SI.recall.geometgfsurfaessetli...
Transcript of SI.recall.geometgfsurfaessetli...
SI.recall.geometgfsurfaessetli_gE.GR?regularsurfaeuc1R2--imfeoning10 coinatecx.ME:
aiented.chwseaunitnormdvectwfeldvtfnstandse.coud fundamentdfumsg= 淳f.jdisdxwheneg.cn = <姿:
,
姿⼦ 〉
Isual īnner pwductofRA⼆
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
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
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
⇒ 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
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
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-
⼀
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
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 )
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*