniented.dxndx.in以 品⇒ dvol =

ITgdxn-ndxntfEECMJRJ.gnedientoffis7f.CH#I7fEXlM)df = 癸 dxi ⇒ 7 f = 8⽉ 炎 ⾼、

2 Thetrmal ) adj.int of -7 isthedivege.nuoperatrf,7--7f Defuedul

VJbyrquirmgeiMilRJ-n-AMjttniyc-tuyffdivMdv.lifefndeMMdiu( V )

(-1 V

Ufwithcompactsupportdivincordnate? V= "忘 ⼆袋

< 7f , V > = < g的 意 嘉 ,

vk, 忘 > = 季。我 意 比 ⼆ 意Üdiv

( V)fsf ,以 dwl =f 器 Vligdxindxn 11

⼆ Sf 意 (Ütgldxidxnff慶義代←_←) dvd

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cu_d.is Èg意 (ÜT ) = 器 + ǖvk = : ⼭ (以

ii) div ( V ) = trgl 。 - 7.V )

iii) fdivlv ) + <f ,V 〉 ) dwl = d ( Lcfndwl )

cntractann-tmAsacorollang.fdivlhdv.li0 by avectufieldM t VhascompactsuppwtreyFwm 7g 三 0

,

ii.!!!䃗SIV.firstvaiationalfrmuladefiui-Eiscalledaminimalsurf.ae

ifitisaaiticalstateoftheareafunctionalfnee.tn/Tbemonepecise.Ufaetiipy

⼆with compactsupport

Et = { p + tftp.up/peEYisalimmersed)suvfacefwtE(-E.E)minimal-武器到 ⼆ 0

(]F 必 , it) = Ilxl.is + tf⼼奶 2必⼒

程⼼ = <器 器 >

t.lt ) = < 器, 嘉 > +1 iaj ) = <選,器 > + ciasjs

稲⼼ = < 忘 (劃_),器 > + ( i - Ī )

= < 忘 ( f 2 ), 發。 >

t ( i - j )

- f < 意,發p + cìg) =

- 2f < 2.器 i >

= -2fhj

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Vollzt) = {nt-dxn.dxnadvantageofcoordinatecomputat.in : 忘 isclearcheckBct) : [0 , I ] → Glcni R)-

⻮ det Blti ) = detl Blti ) ⼤ ( 5化邀-_ tio

⇒ 龀。此回 ) = {sisjosdul-

úmeancuwatue⼆ f-fjhijdwlif-fHdu.ltEli

器 iiǚǘǜiriatimàformula

defin-E-dledaminimalsurfaeifHEOSV.laplace operatr

In R'w G

,

△ ⼀ 到忘了 appearsinphysics.complexanalysis.etc.IOFor f ETTI ; R ) ,

cmsider the Dirichlet energyEG ) = 17fidv.lt

cūticdstate ? Epeet Ei R) with compactsupport0 ⼆点hiftp ) ⼆点h { ( lfitztef.JP + ftp.Ddwl

⼆ 2 Scof.psdv.li -2SpdvcofdvolzEfom 3亚 ,

⇒ divl 7 f ) 三 0

dcfn-divlofjiscdledthelaplacianoff.andisdenoted.bg△ f

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h. locd coordiuate lfom S 亚 ),

△ f = Èg 录 (Tggi 著 )

Wheugij-Sij.it istheusudlaplacianas.in PDE

2° If 三 B compact wīthout boundang and Af = 0,

off Af dool = - Spfidnl ⇒ Jf E 0

⇒ fisaconstantfunu.im

30 zc.IR3

(⼼ , ⼼ w3) :standardcoordiuatefrR3-Icx.is

= ( wi,如 ,⼼⼼⼼

,

⼼必⾏ )

so-siuAssmoothfunctimsmlE.gl ,

△ Ǜ ⼆ Hvi-ci-thcomponent.fi )Pf △⼼ = H i ←→ △⼆ = H 2 ( wmpouentune act )

⇐) 〈 △⼆,

2 > = H

< △E.

2 > = <Ég 嘉 (Tgg吋意. ),

2

>_<gii器- + (濺? 2 >= < g的 (器吋 ,

2 > = _ A) = H*

ECIRBaminimdsurfaafaudmyiftherestric.fim of standardwrdinatefunctimsareharmmic functims (withrespect to g )

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Car-nThuesuochedlompactwithoutb.mn day )

minimalswfaceinlR.pfzsfso.lt⼼ = o => wi= cnstant

⇒ Iīs a censtant map →←

pf ⼟ ) If H = 0 =) K a 0

tneugA detgA relies m dim之 ⼆ 2

Butaclosedsurfae.in 1123 always has some

pwhereKp 〉 0*