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Submitted on 12 Nov 2007

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Mesh Parameterization: Theory and PracticeKai Hormann, Bruno Lévy, Alla Sheffer

To cite this version:Kai Hormann, Bruno Lévy, Alla Sheffer. Mesh Parameterization: Theory and Practice. This documentis the support of a course given at SIGGRAPH 2007. 2007. <inria-00186795>

r♣ ♦rs ♦ts

s Pr♠tr③t♦♥ ♦r② ♥ Prt

♦r♠♥♥

st ❯♥rst②

♦ ♥♦♦②

r♥♦ é②

r

❯♥rst②

♦ rts ♦♠

st

♠♠r②

s ♣r♠tr③t♦♥ s ♣♦r ♦♠tr② ♣r♦ss♥ t♦♦ t ♥♠r♦s ♦♠♣trr♣s ♣♣t♦♥s r♦♠ t①tr ♠♣♣♥ t♦ ♥♠t♦♥ tr♥sr s ♦rs ♦t♥sts ♠t♠t ♦♥t♦♥s srs r♥t ♠t♦s ♦r ♣r♠tr③♥ ♠ss ♦rr♦s ♦♠♥s ssss ♠r♥ t♦♦s ♦ ♣r♠tr③t♦♥ ♥ ♥trsr♠♣♣♥ ♥ ♠♦♥strts rt② ♦ ♣r♠tr③t♦♥ ♣♣t♦♥s

Prrqsts

♥ s♦ s♦♠ ♣r♦r ①♣♦sr t♦ ♠s r♣rs♥tt♦♥ ♦ ♦♠tr♠♦s ♥ ♦r♥ ♥♦ ♦ t♦r s ♠♥tr② ♥r r ♥ t♥♠♥ts ♦ ♦♠♣tr r♣s ♣t♦♥ ♣rrqsts s♦♠ trs ♠② s♦ss♠ s♦♠ ♠rt② t r♥t ♦♠tr② r♣ t♦r② r♠♦♥ ♥t♦♥s♥ ♥♠r ♦♣t♠③t♦♥

♥t♥ ♥

rt st♥ts rsrrs ♥ ♣♣t♦♥ ♦♣rs ♦ s t♦ ♥rst♥ t♦♥♣ts ♥ t♥♦♦s s ♥ ♠s ♣r♠tr③t♦♥ ♥ s t♦ t③ t♠ st♥rs t ♥ ♦r ♦ t s♣tr♠ ♦ ♣r♦ss♥ ♣♣t♦♥s tt ♥t r♦♠♣r♠tr③t♦♥ ♥ r♥ ♦ t♦ t r♥t ♠t♦s ♥ tr♠s ♦ s♣ ♣♣t♦♥ rqr♠♥ts

♦rs

s ♥♦ts r r② s ♦♥ t ♦♦♥ sr② ♣♣rs ♥ ♦♥ ♦♥ tt♦r ♣s

• ♦tr ♥ ♦r♠♥♥ r ♣r♠tr③t♦♥ tt♦r ♥ sr② ♥ ♥s ♥ trs♦t♦♥ ♦r ♦♠tr ♦♥ t♠ts ♥❱s③t♦♥ ♣s ♣r♥r

• r Pr♥ ♥ ♦s s ♣r♠tr③t♦♥ ♠t♦s ♥ tr♣♣t♦♥s ♦♥t♦♥s ♥ r♥s ♥ ♦♠♣tr r♣s ♥ ❱s♦♥

• é② Pr♠tr③t♦♥ ♥ ♦r♠t♦♥ ♥②ss ♦♥ ♠♥♦ ♥r♣♦rt tt♣♦rr♣t♦♥s

• ♦r ♦ r♣t ♣♥ ♥ ♦♥ r♦♠ tt♣♦r

rs♦tr

♣♣♠♥t ♠tr ♥ ♣rs♥tt♦♥ ss r s♦ ♦♥ t ♣rs♥tr ♣s

♦rs r

♦r ♥② t♦ srs t s♠r t♦♣♦♦② tr ①sts t ♠♣♣♥ t♥t♠ ♦♥ ♦ ts srs s tr♥r ♠s t ♣r♦♠ ♦ ♦♠♣t♥ s ♠♣♣♥ s rrr t♦ s ♠s ♣r♠tr③t♦♥ sr tt t ♠s s♠♣♣ t♦ s t②♣② t ♣r♠tr ♦♠♥ Pr♠tr③t♦♥ s ♥tr♦t♦ ♦♠♣tr r♣s ♦r ♠♣♣♥ t①trs ♦♥t♦ srs r t st t sr② ♦♠ qt♦s t♦♦ ♦r ♠♥② ♠s ♣r♦ss♥ ♣♣t♦♥s ♥♥t♠♣♣♥ ttr♥sr ♠♦r♣♥ ♠st♥ ♠s♦♠♣t♦♥ r♠s♥♦♠♣rss♦♥ srtt♥ ♥ s♣♥②ss ♥ ♣r t♦ t ♥rs ♥trst ♥♣♣②♥ ♣r♠tr③t♦♥ r♦s ♠t♦s r ♦♣ ♦r r♥t ♥s ♦ ♣r♠tr ♦♠♥s ♥ ♣r♠tr③t♦♥ ♣r♦♣rts

♦ ♦ ts ♦rs s t♦ ♠r③ t ♥ t t t♦rt ♥ ♣rts♣ts ♦ ♠s ♣r♠tr③t♦♥ ❲ ♠ t♦ ♣r♦ t ss ♥ t♦ ♠♣♠♥t ♦r♠♣r♦ ①st♥ ♠t♦s t♦ ♥stt ♥ ♣♣r♦s ♥ t♦ rt② t tstt② ♦ t t♥qs ♦r ♣rtr ♣♣t♦♥

♦rs strts t ♥ ♥tr♦t♦♥ t♦ t ♥r ♦♥♣t ♦ ♣r♠tr③t♦♥♥ ♥ ♦r ♦ ts ♣♣t♦♥s rst ♦ t ♦rs t♥ ♦ss ♦♥ ♣♥r ♣r♠tr③t♦♥s t s♦♥ rsss ♠♦r r♥t ♣♣r♦s ♦r tr♥t♦♠♥s ♦rs ♦rs t ♠t♠t r♦♥ ♥♥ ♥tt ①♣♥t♦♥s ♦ ♣r♠tr③t♦♥ ♣r♦♣rts tt② ♦♥♦r♠t② strt ♥ r♣rsrt♦♥ stt♦trt s r ② ①♣♥♥ t ♠♥ s ♦ sr♣♣r♦s s♠♠r③♥ tr ♣r♦♣rts ♥ strt♥ t♠ s♥ ♠♦s ♦rs rsss ♣rt s♣ts ♦ ♠♣♠♥t♥ ♠s ♣r♠tr③t♦♥ sss♥♥♠r ♦rt♠s ♦r r♦st② ♥ ♥t② ♣r♠tr③♥ r ♠ss ♥ ♣r♦♥ t♣s ♦♥ ♦ t♦ ♥ t rt② ♦ ♦t♥ ♦♥trt♥ rtr ♥ ♦♦s♥ ♥♣♣r♦♣rt ♣r♠tr③t♦♥ ♠t♦ ♦r s♣ trt ♣♣t♦♥ ❲ ♦♥ ②♣rs♥t♥ st ♦ ♦♣♥ rsr ♣r♦♠s ♥ ♣♦t♥t ♣♣t♦♥s tt ♥ ♥tr♦♠ ♣r♠tr③t♦♥

♣rs

t sr♥ t ❯

t sr♥ s ♥ ss♦t ♣r♦ss♦r t t ♦r♥ ♥sttt ♦ ♥♦♦②t ♥ t ♦♠♣tr ♥ ♣rt♠♥t tr r♥ s P r♦♠ tt♦♥ P♦②t♥ ♥sttt ♦ r♥♦ P s♣♥t t♦ ②rs s ♣♦st♦t♦rrsrr t t ♦r strt♥ rsr r♦♣ t t ❯♥rst② ♦ ♦tr♥♦r♥ ❯ s rsr ♥trsts r♦ r♦♥ ♣♣②♥ srt r♥t♦♠tr② r♥t ②t r②srt③ ♦♠♣tt♦♥ ♦♥t♦♥s t♦ r♥ ♦ s ♥ ♣♣t♦♥s s s ♠s♥ ♣r♠tr③t♦♥ s♠♦♦t♥ ♥s♦ ♠♥s t

♦r♠♥♥ st ❯♥rst② ♦ ♥♦♦② r♠♥②

♦r♠♥♥ s ♥ ssst♥t ♣r♦ss♦r ♦r ♦♠♣tr r♣s ♥ t ♣rt♠♥t ♦♥♦r♠ts t st ❯♥rst② ♦ ♥♦♦② ♥ r♠♥② s rsr ♥trsts r♦ss ♦♥ t ♠t♠t ♦♥t♦♥s ♦ ♦♠tr② ♣r♦ss♥ ♦rt♠s s str ♣♣t♦♥s ♥ ♦♠♣tr r♣s ♥ rt s r ♦r♠♥♥ s ♦t♦rsr ♣♣rs ♦♥ ♣r♠tr③t♦♥ ♠t♦s sr r♦♥strt♦♥ ♥ r②♥tr♦♦r♥ts ♦r♠♥♥ r s P r♦♠ t ❯♥rst② ♦ r♥♥ ♥ ♥ s♣♥t t♦ ②rs s ♣♦st♦t♦r rsr ♦ t t ts ♦♥ r♦♣t t Ps♥ ♥ t ♥sttt ♦ ♥♦r♠t♦♥ ♥ ♥ ♥♦♦s ♥Ps t②

r♥♦ é② r♥

r♥♦ é② s srr t s t ♦ t Pr♦t♠ s ♠♥ ♦♥trt♦♥s ♦♥r♥ t①tr ♠♣♣♥ ♥ ♣r♠tr③t♦♥ ♠t♦s♦r tr♥t srs ♥ r ♥♦ s ② s♦♠ ♠♦♥ s♦tr ♥♥ ② ♦ ♥r ♦ ♥ t ♦t♥ s P ♥ r♦♠ tP ♥sttt t♦♥ P♦②t♥q ♦rr♥ s ♦r ♥tt ♦♠♣tt♦♥♦♣♦♦② ♦♠♥t♦rs ♥ ♠♥ s r t P ♣r ♥ str♥ P tss ♥ ♦♠♣tr ♥s

r ❯♥rst② ♦ rts ♦♠ ♥

r s ♥ ssst♥t ♣r♦ss♦r ♥ t ♦♠♣tr ♥ ♣rt♠♥t t t ❯♥rst② ♦ rts ♦♠ ♥ ♦♥ts rsr ♥ t rs ♦ ♦♠♣trr♣s ♥ ♦♠♣tr ♥♥r♥ r r s ♣r♦♠♥♥t② ♥trst ♥t ♦rt♠ s♣ts ♦ t ♦♠tr② ♣r♦ss♥ ♦s♥ ♦♥ sr ♥♠♥t♣r♦♠s ♦ ♠s ♠♥♣t♦♥ ♥ t♥ r r♥t rsr rsss ♦rt♠s♦r ♠s ♣r♠tr③t♦♥ ♣r♦ss♥ ♦ ♦♣ srs ♠s t♥ ♥ s♠♥tt♦♥ ♦t♦r sr ♣r♠tr③t♦♥ ♠t♦s r s ♥ ♣♦♣r ♠♦rs ♥♥ ♥r ♥ t r r r P r♦♠ t r❯♥rst② ♦ rs♠ ♥ s♣♥t t♦ ②rs s ♣♦st♦t♦r rsrr ♥ t❯♥rst② ♦ ♥♦s t ❯r♥♠♣♥ ♥ t♥ t♦ ②rs s ♥ ssst♥t ♣r♦ss♦rt ♥♦♥ sr

♥ ❩♦ r♦s♦t ❯

♥ ❩♦ s srr ♦ t r♣s r♦♣ t r♦s♦t sr s r s ♥ P rs ♥ ♦♠♣tr ♥ r♦♠ ❩♥ ❯♥rst② ♥ ♥ rs♣t② ♥ s rsr ♥trsts ♥ ♦♠tr② ♣r♦ss♥ t①tr♣r♦ss♥ ♥ rt♠ r♥r♥ ♦s ♦r t♥ r♥t ♥ ♣♥♥ ❯ ♣t♥ts♦♠ ♦ ts t♥qs ♥ ♥trt ♥ ❲♥♦s ❱st rt❳ ♥ ❳❳

②s

♦r♥♥ sss♦♥

• ♥tr♦t♦♥ ❬ ♠♥ ❪

• r♥t ♦♠tr② Pr♠r ❬ ♠♥ ❪

• r②♥tr ♣♣♥s ❬ ♠♥ ❪

• tt♥ t ♦♥r② r ❬ ♠♥ r♥♦❪

• ♥rt ♠t♦s ♥ r Pttr♥s ❬ ♠♥ ❪

• ♥ t ♦r ♥ ♣rt ♠♥tt♦♥ ♥ ♦♥str♥ts ❬ ♠♥ ♥ ❩♦❪

• ♦♠♣rs♦♥ ♥ ♣♣t♦♥s ♦ P♥r t♦s ❬ ♠♥ ❪

tr♥♦♦♥ sss♦♥

• ♣r Pr♠tr③t♦♥ ❬ ♠♥ ❪

• srt ①tr♦r s ♥ ts ❬ ♠♥ t❪

• ♦ Pr♠tr③t♦♥ ❬ ♠♥ t❪

• r♦ssPr♠tr③t♦♥♥trsr ♠♣♣♥ ❬ ♠♥ ❪

• ♥ t ♦r ♥ ♣rt ♠r s♣ts ❬ ♠♥ r♥♦❪

• ♦♠♣rs♦♥ ♥ ♣♣t♦♥s ♦ ♦ t♦s ❬ ♠♥ r♥♦❪

• ♣♥ Pr♦♠s ♥ ❬ ♠♥ ❪

♦♥t♥ts

♥tr♦t♦♥ ♣♣t♦♥s

r♥t ♦♠tr② Pr♠r s ♥t♦♥s ♥tr♥s r Pr♦♣rts tr st♦rt♦♥

r②♥tr ♣♣♥s r♥ ss Pr♠tr③t♦♥ ② ♥ ♦♠♥t♦♥s r②♥tr ♦♦r♥ts ♦♥r② ♣♣♥

tt♥ t ♦♥r② r ♦r♠t♦♥ ♥②ss Pr♠tr③t♦♥ ♠t♦s s ♦♥ ♦r♠t♦♥ ♥②ss ♦♥♦r♠ ♠t♦s

♥♣ t♦s

♠♥tt♦♥ ♥ ♦♥str♥ts ♠♥tt♦♥ ♦♥str♥ts

♦♠♣rs♦♥ ♦ P♥r t♦s

tr♥t s ♦♠♥s ❯♥t ♣r ♠♣ ♥ qrtr ♦♠♣①s ♦ Pr♠tr③t♦♥

r♦ssPr♠tr③t♦♥ ♥trsr ♣♣♥ s ♦♠♣① t♦s ♥r② r♥ ♠t♦s

♦♠♣t ♠s♥

♠r ♣t♠③t♦♥ t strtrs ♦r s♣rs ♠trs ♦♥ ♥r ②st♠s ♥t♦♥s ♦ ♠♥② rs rt ♣t♠③t♦♥ ♦♥♥r ♦♣t♠③t♦♥ ♦♥str♥ ♣t♠③t♦♥ r♥ t♦

♦r♣②

♣tr

♥tr♦t♦♥

♥ ♥② t♦ srs t s♠r t♦♣♦♦② t s ♣♦ss t♦ ♦♠♣t ♦♥t♦♦♥ ♥♦♥t♦ ♠♣♣♥ t♥ t♠ ♦♥ ♦ ts srs s r♣rs♥t ② tr♥r ♠st ♣r♦♠ ♦ ♦♠♣t♥ s ♠♣♣♥ s rrr t♦ s ♠s ♣r♠tr③t♦♥ sr tt t ♠s s ♠♣♣ t♦ s t②♣② rrr t♦ s t ♣r♠tr ♦♠♥Pr♠tr③t♦♥s t♥ sr ♠ss ♥ rt② ♦ ♦♠♥s ♥♠r♦s♣♣t♦♥s ♥ ♦♠♣tr r♣s ♥ ♦♠tr② ♣r♦ss♥ s sr ♦ ♥ r♥t②rs ♥♠r♦s ♠t♦s ♦r ♣r♠tr③♥ ♠ss r ♦♣ trt♥ rs♣r♠tr ♦♠♥s ♥ ♦s♥ ♦♥ r♥t ♣r♠tr③t♦♥ ♣r♦♣rts

s ♦rs rs t r♦s ♣r♠tr③t♦♥ ♠t♦s s♠♠r③♥ t ♠♥s ♦ t♥q ♥ ♦s♥ ♦♥ t ♣rt s♣ts ♦ t ♠t♦s t s♦♣r♦s ①♠♣s ♦ t rsts ♥rt ② ♠♥② ♦ t ♠♦r ♣♦♣r ♠t♦s ❲♥sr ♠t♦s rss t s♠ ♣r♠tr③t♦♥ ♣r♦♠ t sr② strs t♦ ♣r♦ ♥ ♦t ♦♠♣rs♦♥ t♥ t♠ s ♦♥ rtr s s ♣r♠tr③t♦♥qt② ♥② ♥ r♦st♥ss ❲ s♦ sss ♥ t t ♣♣t♦♥s tt ♥tr♦♠ ♣r♠tr③t♦♥ ♥ t ♣rt sss ♥♦ ♥ ♠♣♠♥t♥ t r♥tt♥qs

♣♣t♦♥s

r ♣r♠tr③t♦♥ s ♥tr♦ t♦ ♦♠♣tr r♣s s ♠t♦ ♦r ♠♣♣♥t①trs ♦♥t♦ srs ❬♥♥s t ♦t t ❪ r t st ts r② ♦♠ qt♦s t♦♦ s ♦r ♠♥② ♠s ♣r♦ss♥ ♣♣t♦♥ssss ♦

t ♣♣♥

t ♦ts ♥ ♥t② r♣rs♥t ② ♦rs ♦♠tr s♣ ♣♦②♦♥♠s ♦r ss♦♥ sr t t ts ♦rrs♣♦♥♥ t♦ tr♥ st♦r ♥ s♣rt rr② ♥ trt♦♥ t①tr ♠♣♣♥ t ts r t ♦♦rs ♦ trs♣t ♣①s ♦s ♥ rtr ♥r ② st♦r♥ ♠♣ ♥♦r♠ ♦r s♣♠♥t ♠♣s ♥t t♥qs ❬P♥ t P♦r♠s t ❪ ♠♦ tr♦♥ ♦ s♣ ♥ t ♥♦r♦♦ ♦ t sr ② s♥ ♦♠tr t①tr rtr

①tr ♣♣♥ ♦r♠ ♣♣♥ t r♥sr

♦r♣♥ s ♦♠♣t♦♥ t♥

tss ♠s♥ r tt♥

r Pr♠tr③t♦♥ ♣♣t♦♥s

r ♣♣t♦♥ ♦ ♣r♠tr③t♦♥ t①tr ♠♣♣♥ st qrs ♦♥♦r♠♣s ♠♣♠♥t ♥ t ♣♥♦r ♥r ♠♦r

r ♣♣t♦♥ ♦ ♣r♠tr③t♦♥ ♣♣r♥♣rsr♥ s♠♣t♦♥ t ts r ♥♦ ♥ ♥♦r♠ ♠♣ ♣♣ ♦♥t♦ r♠t② s♠♣ rs♦♥♦ t ♠♦ ♦ t ♦r♥ s③

r ♦ ♣r♠tr③t♦♥ r③s ♥ strt♦♥ ♦ t ♥t ♦♠tr②s strt♦♥ ♥ t♥ r♥st♥t ♥t♦ tr♥t s♣ r♣rs♥tt♦♥s

t♥ ♦♥ t♥qs r ♥ ♥ ♦rr t♦ ♠♦ t t ♦♠♣tt♦♣♦♦② ♦r t tt ♥♥♦t s② ♣♣r♦①♠t ♦② ② t s ss♣rs② ♥tr♦♥ strtrs ♦r ♥♠ r ♥tr ② t♦ ♠♣ ts t♦ srss s♥ ♣♥r ♣r♠tr③t♦♥

t ②♥tss

❲ t ♦ ♦ t①tr ♠♣♣♥ s t♦ r♣rs♥t t ♦♠♣t ♣♣r♥ ♦ ♦ts sr ♠t♦s ♠ s ♦ ♠s ♣r♠tr③t♦♥ t♦ rt t ♦ t♥ssr② ♦r r ♣♣r♥ t♥qs ♥ s s ♥♣t t ♣ts ts♠♣ t ❬♦r t ❪ ♣r♠tr ♦r ♣r♦r ♠♦s ♦r rt sr♥♣t ♥ t♥ ❬rr ♥ rt ❪ t②♣ ♦ t ♥ qt r ♥ t♥tr♠t r♣rs♥tt♦♥s s t♦ rt t ♣r t ♥ r♣rs♥tt♦♥s s t♦st♦r t

♦r♣♥ ♥ t r♥sr

♠♣ t♥ t srs ♦ t♦ ♦ts ♦s t tr♥sr ♦ t r♦♠ ♦♥ ♦tt♦ ♥♦tr ❬Pr♥ t ❪ ♦r t ♥tr♣♦t♦♥ t♥ t s♣ ♥ ♣♣r♥ ♦ sr ♦ts ❬① r♦② ♥ r r♥r t ❪② r②♥ t ♥tr♣♦t♦♥ rt♦s ♦r t♠ ♦♥ ♥ ♣r♦ ♠♦r♣♥ ♥♠t♦♥s♥ s♣t②r②♥ ♥ rq♥②r②♥ ♠♦r♣s t rt ♦ ♥ ♥ r♥t♦r r♥t ♣rts ♦ t ♦ts ♦r r♥t rq♥② ♥s ♦rs♥ss ♦ t trs♥ tr♥s♦r♠ ❬♥ t r♦② ♥ r ❪ ♠♣ ♥tr ♦♠♣t rt② ♦r s ♠♦r ♦♠♠♦♥② ♦♥ ♦♠♣t ② ♠♣♣♥ ♦t♦t srs t♦ ♦♠♠♦♥ ♦♠♥ ♥ t♦♥ t♦ tr♥srr♥ t stt ♣♣r♥♦ srs ♥trsr ♣r♠tr③t♦♥s ♦ t tr♥sr ♦ ♥♠t♦♥ t t♥s♣s tr ② tr♥srr♥ t ♦ sr ♥♥ r♦♠ ♦♥s ♦ ♥ ♥♠t♦♥ r♦r ② rt② tr♥srr♥ t ♦ ♥ tr♥s♦r♠t♦♥ ♦ tr♥ ♥ t ♠s❬♠♥r ♥ P♦♣♦ ❪

s ♦♠♣t♦♥

ss r♦♠ r♥ s♥s ♦t♥ ♦♥t♥ ♦s ♥ ♠t♣ ♦♠♣♦♥♥ts é② ❬❪ss ♣♥r ♣r♠tr③t♦♥ t♦ ♦t♥ t ♥tr s♣ ♦r ♦ ♦♥rs ♥ t♦tr♥t t♦s ♥ ♠♥② ss ♣r♦r ♥♦ ♦t t ♦r s♣ ♦ t s♥♥♠♦s ①sts ♦r ♥st♥ ♦r ♠♥ s♥s t♠♣ts ♦ ♥r ♠♥ s♣ rr② ♥ t ❬❪ ♥ ♥♦ t ❬❪ s ts ♣r♦r ♥♦ t♦tt ♦♠♣t♦♥ ♦ s♥s ② ♦♠♣t♥ ♠♣♣♥ t♥ t s♥ ♥ t♠♣t♠♥ ♠♦ r♦② ♥ r ❬❪ ♦♣ ♠♦r ♥r ♥ r♦st t♠♣ts ♣♣r♦ ♦r ♦♠♣t♦♥ ♦ ♥② t②♣ ♦ s♥s t♥qs t②♣② s ♥♥trsr ♣r♠tr③t♦♥ t♥ t t♠♣t ♥ t s♥

s t♥

t♥ ♦♣rt♦♥s ♦t♥ ♥t r♦♠ ♦ ♣r♠tr③t♦♥ t♥ ♣rs ♦ ♠♦sr♠♥♥ t ❬❪ s ♦ ♣r♠tr③t♦♥ t♦ tt t♥♣st tr♥sr ♦ts t♥ ♠♦s ② ♦② ♣r♠tr③ t r♦♥s ♦ ♥trst ♦♥ t t♦♠♦s ♥ ♥ ♦r♣ t t♦ ♣r♠tr③t♦♥s ② s t ♣r♠tr③t♦♥ t♦tr♥sr s♣ ♣r♦♣rts r♦♠ ♦♥ ♠♦ t♦ t ♦tr é② ❬❪ ss ♦ ♣r♠tr③t♦♥ ♦r ♠s ♦♠♣♦st♦♥ ♥ s♠r ♠♥♥r ② ♦♠♣t ♥ ♦r♣♣♥ ♣♥r♣r♠tr③t♦♥ ♦ t r♦♥s ♥r t ♦♠♣♦st♦♥ ♦♥r② ♦♥ t ♥♣t ♠♦s ♥s t t♦ ①trt ♥ s♠♦♦t② ♥ s♣ ♥♦r♠t♦♥ r♦♠ t t♦ ♠♦s

rt♦♥ ♦ t tss

♥ r ♥♠r ♦ ♠♦s r ♣r♠tr③ ♦♥ ♦♠♠♦♥ ♦♠♥ ♦♥ ♥ ♣r♦r♠♥ ♥②ss tr♠♥♥ t ♦♠♠♦♥ t♦rs t♥ ♦ts ♥ tr st♥s♥trts ♦r ①♠♣ ♦♥ ts ♦ ♠♥ s♣s ❬♥ t ❪ t st♥s♥trts ♠② ♥r t ♥ t ts ♥ ♦♠♣r ♥st t ts♥ s♦r ♥st ♦ ts ♠♥s♦♥s ♥ t ts ♥ s t♦ rt ♥♣s ♦t ♥st♥s ② ♥tr♣♦t♦♥ ♦r ①tr♣♦t♦♥ ♦ ①st♥ ♦♥s

♠s♥

r r ♠♥② ♣♦ss tr♥t♦♥s tt r♣rs♥t t s♠ s♣ t s♠r s♦ r② ♦♠ tr♥t♦♥ ♠② ♠♦r sr t♥ ♦trs ♦r r♥t ♣♣t♦♥s ♦r ①♠♣ ♦r ♥♠r s♠t♦♥s ♦♥ srs tr♥s t ♦♦ s♣trt♦ tt r ♥♦t t♦♦ s♠ ♦r t♦♦ s♥♥② r ♠♣♦rt♥t ♦r ♦♥r♥ ♥ ♥♠rr② ♥ ♦♠♠♦♥ ② t♦ r♠s srs ♦r t♦ r♣ ♦♥ tr♥t♦♥ ② ♥♦tr s t♦ ♣r♠tr③ t sr t♥ ♠♣ sr ♥rst♦♦ ♥ s②t♦ rt tr♥t♦♥ ♦ t ♦♠♥ t♦ t ♦r♥ sr ♦r ①♠♣ t ❬❪ s rr r s♠♣♥ ♦ ♣♥r sqr ♦♠♥ ♦tr ♠t♦s ❬s♦ t ❪ s rr ss♦♥ s② t♦ tr♥ s♣ts ♦♥ ts ♦ s♠♣ ♦♠♥ ♦② rr ♠ss ♥ s② s♣♣♦rt t rt♦♥ ♦ s♠♦♦t srs s t ♠t ♣r♦ss ♦ ♣♣②♥ ss♦♥ rs ♦ ♥rt qt② tr♥t♦♥s sr♥ t ❬❪ ♣r♠tr③ t ♥♣t ♠s ♥ t♣♥ ♥ t♥ s ♣♥r ♥② tr♥t♦♥ t♦ ♦t♥ qt② r♠s♥ ♦t sr ♥ ♣r♦♠ ts ♠t♦s s t ♣♣r♥ ♦ s s♦♥t♥ts♦♥ t ts rt t♦ tt t ♣r♠tr③t♦♥ r③s② ♥ ♦ts♠♥ ❬❪♦ ♦ ♣r♠tr③t♦♥ ♥ ♥st s ♦ ♣r♠tr③t♦♥ t♦ ♠♦ rts♦♥ t ♠s s ♣rt ♦ ♥ ①♣t r♠s♥ s♠ ♥t ♠t♦s s s ❬②t ❪ s ♦ ♣r♠tr③t♦♥ t♦ ♥rt ♣r♦♠♥♥t② qrtr ♠srt② ♦♥ t sr

s ♦♠♣rss♦♥

s ♦♠♣rss♦♥ s s t♦ ♦♠♣t② st♦r ♦r tr♥s♠t ♦♠tr ♠♦s s t♦tr t ♦♠♣rss♦♥ rts r ♥rs② ♣r♦♣♦rt♦♥ t♦ t t ♥tr♦♣② s r♦♠♣rss♦♥ rts ♥ ♦t♥ ♥ ♠♦s r r♣rs♥t ② ♠ss tt r srr s ♣♦ss ♦t t♦♣♦♦② ♥ ♦♠tr② ♦♣♦♦ rrt② rrst♦ ♠ss r ♠♦st rts t s♠ r ♦♠tr rrt② ♠♣stt tr♥s r s♠r t♦ ♦tr ♥ tr♠s ♦ s♣ ♥ s③ ♥ rts r ♦st♦ t ♥tr♦ ♦ tr ♥♦rs ♠ss ♥ ♦t♥ ② ♣r♠tr③♥ t♦r♥ ♦ts ♥ t♥ r♠s♥ t rr s♠♣♥ ♣ttr♥s ❬ t ❪ qt② ♦ t ♣r♠tr③t♦♥ rt② ♠♣ts t ♦♠♣rss♦♥ ♥②

r tt♥

♥ ♦ t rr ♣♣t♦♥s ♦ ♠s ♣r♠tr③t♦♥ s sr tt♥ ❬♦tr ❪♥② ♣♣t♦♥s ♥ ♦♠tr② ♣r♦ss♥ rqr s♠♦♦t ♥②t sr t♦ ♦♥strt r♦♠ ♥ ♥♣t ♠s ♣r♠tr③t♦♥ ♦ t ♠s ♦r s ♦♠♥s♥♥t② s♠♣s ts ts rr ♠t♦s tr ♣r♠tr③ t ♥tr ♠s♥ t ♣♥ ♦r s♠♥t t ♥ ♣r♠tr③ ♣t ♥♣♥♥t② ♦r r♥t♠t♦s ❬ t ❪ ♦s ♦♥ ♦♥strt♥ s♠♦♦t ♦ ♣r♠tr③t♦♥s♥ s t♦s ♦r tt♥ ♥ ♦ ♦♥t♥t② ♦ t ♦♥strt srs

♦♥ r♦♠ tr ts

❲ ♦♠♣tr r♣s ♦ss ♦♥ rt ♠♦s ♦♠tr② ♣r♦ss♥ s ♥♠r♦sr♦r ♥♥r♥ ♣♣t♦♥s Prtr② ♣♥r ♠s ♣r♠tr③t♦♥ s ♥♠♣♦rt♥t t♦♦ ♥ ♠♦♥ ♦ts r♦♠ sts ♦ ♠tr r♥♥ r♦♠ r♠♥t♠♦♥ t♦ ♠t ♦r♠♥ ♦r ♦r♥ ❬♥♥s t s t ❪ ♦ts ♣♣t♦♥s rqr t ♦♠♣tt♦♥ ♦ ♣♥r ♣ttr♥s t♦ ♦r♠ t sr s♣s ②♣② ♠♦s r rst s♠♥t ♥t♦ ♥r② ♦♣ rts ♥ tsrts r t♥ ♣r♠tr③ ♥ t ♣♥

❱s③t♦♥

♦♠♣① ♦♠tr strtrs r ♦t♥ ttr s③ ♥ ♥②③ ② ♠♣♣♥ tsr ♥♦r♠♠♣ ♦♦r ♥ ♦tr ♣r♦♣rts t♦ s♠♣r ♥♦♥ ♦♠♥ ♥ ♦t strtrs ♦r s ♠♣♣♥ s ♣rtr② s s t ♠♥ r♥ ❬rt r t ❪ ♦st ♠t♦s ♦r r♥ ♠♣♣♥ s t t tt tr♥ s ♥s ③r♦ ♥ s③ t tr♦ s♣r ❬r t ❪ ♦r ♣♥r❬r t ❪ ♣r♠tr③t♦♥

♣tr

r♥t ♦♠tr② Pr♠r

♦r ♦ ♥t♦ t ts ♦ ♦ t♦ ♦♠♣t ♠s ♣r♠tr③t♦♥ ♥ t t♦♦ t t t s q② r s♦♠ ♦ t s ♣r♦♣rts r♦♠ r♥t ♦♠tr②tt ss♥t ♦r ♥rst♥♥ t ♠♦tt♦♥ ♥ t ♠t♦s srtr ♦r ♠♦r ts ♥ ♣r♦♦s ♦ ts ♣r♦♣rts rr t ♥trst rrt♦ t st♥r trtr ♦♥ r♥t ♦♠tr② ♥ ♥ ♣rtr t♦ t ♦♦s ②♦ r♠♦ ❬❪ ♥♥r ❬❪ r②s③ ❬❪ ♥ ♦r♥ ❬❪

s ♥t♦♥s

♣♣♦s tt Ω ⊂ R2 s s♦♠ s♠♣② ♦♥♥t r♦♥ t♦t ♥② ♦s ♦r

①♠♣

t ♥t sqr Ω = (u, v) ∈ R2 : u, v ∈ [0, 1], ♦r

t ♥t s Ω = (u, v) ∈ R2 : u2 + v2 ≤ 1,

♥ tt t ♥t♦♥ f : Ω → R3 s ♦♥t♥♦s ♥ ♥ ♥t♦♥ ♥♦ t♦ st♥t

♣♦♥ts ♥ Ω r ♠♣♣ t♦ t s♠ ♣♦♥t ♥ R3 ❲ t♥ t ♠ S ♦ Ω ♥r

f srS = f(Ω) = f(u, v) : (u, v) ∈ Ω,

♥ s② tt f s ♣r♠tr③t♦♥ ♦ S ♦r t ♣r♠tr ♦♠♥ Ω t ♦♦s r♦♠t ♥t♦♥ ♦ S tt f s t② t♦♥ t♥ Ω ♥ S ♥ ts ♠ts t♦♥ ts ♥rs f−1 : S → Ω r r s♦♠ ①♠♣s

s♠♣ ♥r ♥t♦♥

♣r♠tr ♦♠♥ Ω = (u, v) ∈ R2 : u, v ∈ [0, 1]

sr S = (x, y, z) ∈ R3 : x, y, z ∈ [0, 1], x + y = 1

♣r♠tr③t♦♥ f(u, v) = (u, 1− u, v)

♥rs f−1(x, y, z) = (x, z)

②♥r

♣r♠tr ♦♠♥ Ω = (u, v) ∈ R2 : u ∈ [0, 2π), v ∈ [0, 1]

sr S = (x, y, z) ∈ R3 : x2 + y2 = 1, z ∈ [0, 1]

♣r♠tr③t♦♥ f(u, v) = (cos u, sin u, v)

♥rs f−1(x, y, z) = (arccos x, z)

♣r♦♦

♣r♠tr ♦♠♥ Ω = (u, v) ∈ R2 : u, v ∈ [−1, 1]

sr S = (x, y, z) ∈ R3 : x, y ∈ [−2, 2], z = 1

4(x2 + y2)

♣r♠tr③t♦♥ f(u, v) = (2u, 2v, u2 + v2)

♥rs f−1(x, y, z) = (x2, y

2)

♠s♣r ♦rt♦r♣

♣r♠tr ♦♠♥ Ω = (u, v) ∈ R2 : u2 + v2 ≤ 1

sr S = (x, y, z) ∈ R3 : x2 + y2 + z2 = 1, z ≥ 0

♣r♠tr③t♦♥ f(u, v) = (u, v,√

1− u2 − v2)

♥rs f−1(x, y, z) = (x, y)

♥ ♥ sr S tt s♦ ♥♦t tt t ♥t♦♥ f s ② ♥♦♠♥s t ♦♥② ♣r♠tr③t♦♥ ♦ S ♦r Ω ♥ t ♥ ♥② t♦♥ ϕ : Ω → Ωt s s② t♦ r② tt t ♦♠♣♦st♦♥ ♦ f ♥ ϕ t ♥t♦♥ g = f ϕ s ♣r♠tr③t♦♥ ♦ S ♦r Ω t♦♦ ♦r ①♠♣ ♥ s② ♦♥strt s r♣r♠tr③t♦♥ ϕ r♦♠ ♥② t♦♥ ρ : [0, 1]→ [0, 1] ② ♥♥

♦r t ♥t sqr ϕ(u, v) = (ρ(u), ρ(v)), ♦r

♦r t ♥t s ϕ(u, v) = (uρ(u2 + v2), vρ(u2 + v2)).

♥ ♣rtr t♥ t ♥t♦♥ ρ(x) = 21+x

♥ ♣♣②♥ ts r♣r♠tr③t♦♥ ♦t ♥t s t♦ t ♣r♠tr③t♦♥ ♦ t ♠s♣r ♥ t ①♠♣ ♦ s t♦♦♥ tr♥t ♣r♠tr③t♦♥

♠s♣r str♦r♣

♣r♠tr ♦♠♥ Ω = (u, v) ∈ R2 : u2 + v2 ≤ 1

sr S = (x, y, z) ∈ R3 : x2 + y2 + z2 = 1, z ≥ 0

♣r♠tr③t♦♥ f(u, v) = ( 2u1+u2+v2 ,

2v1+u2+v2 ,

1−u2−v2

1+u2+v2 )

♥rs f−1(x, y, z) = ( x1+z

, y

1+z)

♥tr♥s r Pr♦♣rts

t♦ t ♣r♠tr③t♦♥ ♦ sr s ♥♦t ♥q♥ tr sss ♦t♦ t t st ♣r♠tr③t♦♥ t rs♣t t♦ rt♥ rtrt ♥rtss s r② ♥② t♥ t♦ s t ♦s t♦ ♦♠♣t rt② ♦ ♣r♦♣rts ♦ t sr♦r ①♠♣ f s r♥t t♥ ts ♣rt rts

fu =∂f

∂u♥ fv =

∂f

∂v

s♣♥ t ♦ t♥♥t ♣♥ ♥ ② s♠♣② t♥ tr r♦ss ♣r♦t ♥ ♥♦r♠③♥t rst t t sr ♥♦r♠

nf =fu × fv

‖fu × fv‖.

♦ s♠♣② t ♥♦tt♦♥ ♦t♥ s♣ ♦ fu ♥ fv s t rts ♥ ♦ nf

s t sr ♥♦r♠ t s♦ ♣ ♥ ♠♥ tt ♦r♠② tr r ♥t♦♥sr♦♠ R

2 t♦ R3 ♥ ♦tr ♦rs ♦r ♥② ♣♦♥t (u, v) ∈ Ω ♥ t ♣r♠tr ♦♠♥ t

t♥♥t ♣♥ t t sr ♣♦♥t f(u, v) ∈ S s s♣♥♥ ② t t♦ t♦rs fu(u, v)♥ fv(u, v) ♥ nf (u, v) s t ♥♦r♠ t♦r t ts ♣♦♥t ♥ t s r② ts② ♦♥sr♥ t♦ ①♠♣s

♦r t s♠♣ ♥r ♥t♦♥ f(u, v) = (u, 1− u, v) t

fu(u, v) = (1,−1, 0) ♥ fv(u, v) = (0, 0, 1)

♥ rtrnf (u, v) = (−1√

2, 1√

2, 0),

s♦♥ tt t ♥♦r♠ t♦r s ♦♥st♥t ♦r ♣♦♥ts ♦♥ S

♦r t ♣r♠tr③t♦♥ ♦ t ②♥r f(u, v) = (cos u, sin u, v) t

fu(u, v) = (− sin u, cos u, 0) ♥ fv(u, v) = (0, 0, 1)

♥ rtrnf (u, v) = (cos u, sin u, 0),

s♦♥ tt t ♥♦r♠ t♦r t ♥② ♣♦♥t (x, y, z) ∈ S s st (x, y, 0)

♦t tt ♥ ♦t ①♠♣s t sr ♥♦r♠ s ♥♣♥♥t ♦ t ♣r♠tr③t♦♥♥ t ts ♦s ♦r srs ♥ s tr♦r ♥ ♥tr♥s ♣r♦♣rt② ♦ tsr ♦r♠② ♥ s♦ s② tt t sr ♥♦r♠ s ♥t♦♥ n : S → S

2r S

2 = (x, y, z) ∈ R3 : x2 + y2 + z2 = 1 s t ♥t s♣r ♥ R

3 s♦ tt

n(p) = nf (f−1(p))

❲ tt② ss♠ tt t ♣r♠tr③t♦♥ s rr fu ♥ fv r ②s ♥r② ♥♣♥♥t ♥ tr♦r nf s ♥♦♥③r♦

♦r ♥② p ∈ S ♥ ♥② ♣r♠tr③t♦♥ f s ♥ ①rs ②♦ ♠② ♥t t♦ r② ts♦r t t♦ tr♥t ♣r♠tr③t♦♥s ♦ t ♠s♣r ♥ ♦ tr ♥tr♥ssr ♣r♦♣rts r t ss♥ rtr K(p) ♥ t ♠♥ rtr H(p) s s t t♦t r ♦ t sr A(S) ♦ ♦♠♣t t ttr ♥ t rst♥♠♥t ♦r♠

If =

(fu · fu fu · fv

fv · fu fv · fv

)

=

(E FF G

)

,

r t ♣r♦t t♥ t ♣rt rts s t s ♦t ♣r♦t ♥ R3 t

♦♦s ♠♠t② r♦♠ t ②r③ ♥qt② tt t tr♠♥♥t ♦ tss②♠♠tr 2 × 2 ♠tr① s ②s ♥♦♥♥t s♦ tt ts sqr r♦♦t s ②s r r ♦ t sr s t♥ ♥ s

A(S) =

∫

Ω

√

det If du dv.

♦r ①♠♣ t ♦rt♦r♣ ♣r♠tr③t♦♥ f(u, v) = (u, v,√

1− u2 − v2)♦ t ♠s♣r ♦r t ♥t s tr s♦♠ s♠♣t♦♥s ♥ tt

det If =1

1− u2 − v2

♥ ♥ ♦♠♣t t r ♦ t ♠s♣r s ♦♦s

A(S) =

1∫

−1

√1−v2∫

−√

1−v2

1√1− u2 − v2

du dv

=

1∫

−1

[

arcsinu√

1− v2

]√

1−v2

−√

1−v2

dv

=

1∫

−1

π dv

= 2π,

s ①♣t ♦rs t t s♠ rst s t str♦r♣ ♣r♠tr③t♦♥ ♥ ②♦ ♠② ♥t t♦ tr② tt s ♥ ①rs

♥ ♦rr t♦ ♦♠♣t t rtrs ♠st rst ss♠ t ♣r♠tr③t♦♥ t♦ t r♥t s♦ tt ts s♦♥ ♦rr ♣rt rts

fuu =∂2f

∂u2, fuv =

∂2f

∂u∂v, ♥ fvv =

∂2f

∂v2

r ♥ ♥ t ♦t ♣r♦ts ♦ ts rts t t sr ♥♦r♠t♥ s t s②♠♠tr 2× 2 ♠tr① tt s ♥♦♥ s t s♦♥ ♥♠♥t ♦r♠

IIf =

(fuu · nf fuv · nf

fuv · nf fvv · nf

)

=

(L MM N

)

.

ss♥ ♥ ♠♥ rtr r ♥② ♥ s t tr♠♥♥t ♥ t tr♦ t ♠tr① I−1

f IIf rs♣t②

K = det(I−1f IIf ) =

det IIf

det If

=LN −M2

EG− F 2

♥

H = 12trace(I−1

f IIf ) =LG− 2MF + NE

2(EG− F 2).

♦r ①♠♣ rr②♥ ♦t ts ♦♠♣tt♦♥s rs tt t rtrs r ♦♥st♥t♦r ♠♦st ♦ t srs r♦♠ ♦

s♠♣ ♥r ♥t♦♥ K = 0, H = 0,

②♥r K = 0, H = 12,

♠s♣r K = 1, H = −1.

s ♥ ①rs s♦ tt t rtrs t ♥② ♣♦♥t p = (x, y, z) ♦ t ♣r♦♦r♦♠ ♦ r K(p) = 1

4(1+z)2♥ H(p) = 2+z

4(1+z)3/2

tr st♦rt♦♥

♣rt r♦♠ ts ♥tr♥s sr ♣r♦♣rts tr r ♦trs tt ♣♥ ♦♥ t ♣r♠tr③t♦♥ ♠♦st ♠♣♦rt♥t② t ♠tr st♦rt♦♥ ♦♥sr ♦r ①♠♣ t t♦♣r♠tr③t♦♥s ♦ t ♠s♣r ♦ ♥ ♦t ss t ♠ ♦ t sr ♦♥t rt s ♦r ② rr r t② s t ♠ ♦ t ♦rrs♣♦♥♥r ♥ t ♣r♠tr ♦♠♥ s♦♥ ♦♥ t t ❨♦ ♥♦t tt t sr r♦♦s ♠♦r rr ♦r t str♦r♣ t♥ ♦r t ♦rt♦r♣ ♣r♦t♦♥ ♥ ttt ttr ♦♥sr② strts t r ♥ t r rt♦♥ ♥r t ♦♥r②

♦ ttr ♥rst♥ ts ♥ ♦ strt♥ t s s t ♣♣♥s t♦ t sr♣♦♥t f(u, v) s ♠♦ t♥② tt t ② r♦♠ (u, v) ♥ t ♣r♠tr ♦♠♥ ♥♦t ts ♥♥ts♠ ♣r♠tr s♣♠♥t ② (∆u, ∆v) t♥ t ♥ sr♣♦♥t f(u + ∆u, v + ∆v) s ♣♣r♦①♠t② ♥ ② t rst ♦rr ②♦r ①♣♥s♦♥ f ♦f r♦♥ (u, v)

f(u + ∆u, v + ∆v) = f(u, v) + fu(u, v)∆u + fv(u, v)∆v.

s ♥r ♥t♦♥ ♠♣s ♣♦♥ts ♥ t ♥t② ♦ u = (u, v) ♥t♦ t t♥♥t ♣♥Tp t p = f(u, v) ∈ S ♥ tr♥s♦r♠s rs r♦♥ u ♥t♦ ♣ss r♦♥ p sr ttr ♣r♦♣rt② ♦♠s ♦♦s rt t ②♦r ①♣♥s♦♥ ♠♦r♦♠♣t② s

f(u + ∆u, v + ∆v) = p + Jf (u)(∆u

∆v

),

r Jf = (fu fv) s t ♦♥ ♦ f t 3×2 ♠tr① t t ♣rt rts♦ f s ♦♠♥ t♦rs ♥ s♥ t s♥r ♦♠♣♦st♦♥ ♦ t ♦♥

Jf = UΣV T = U

(σ1 0

0 σ2

0 0

)

V T ,

r rst ♦rr ②♦r ①♣♥s♦♥ f ♦ t ♣r♠tr③t♦♥ f

r ❱ ♦♠♣♦st♦♥ ♦ t ♠♣♣♥ f

t s♥r s σ1 ≥ σ2 > 0 ♥ ♦rt♦♥♦r♠ ♠trs U ∈ R3×3 ♥ V ∈ R

2×2

t ♦♠♥ t♦rs U1, U2, U3 ♥ V1, V2 rs♣t② ♥ s♣t ♣ t ♥r tr♥s♦r♠t♦♥ f s s♦♥ ♥ r

tr♥s♦r♠t♦♥ V T rst r♦tts ♣♦♥ts r♦♥ u s tt t t♦rs V1

♥ V2 r ♥ ♥♠♥t t t u ♥ t v①s trrs

tr♥s♦r♠t♦♥ Σ t♥ strts r②t♥ ② t t♦r σ1 ♥ t u ♥ ②σ2 ♥ t vrt♦♥

tr♥s♦r♠t♦♥ U ♥② ♠♣s t ♥t t♦rs (1, 0) ♥ (0, 1) t♦ t t♦rsU1 ♥ U2 ♥ t t♥♥t ♣♥ Tp t p

s ♦♥sq♥ ♥② r ♦ rs r r♦♥ u ♠♣♣ t♦ ♥ ♣s ts♠①s ♦ ♥t rσ1 ♥ rσ2 r♦♥ p ♥ t ♦rt♦♥♦r♠ r♠ [V1, V2] s ♠♣♣t♦ t ♦rt♦♦♥ r♠ [σ1U1, σ2U2]

s tr♥s♦r♠t♦♥ ♦ rs ♥t♦ ♣ss s ♦ ♠tr st♦rt♦♥ ♦ t♣r♠tr③t♦♥ s t s♦s ♦ f s ♦② r♦♥ s♦♠ ♣r♠tr ♣♦♥t u ∈ Ω♥ t ♦rrs♣♦♥♥ sr ♣♦♥t p = f(u) ∈ S ♦r♦r ♥♦r♠t♦♥ ♦tts ♦ ♠tr st♦rt♦♥ s ♥ ♥ t s♥r s σ1 ♥ σ2 ♦r ①♠♣ ♦t s r ♥t t♥ Jf s st r♦tt♦♥ ♣s ♥♦r♠ s♥ ♥ f ♦s ♥♦tst♦rt ♥s r♦♥ u s t ♣r♦t ♦ t s♥r s s 1 t♥ tr ♦ ♥② r ♥ t ♣r♠tr ♦♠♥ s ♥t t♦ t r ♦ t ♦rrs♣♦♥♥♣s ♥ t t♥♥t ♣♥ ♥ s② tt f s ♦② r♣rsr♥

♦♠♣t♥ t s♥r s rt② s t t♦s s♦ tt ttr rs♦rt t♦t t tt t s♥r s ♦ ♥② ♠tr① A r t sqr r♦♦ts ♦ t ♥s♦ t ♠tr① AT A ♥ ♦r s t ♠tr① Jf

T Jf s ♥ ♦ q♥t♥ ♥♠② t

rst ♥♠♥t ♦r♠

JfT Jf =

(fu

T

fvT

)

(fu fv) = If =

(E FF G

)

,

♥ ♥ s② ♦♠♣t t t♦ ♥s λ1 ♥ λ2 ♦ ts s②♠♠tr ♠tr① ②s♥ t ♥t② tt ♦r♠

λ1,2 = 12

((E + G)±

√

4F 2 + (E −G)2).

❲ ♥♦ s♠♠r③ t ♠♥ ♣r♦♣rts tt ♣r♠tr③t♦♥ ♥ ♦②

f s s♦♠tr ♦r ♥t♣rsr♥ ⇐⇒ σ1 = σ2 = 1 ⇐⇒ λ1 = λ2 = 1,

f s ♦♥♦r♠ ♦r ♥♣rsr♥ ⇐⇒ σ1 = σ2 ⇐⇒ λ1 = λ2,

f s qr ♦r r♣rsr♥ ⇐⇒ σ1σ2 = 1 ⇐⇒ λ1λ2 = 1.

♦s② ♥② s♦♠tr ♠♣♣♥ s ♦♥♦r♠ ♥ qr ♥ r② ♠♣♣♥ tts ♦♥♦r♠ ♥ qr s s♦ s♦♠tr ♥ s♦rt

s♦♠tr ⇐⇒ ♦♥♦r♠ + qr.

s q♣♣ t s ♦ t♦ t ①♠♣s ♦ ♥ tr ♣r♦♣rts

s♠♣ ♥r ♥t♦♥

♣r♠tr③t♦♥ f(u, v) = (u, 1− u, v)

♦♥ Jf =(

1 0−1 00 1

)

rst ♥♠♥t ♦r♠ If =(

2 00 1

)

♥s λ1 = 2, λ2 = 1

s ♣r♠tr③t♦♥ s ♥tr ♦♥♦r♠ ♥♦r qr

②♥r

♣r♠tr③t♦♥ f(u, v) = (cos u, sin u, v)

♦♥ Jf =(

cos u 0− sin u 0

0 1

)

rst ♥♠♥t ♦r♠ If =(

1 00 1

)

♥s λ1 = 1, λ2 = 1

s ♣r♠tr③t♦♥ s s♦♠tr

♣r♦♦

♣r♠tr③t♦♥ f(u, v) = (2u, 2v, u2 + v2)

♦♥ Jf =(

2 00 22u 2v

)

rst ♥♠♥t ♦r♠ If =(

4+4u2 4uv

4uv 4+4v2

)

♥s λ1 = 4, λ2 = 4(1 + u2 + v2)

s ♠♣♣♥ s ♥♦t qr ♥ ♦♥♦r♠ ♦♥② t (u, v) = (0, 0)

♠s♣r ♦rt♦r♣

♣r♠tr③t♦♥ f(u, v) = (u, v, 1d) t d = 1√

1−u2−v2

♦♥ Jf =( 1 0

0 1−ud −vd

)

rst ♥♠♥t ♦r♠ If =(

1+u2d2 uvd2

uvd2 1+v2d2

)

♥s λ1 = 1, λ2 = d2

s ♠♣♣♥ s s♦♠tr t (u, v) = (0, 0) t ♥tr ♦♥♦r♠ ♥♦r qrsr

♠s♣r str♦r♣

♣r♠tr③t♦♥ f(u, v) = (2ud, 2vd, (1− u2 − v2)d) t d = 11+u2+v2

♦♥ Jf =

(2d−4u2d2 −4uvd2

−4uvd2 2d−4v2d2

−4ud2 −4vd2

)

rst ♥♠♥t ♦r♠ If =(

4d2 00 4d2

)

♥s λ1 = 4d2, λ2 = 4d2

s ♠♣♣♥ s ②s ♦♥♦r♠ t qr ♥ ts s♦♠tr ♦♥② t t♦♥r② ♦ Ω ♦r u2 + v2 = 1

t tr♥s ♦t tt t ♦♥② ♣r♠tr③t♦♥ tt s ♦♣t♠ ♥ t s♥s tt t ss♦♠tr r②r ♥ ts ♦s ♥♦t ♥tr♦ ♥② st♦rt♦♥ t s t ♦♥ ♦r t②♥r ♥ t t s s♦♥ ② ÿ ❬❪ tt ♦② s♦♠tr ♣r♠tr③t♦♥ ①sts ♦♥② ♦r ♦♣ srs ♣♥s ♦♥s ♥ ②♥rs t ♥s♥ss♥ rtr K(p) = 0 t sr ♣♦♥ts p ∈ S s ♥ ①rs ②♦ ♥ tr②t♦ ♥ s ♦② s♦♠tr ♣r♠tr③t♦♥ ♦r t ♣♥r sr ♣t r♦♠ trst ①♠♣

tr ♥trst♥ ♣r♠tr③t♦♥s r t♦s tt r ♦② ♦♥♦r♠ tstr♦r♣ ♣r♦t♦♥ ♦r t ♠s♣r ♥ t s s♦♥ ② ♠♥♥ ❬❪ tt

s ♣r♠tr③t♦♥ ①sts ♦r ♥② sr tt s t♦♣♦♦② q♥t t♦ s♥ ♥② s♠♣② ♦♥♥t ♣r♠tr ♦♠♥

♦r ♥r② t st ♣r♠tr③t♦♥ f ♦ sr S ♦r ♣r♠tr ♦♠♥Ω s ♦♥ s ♦♦s ❲ rst ♥ rt ♥♦♥♥t ♥t♦♥ E : R

2+ → R+

tt ♠srs t ♦ st♦rt♦♥ ♦ ♣r♠tr③t♦♥ t s♥r s σ1 ♥σ2 ❯s② ts ♥t♦♥ s ♦ ♠♥♠♠ t (1, 1) s♦ s t♦ ♦r s♦♠tr② t♣♥♥ ♦♥ t ♣♣t♦♥ t ♠② s♦ ♥ s tt t ♠♥♠ st♥ ♦♥ t ♦ ♥ (x, x) ♦r x ∈ R+ ♦r ①♠♣ ♦♥♦r♠ ♠♣♣♥s s ♣rrr ♦r st♦rt♦♥ ♦ ♣rtr ♣r♠tr③t♦♥ f s t♥ ♠sr② s♠♣② r♥ t ♦ st♦rt♦♥ ♦r t ♦ ♦♠♥

E(f) =

∫

Ω

E(σ1(u, v), σ2(u, v)) du dv/

A(Ω),

♥ t st ♣r♠tr③t♦♥ t rs♣t t♦ E s t♥ ♦♥ ② ♠♥♠③♥ E(f) ♦rt s♣ ♦ ♠ss ♣r♠tr③t♦♥s

♣tr

r②♥tr ♣♣♥s

♥ ♠♥② ♣♣t♦♥s ♥ ♥ ♣rtr ♥ ♦♠♣tr r♣s t s ♥♦②s ♦♠♠♦♥t♦ ♦r t ♣s ♥r srs ♥ t ♦r♠ ♦ tr♥ ♠ss ♥ ♠♥②st t♦ ts t②♣ ♦ sr ♦r t r♠♥r ♦ ts ♦rs ♥♦ts

r♥ ss

s ♥ t ♣r♦s ♣tr t s ♥♦t ♣♦♥ts ♥ R3 ② p = (x, y, z) ♥ ♣♦♥ts ♥

R2 ② u = (u, v) ♥ s t♥ ♥ s t ♦♥① ♦ ♦r q♥t② t

♥ s♠♥t t♥ t♦ st♥t ♣♦♥ts ♥ tr♥ s t ♦♥① ♦ tr♥♦♥♦♥r ♣♦♥ts ❲ ♥♦t s ♥ tr♥s ♥ R

3 t ♣t ttrs ♥t♦s ♥ R

2 t s♠ ttrs ♦r ①♠♣ e = [u1, u2] ♥ T = [p1, p2, p3] tr♥ ♠s ST s t ♥♦♥ ♦ st ♦ sr tr♥s T = T1, . . . , Tm

♥trst ♦♥② t ♦♠♠♦♥ s E = E1, . . . , El ♥ rts V = p1, . . . ,pn+b ♦rs♣② t st ♦ rts ♦♥ssts ♦ n ♥tr♦r rts VI = p1, . . . ,pn ♥ b♦♥r② rts VB = pn+1, . . . ,pn+b ♦ st♥t rts pi, pj ∈ V r ♥♦rs t② r t ♥ ♣♦♥ts ♦ s♦♠ E = [pi, pj] ∈ E ♥ ♦r ♥② pi ∈ V t Ni = j : [pi, pj] ∈ E t st ♦ ♥s ♦ ♥♦rs ♦ pi

♣r♠tr③t♦♥ f ♦ ST s s② s♣ t ♦tr ② r♦♥ tt s ②♥♥ t ♥rs ♣r♠tr③t♦♥ g = f−1 s ♠♣♣♥ g s ♥q② tr♠♥② s♣②♥ t ♣r♠tr ♣♦♥ts ui = g(pi) ♦r rt① pi ∈ V ♥ ♠♥♥ ttg s ♦♥t♥♦s ♥ ♥r ♦r tr♥ ♥ ts stt♥ g|T s t ♥r ♠♣ r♦♠ sr tr♥ T = [pi, pj, pk] t♦ t ♦rrs♣♦♥♥ ♣r♠tr tr♥ t = [ui, uj, uk]

♥ f |t = (g|T )−1 s t ♥rs ♥r ♠♣ r♦♠ t t♦ T ♣r♠tr ♦♠♥ Ω ♥②s t ♥♦♥ ♦ ♣r♠tr tr♥s s r

Pr♠tr③t♦♥ ② ♥ ♦♠♥t♦♥s

rtr s♠♣ ♦r ♦♥strt♥ ♣r♠tr③t♦♥ ♦ tr♥ ♠s s s ♦♥t ♦♦♥ ♣②s ♠♦ ♠♥ tt t s ♦ t tr♥ ♠s r s♣r♥stt r ♦♥♥t t t rts ♥♦ ① t ♦♥r② ♦ ts s♣r♥ ♥t♦rs♦♠r ♥ t ♣♥ t♥ t ♥tr♦r ♦ ts ♥t♦r r① ♥ t ♥rt②

ST

Ω

f|t

t T

g|T

r Pr♠tr③t♦♥ ♦ tr♥ ♠s

♠♦st ♥t ♦♥rt♦♥ ♥ ♥ s♠♣② ss♥ t ♣♦st♦♥s r t ♦♥ts ♦t ♥t♦r ♦♠ t♦ rst s ♣r♠tr ♣♦♥ts

ss♠ s♣r♥ t♦ ♥ t s♥s tt t rst ♥t s ③r♦ ♥ t♣♦t♥t ♥r② s st 1

2Ds2 r D s t s♣r♥ ♦♥st♥t ♥ s t ♥t ♦ t

s♣r♥ t♥ ♥ ♦r♠③ ts ♣♣r♦ s ♦♦s ❲ rst s♣② t ♣r♠tr♣♦♥ts ui = (ui, vi) i = n + 1, . . . , n + b ♦r t ♦♥r② rts pi ∈ VB ♦ t ♠s♥ s♦♠ ② s t♦♥ ♥ ♠♥♠③ t ♦r s♣r♥ ♥r②

E = 12

n∑

i=1

∑

j∈Ni

12Dij‖ui − uj‖2,

r Dij = Dji s t s♣r♥ ♦♥st♥t ♦ t s♣r♥ t♥ pi ♥ pj t rs♣tt♦ t ♥♥♦♥ ♣r♠tr ♣♦st♦♥s ui = (ui, vi) ♦r t ♥tr♦r ♣♦♥ts

s t ♣rtrt ♦ E t rs♣t t♦ ui s

∂E

∂ui

=∑

j∈Ni

Dij(ui − uj),

t ♠♥♠♠ ♦ E s ♦t♥ ∑

j∈Ni

Dijui =∑

j∈Ni

Dijuj

♦s ♦r i = 1, . . . , n s s q♥t t♦ s②♥ tt ♥tr♦r ♣r♠tr ♣♦♥tui s ♥ ♥ ♦♠♥t♦♥ ♦ ts ♥♦rs

ui =∑

j∈Ni

λijuj,

t ♥♦r♠③ ♦♥tsλij = Dij

/∑

k∈Ni

Dik

tt ♦♦s② s♠ t♦ 1

t♦♥ t♦r 1

2♣♣rs s s♠♠♥ ♣ t s ♥ ts ② ♦♥ts r② t

② s♣rt♥ t ♣r♠tr ♣♦♥ts ♦r t ♥tr♦r ♥ t ♦♥r② rts ♥ ts♠ ♦♥ t rt ♥ s ♦ t

ui −∑

j∈Ni,j≤n

λijuj =∑

j∈Ni,j>n

λijuj,

♥ s tt ♦♠♣t♥ t ♦♦r♥ts ui ♥ vi ♦ t ♥tr♦r ♣r♠tr ♣♦♥ts ui

rqrs t♦ s♦ t ♥r s②st♠s

AU = U ♥ AV = V ,

r U = (u1, . . . , un) ♥ V = (v1, . . . , vn) r t ♦♠♥ t♦rs ♦ ♥♥♦♥ ♦♦r♥ts U = (u1, . . . , un) ♥ V = (v1, . . . , vn) r t ♦♠♥ t♦rs t ♦♥ts

ui =∑

j∈Ni,j>n

λijuj ♥ vi =∑

j∈Ni,j>n

λijvj

♥ A = (aij)i,j=1,...,ns t n× n ♠tr① t ♠♥ts

aij =

1 i = j,

−λij j ∈ Ni,

0 ♦trs.

t♦s ♦r ♥t② s♦♥ ts s②st♠s r sr ♥ ♣tr ♦ ts ♦rs♥♦ts

r②♥tr ♦♦r♥ts

qst♦♥ r♠♥s ♦ t♦ ♦♦s t s♣r♥ ♦♥st♥ts Dij ♥ t s♣r♥ ♠♦ ♦r♠♦r ♥r② t ♥♦r♠③ ♦♥ts λij ♥ s♠♣st ♦ ♦ ♦♥st♥ts♣r♥ ♦♥st♥ts Dij = 1 ♦s t♦ t ♦r ♦ tt ❬ ❪ ♦ s t ♥ ♠♦r strt r♣t♦rt stt♥ t♦ ♦♠♣t strt ♥ ♠♥s ♦ ♣♥rr♣s ♥ t ♦ t♥ s♣r♥ ♦♥st♥ts tt r ♣r♦♣♦rt♦♥ t♦ t ♥ts ♦t ♦rrs♣♦♥♥ s ♥ t tr♥ ♠s s s ② r♥r ♥ ♦r♠♥♥ ❬❪ ♠♥ r ♦ ♦t ♣♣r♦s s tt t② ♦ ♥♦t t ♦♦♥ ♠♥♠♠rqr♠♥t tt s♦ ①♣t r♦♠ ♥② ♣r♠tr③t♦♥ ♠t♦

♥r r♣r♦t♦♥ ♣♣♦s tt ST s ♦♥t♥ ♥ ♣♥ s♦ tt ts rts ♦♦r♥ts pi = (xi, yi, 0) t rs♣t t♦ s♦♠ ♣♣r♦♣rt② ♦s♥ ♦rt♦♥♦r♠♦♦r♥t r♠ ♥ ♦② s♦♠tr ♥ ts ♦♣t♠ ♣r♠tr③t♦♥ ♥ ♥ ② st s♥ t ♦ ♦♦r♥ts xi = (xi, yi) s ♣r♠tr ♣♦♥ts t♠sstt s ② stt♥ ui = xi ♦r i = 1, . . . , n + b s t ♦r ♣r♠tr③t♦♥ t♥ s ♥r ♥t♦♥ s② tt ♣r♠tr③t♦♥ ♠t♦ s ♥r r♣r♦t♦♥ t♣r♦s s ♥ s♦♠tr ♠♣♣♥ ♥ ts stt♥

r ♦tt♦♥ ♦r t ♦♥strt♦♥ ♦ r②♥tr ♦♦r♥ts

♥ t stt♥ r♦♠ t ♣r♦s st♦♥ ♥r r♣r♦t♦♥ ♥ t♣r♠tr ♣♦♥ts ♦r t ♦♥r② rts r st ♦rrt② ♥ t s λij r ♦s♥s tt

xi =∑

j∈Ni

λijxj ♥∑

j∈Ni

λij = 1

♦r ♥tr♦r rts ❱s λij t ♦t ts ♣r♦♣rts r s♦ r②♥tr♦♦r♥ts ♦ xi t rs♣t t♦ ts ♥♦rs xj j ∈ Ni s♦♠ xi s ①t② tr♥♦rs t♥ t λij r ♥q② ♥ ♥ ts r②♥tr ♦♦r♥ts ♥str♥s t② ♠♥② s ♣♣t♦♥s ♥ ♦♠♣tr r♣s ♦r ♥P♦♥ s♥ r②tr♥♥trst♦♥ ♦♠tr ♠♦♥ tr♥r é③r♣ts s♣♥s ♦r tr♥t♦♥s ♥ ♠♥② ♦tr s t ♥t ♠♥t♠t♦ trr♥ ♠♦♥

♦r ♣♦②♦♥s t ♠♦r t♥ tr rts t r②♥tr ♦♦r♥ts ♦ ♣♦♥t♥ t ♥tr♦r r ♦r ♥♦t ♥q ♥②♠♦r ♥ tr r sr ②s ♦ ♥♥t♠ ♠♦st ♣♦♣r ♦ t♠ ♥ sr ♥ ♦♠♠♦♥ r♠♦r ❬♦trt ❪ tt s r② r ♦r ♥② ♥tr♦r ♣♦♥t xi ♥ ♦♥ ♦ ts ♥♦rs xj t rij = ‖xi − xj‖ t ♥t ♦ t eij = [xi, xj] t♥ t t♦♣♦♥ts ♥ t t ♥s t t ♦r♥rs ♦ t tr♥s ♥t t♦ eij ♥♦t ss♦♥ ♥ r r②♥tr ♦♦r♥ts λij ♦ xi t rs♣t ts ♥♦rsxj j ∈ Ni ♥ t♥ ♦♠♣t ② t ♥♦r♠③t♦♥ λij = wij

/∑

k∈Niwik r♦♠ ♥②

♦ t ♦♦♥ ♦♠♦♥♦s ♦♦r♥ts wij

• ❲s♣rss ♦♦r♥ts rst ♥r③t♦♥ ♦ r②♥tr ♦♦r♥ts ♦s t♦ ❲s♣rss ❬❪ ♦ sst t♦ st

wij =cot αji + cot βij

rij2

.

❲ s ♠♥② ♥trst ♥ ♣♣②♥ ts ♦♦r♥ts ♥ ♥t ♠♥t♠t♦s sr♥ t ❬❪ s t♠ ♦r ♣r♠tr③♥ tr♥ ♠ss ♥②r t ❬❪ ♦r ♥tr♣♦t♥ ♦♦r s ♥s ♦♥① ♣♦②♦♥s♦r♦r s♠♣ ♦♠tr ♦♥strt♦♥ ♦ ts ♦♦r♥ts s ♥ ② t ❬❪

p2 = (-1,-1,0) p3 = (1,-1,0)

p4 = (1,1,0)

p5 = (-1,1,0)

p1 = (3,0,1)

r ①♠♣ ♦ tr♥ ♠s ♦r ♦♥② t r②♥tr ♠♣♣♥ t♠♥ ♦♦r♥ts s t♦♥

• srt r♠♦♥ ♦♦r♥ts ♥♦tr t②♣ ♦ r②♥tr ♦♦r♥ts tt st♠r♦♠ ♥t ♠♥t ♠t♦s ♥ t② rs r♦♠ t st♥r ♣s ♥r♣♣r♦①♠t♦♥ t♦ t ♣ qt♦♥ r ♥ ②

wij = cot γij + cot γji.

♥ t ♦♥t①t ♦ ♠s ♣r♠tr③t♦♥ ts ♦♦r♥ts r rst s ② t ❬❪ t t② s♦ ♥ s t♦ ♦♠♣t srt ♠♥♠ srs❬P♥ ♥ P♦tr ❪

• ♥ ♦♦r♥ts ② srt③♥ t ♠♥ t♦r♠ ♦tr ❬❪♦♥ ②t ♥♦tr st ♦ r②♥tr ♦♦r♥ts t

wij =tan

αij

2+ tan

βji

2

rij

.

❲ s ♠♥ ♣♣t♦♥ s ♠s ♣r♠tr③t♦♥ ♦r♠♥♥ ♥ r♥ ❬❪♥ ♦r♠♥♥ ♥ ♦tr ❬❪ tr s♦ tt t② ♠♥② ♦tr s♣♣t♦♥s ♥ ♣rtr ♥ ♦♠♣tr r♣s

t② ♦ tr ♦s s tt t ts wij ♣♥ ♦♥ ♥s ♥ st♥s♦♥② s♦ tt t② ♥ ♥♦t ♦♥② ♦♠♣t xi ♥ ts ♥♦rs r ♦♣♥r t♠♦r ♥r② ♦r ♥② ♥tr♦r rt① pi ∈ VI ♦ tr♥ ♠s ts ♥s ♥st♥s r st t♥ r♦♠ t tr♥s r♦♥ pi ♦rs ♥ tr♥t ♣♣r♦tt s ♥tr♦ ② ♦tr ❬❪ s t♦ ♦② tt♥ t ♦♥r♥ ♦ tr♥s r♦♥pi ♥t♦ t ♣♥ t ♥ ①♣♦♥♥t ♠♣ ♥ t♥ t♦ ♦♠♣t t ts wij

r♦♠ ts ♣♥r ♦♥rt♦♥ tr♥ ♠s ♣r♠tr③t♦♥ tt s ♦♠♣t ② s♦♥ t ♥r s②st♠s

t ♥② st ♦ r②♥tr ♦♦r♥ts λij s r②♥tr ♠♣♣♥ ♥ ♦♦s②s t ♥r r♣r♦t♦♥ ♣r♦♣rt② ♣r♦ tt ♥ ♣♣r♦♣rt ♠t♦ ♦r ♦♠♣t♥t ♣r♠tr ♣♦♥ts ♦r t ♦♥r② rts ♠♣♣♥ t♠ t♦ t st sqrs♣♥ s t♦♥ s s

s♣t ts ♣r♦♣rt② t ♠② ♣♣♥ tt r②♥tr ♠♣♣♥ ♥ ♦♥strt♦r ♥♦♥♣♥r ♠s s ♥ ♥①♣t rst s t s♠♣ ①♠♣ ♥ r

x4 = (-1,-1)

x5 = (1,0)

x6 = (-1,1)

x7 = (-1/3,0)

x1 = (-1/5,0)

x3 = (1/3,0)

x2 = (0,0)

r ①♠♣ ♦ tr♥ ♠s ♦r t ♥r s②st♠ t ❲s♣rss♦♦r♥ts s s♥r

strts s u2 = (−1,−1) u3 = (1,−1) u4 = (1, 1) u5 = (−1, 1) s ♣r♠tr♣♦♥ts ♦r t ♦r ♦♥r② rts ♥ ♦♠♣t t r②♥tr ts λ12 λ13 λ14λ15 t t ♦r♠s sr ♦ t♥ t t ♦♦♥ ♣♦st♦♥s ♦r u1

❲s♣rss ♦♦r♥ts u1 = (−35.1369, 0),

srt r♠♦♥ ♦♦r♥ts u1 = (2.1138, 0),

♠♥ ♦♦r♥ts u1 = (0.4538, 0).

t s ♦♥② t ♠♥ ♦♦r♥ts ② ♣♦st♦♥ ♦r u1 tt s ♦♥t♥ ♥t ♦♥① ♦ t ♦tr ♦r ♣r♠tr ♣♦♥ts ♥ s♥ t ♦tr ♦♦r♥ts rt ♣r♠tr tr♥s tt ♦r♣ ts ♦t♥ t tt② ♣r♦♣rt② tt ♥②♣r♠tr③t♦♥ s♦

rs♦♥ ♥ ts ♦r s tt t ❲s♣rss ♥ srt r♠♦♥ ♦♦r♥ts ♥ ss♠ ♥t s ♥ rt♥ ♦♥rt♦♥s t ♦♥ ♥ r rs t ♠♥ s ♦♦r♥ts r ②s ♣♦st ♥ ♦r♣♣♥ tr♥s ♠② ♦r ♦r ♥t ts ts ♥r ♣♣♥s ts r ♣♦st ♥t ♣r♠tr ♣♦♥ts ♦ t ♦♥r② rts ♦r♠ ♦♥① s♣ ttr t srst ♥ ♣r♦♥ ② tt ❬❪ ♦r t s♣ s ♦ λij = 1/ηi r ηi = #Ni st ♥♠r ♦ pis ♥♦rs r ♥♦t tr r②♥tr ♦♦r♥ts t ♦tr❬❪ ♦sr tt t ♣r♦♦ rrs ♦r t♦ rtrr② ♣♦st ts λij ♥t②♦rtr t ❬❪ ♦ ♥ s♦ tt t rstrt♦♥ t♦ ♦♥① ♦♥r② ♥ ♦♥sr② r① t ts rqrs t♦ s♦ ♥♦♥♥r ♣r♦♠

♥♦tr ♠♣♦rt♥t s♣t ♦♥r♥s t s♦t② ♦ t ♥r s②st♠s ♥ ts ♥ s♦♥ tt t ♠tr① A s ②s r♥t t♦ ♥♦♥s♥r ♦r srtr♠♦♥ ❬P♥ ♥ P♦tr ❪ ♥ ♠♥ ♦♦r♥ts ❬♦tr ❪ ♦r❲s♣rss ♦♦r♥ts ♦r t ♠② ♣♣♥ tt t s♠ ♦ ♦♠♦♥♦s ♦♦r♥ts Wi =

∑

k∈Niwik s ③r♦ s♦ tt t ♥♦r♠③ ♦♦r♥ts λij ♥ ts t ♠tr①

A r ♥♦t ♥ ♥ ♥ t ①♠♣ s♦♥ ♥ r ts t② ♣♣♥s♦r ♥tr♦r rts x1 x2 x3 t ♥ s♣ t ♥♦r♠③t♦♥ ♥ tr② t♦ s♦t q♥t ♥ ♥ ♦♠♦♥♦s s②st♠s WAU = WU ♥ WAV = WVt W = diag(W1, . . . ,Wn) ♥st ♥ tt t ♠tr① WA s s♥r ♥ ts

♣rtr ①♠♣ ♥♠② WA =(

0 −50 040 0 −240 18 0

)

♦♥r② ♣♣♥

rst st♣ ♥ ♦♥strt♥ r②♥tr ♠♣♣♥ s t♦ ♦♦s t ♣r♠tr ♣♦♥ts♦r t ♦♥r② rts ♥ t s♠♣st ② ♦ ♦♥ t s t♦ st ♣r♦t t ♦♥r② rts ♥t♦ t ♣♥ tt ts t ♦♥r② rts st ♥ st sqrs s♥s♦r ♦r ♠ss t ♦♠♣① ♦♥r② ts s♠♣ ♣r♦r ♠② t♦ ♥sr ♦♦rs ♥ t ♦♥r② ♣♦②♦♥ ♥ ♥♥♦t s ♥ ♥r tr rt♦ sss t♦ t ♥t♦ ♦♥t r ♦♦s♥ t s♣ ♦ t ♦♥r② ♦ t♣r♠tr ♦♠♥ ♥ ♦♦s♥ t strt♦♥ ♦ t ♣r♠tr ♣♦♥ts r♦♥ t♦♥r②

♦♦s♥ t s♣

♥ ♠♥② ♣♣t♦♥s t s s♥t ♦r ♥ sr t♦ t rt♥ ♦r rs ♣r♠tr ♦♠♥ t t ♥t tt s ♦♥① s♣ r♥ts ttt② ♦ t ♣r♠tr③t♦♥ ♣♦st r②♥tr ♦♦r♥ts t ♠♥ ♦♦r♥ts r s t♦ ♦♠♣t t ♣r♠tr ♣♦♥ts ♦r t ♥tr♦r rts ♦♥①t② rstrt♦♥ ♠② ♦r ♥rt st♦rt♦♥s ♥r t ♦♥r② ♥t ♦♥r② ♦ t tr♥ ♠s ST ♦s ♥♦t rs♠ ♦♥① s♣ ♥ ♣rts♦t♦♥ t♦ ♦ s st♦rt♦♥s s t♦ rt ♦♥r② t♦ ♠♥t t♥ ♠s t ①tr tr♥s r♦♥ t ♦♥r② s♦ s t♦ ♦♥strt ♥ ①t♥♠s t ♥ ♦♥r② s ♣♣r♦ s ♥ sss② s ② t ❬❪ ♥ ós ♥ ❱ár② ❬❪

♦♦s♥ t strt♦♥

s ♣r♦r ♠♥t♦♥ ♥ t trtr s t♦ s s♠♣ ♥rt ♣r♠tr③t♦♥ ♠t♦ s s ♦r ♥t ❬r t ❪ ♦r ♥tr♣t ♣r♠tr③t♦♥❬ ❪ ♦r ♣♥ t ♣r♠tr ♣♦♥ts tr r♦♥ t ♦ ♦♥r② ♦r ♦♥ s ♦ t ♦♥r② ♥ ♦r♥ t rt♥r ♦♠♥ ❬♦r♠♥♥ t♦♥ ❪

s♣t ts rsts ♦r♥ ♣rtt② ♥ s♦♠ ss ♥ t♦ ① t ♦♥r②rts ♠② sr ♠tt♦♥ ♥ ♦trs ♥ t ♥①t ♣tr sts ♣r♠tr③t♦♥ ♠t♦s tt ♥ ♥ t ♣♦st♦♥ ♦ t ♦♥r② ♣r♠tr ♣♦♥ts ♥ t♦♣t♠③t♦♥ ♣r♦ss ♥ ts ② ♣r♠tr③t♦♥s t ss st♦rt♦♥

♣tr

tt♥ t ♦♥r② r

s ①♣♥ ♥ t ♣r♦s st♦♥ tts t♦r♠ ♦♠♥ t ♠♥ ts♣r♦s ♣r♦② ♦rrt ② ♦ ♦♥strt♥ ♣r♠tr③t♦♥ ♦r ssr ♦r ♦r s♦♠ srs t ♥sst② t♦ ① t ♦♥r② ♦♥ ♦♥①♣♦②♦♥ ♠② ♣r♦♠t r ♦r t ♦♦♥ rs♦♥s ♥ ♥rt s t t♦ ♥ ♥tr ② ♦ ①♥ t ♦rr ♦♥ ♦♥① ♣♦②♦♥ ♥ ♦r s♦♠ srs t s♣ ♦ t ♦♥r② s r r♦♠ ♦♥① r♦r t♦t♥ ♣r♠tr③t♦♥ s♦s ♦r♠t♦♥s ♥ ♦♥ ♥ ♠♥ r♥t②s ♦ ♠♣r♦♥ t rst s♦♥ ♥ t r t s♦♦t♥ ♣r♠tr③t♦♥ ♣r♦② ♥♦t s ♦♦ s t ♦♥ s♦♥ ♥ r tt ttr ♠ts t t♥♥r ♦ ①♣t ♦r s ♠s ♦r ts rs♦♥s t ♥①t st♦♥ stst ♠t♦s tt ♥ ♦♥strt ♣r♠tr③t♦♥s t r ♦♥rs tt ♠♥♠③♦r♠t♦♥s ♥ s♠r ② ❲ strt ② ♥ ♥ ♥tt♦♥ ♦ ♦ t♦ s t ♥♦t♦♥sr♦♠ r♥t ♦♠tr② ①♣♥ ♥ ♣tr ♥ ♦r ♦♥t①t ♦ ♣r♠tr③t♦♥t r ♦♥rs

♦r♠t♦♥ ♥②ss

♦ s ♦ t♦ ♣♣② t t♦rt ♦♥♣ts ①♣♥ ♥ ♣tr rst ♥ t♦ rs♣tr ♥tt ♠♥♥

♦♥ ♠tr①

rst rts ♦ t ♣r♠tr③t♦♥ r ♥♦ ♥ ♦r♠t♦♥ ♥②ss t st♥ ♥ssr② t♦ ♥ ♥tt♦♥ ♦ tr ♦♠tr ♠♥♥ ♥ ♣②ss ♠tr ♣♦♥t♠♥s sts t ♠♦♠♥t ♦ ♥ ♦t ♣♣r♦①♠t ② ♣♦♥t p ♥ ♦rsr ♣♣ t♦ t trt♦r② s t r sr ② t ♣♦♥t p ♥ t rs r♦♠t0 t♦ t1 r t ♥♦ts t♠ ♥t♦♥ ♣tt♥ ♥ t♠ t ♥ ♦rrs♣♦♥♥t t ♣♦st♦♥ p(t) = x(t), y(t), z(t) ♦ t ♣♦♥t p s ♣r♠tr③t♦♥ ♦ ttrt♦r② ♣r♠tr③t♦♥ ♦ r t s ♥♦♥ tt t t♦r ♦ trts v(t) = ∂p/∂t = ∂x/∂t, ∂y/∂t, ∂z/∂t ♦rrs♣♦♥s t♦ t s♣ ♦ p tt♠ t

r ♠s t ♥ ② tt ♠s t ♦♠♦♠♦r♣ t♦ s s♥ ts♠str ♦rt♠ ❬r ♥ rt ❪ tt♦tr ♣r♠tr③t♦♥ ♦t♥② ①♥ t ♦rr ♦♥ sqr ♣r♠tr③t♦♥ ♦t♥ t r♦♥r②♣r♠tr③t♦♥ ❬r ♥ trr ❪

u

v

ΩRI2 RI

3

S

∂x∂u

∂x∂v

dudv

x(u,v)

ww’

Cu

Cv

Cw

v0

u0

r ♠♥tr② s♣♠♥ts r♦♠ ♣♦♥t (u, v) ♦ Ω ♦♥ t u ♥ t v ①sr tr♥s♦r♠ ♥t♦ t t♥♥t t♦rs t♦ t s♦u ♥ s♦v rs ♣ss♥ tr♦t ♣♦♥t f(u, v)

s s♦♥ ♥ r ♦♥sr ♥♦ ♥t♦♥ f : (u, v) 7→ (x, y, z) ♣tt♥ ss♣ Ω ♦ R

2 ♥ ♦♥t♦♦♥ ♦rrs♣♦♥♥ t sr S ♦ R3 srs (u, v)

r t ♦♦r♥ts ♥ ♣r♠tr s♣ ♥ t s ♦ r ♣r♠tr③t♦♥ tr s sr ② s♥ ♣r♠tr t ♥ ♦♥trst ♥ ♦r s ♦♥sr sr♣r♠tr③t♦♥ f(u, v) = x(u, v), y(u, v), z(u, v) ♥ tr r t♦ ♣r♠trs u♥ v r♦r t ♥ ♣♦♥t (u0, v0) ♦ t ♣r♠tr s♣ Ω tr r t♦s♣ t♦rs t♦ ♦♥sr fu = (∂f/∂u)(u0, v0) ♥ fv = (∂f/∂v)(u0, v0) t s s② t♦ tt fu s t s♣ t♦r ♦ t r Cu : t 7→ f(u0 + t, v0) t X (u0, v0) ♥tt fv s t s♣ t♦r ♦ t r Cv : t 7→ f(u0, v0 + t) r Cu rs♣Cv s t s♦u rs♣ t s♦v r ♣ss♥ tr♦ f(u0, v0) t ♠ tr♦f ♦ t ♥ ♦ qt♦♥ u = u0 rs♣ v = v0

1st ♥♠♥t ♦r♠ ♥ t ♥s♦tr♦♣② ♣s

t tt ♣♦♥t ♦♥ ♠② t♥ tt t ♥♦r♠t♦♥ ♣r♦ ② t t♦ t♦rs fu(u0, v0)♥ fv(u0, v0) s ♥♦t s♥t t♦ rtr③ t st♦rt♦♥s t♥ Ω ♥ S ♥ t♥♦r♦♦ ♦ (u0, v0) ♥ f(u0, v0) ♥ t t② ♥ s t♦ ♦♠♣t ♦ ♥rtrr② t♦r w = (a, b) ♥ ♣r♠tr s♣ s tr♥s♦r♠ ♥t♦ t♦r w′ ♥ t♥♦r♦♦ ♦ (u0, v0) ♥ ♦tr ♦rs ♥t t♦ ♦♠♣t t s♣ t♦r w′ =∂f(u0 + t.a, v0 + t.b)/∂t ♦ t r ♦rrs♣♦♥♥ t♦ t ♠ ♦ t strt ♥(u, v) = (u0, v0) + t.w t♦r w′ t t♥♥t t♦ t r Cw ♥ s♠♣②♦♠♣t ② ♣♣②♥ t ♥ r ♥ ♦♥ ♥ tt t ♥ ♦♠♣t r♦♠t rts ♦ f s ♦♦s w′ = afu(u0, v0) + bfv(u0, v0) t♦r w′ s rrrt♦ s t rt♦♥ rt ♦ f t (u0, v0) rt t♦ t rt♦♥ w

♥ ♠tr① ♦r♠ w′ s ♦t♥ ② w′ = J(u0, v0)w r J(u0, v0) s t ♠tr① ♦ t ♣rt rts ♦ f

J(u0, v0) =

∂x∂u

(u0, v0)∂x∂v

(u0, v0)

∂y

∂u(u0, v0)

∂y

∂v(u0, v0)

∂z∂u

(u0, v0)∂z∂v

(u0, v0)

=

[

fu(u0, v0) fv(u0, v0)

]

s r② s ♥ ♣tr t ♠tr① J(u0, v0) s rrr t♦ s t ♦♥♠tr① ♦ f t (u0, v0)

♥♦t♦♥ ♦ rt♦♥ rt ♠s t ♣♦ss t♦ ♥♦ t ♥ ♠♥tr②s♣♠♥t w r♦♠ ♣♦♥t (u0, v0) ♥ ♣r♠tr s♣ ♦♠s ♥ t s tr♥s♦r♠② t ♥t♦♥ f ♦♥ ♠tr① ♣s s♦ ♦♠♣t♥ ♦t ♣r♦ts ♥ t♦r♥♦r♠s ♦♥t♦ t sr S s ♥ ♦♥ s♥ t ♠tr① JTJ rrr t♦ s t 1st

♥♠♥t ♦r♠ ♦ f s♦ sr ♥ t r♥t ♦♠tr② st♦♥ s ♠tr①s ♥♦t ② I ♥ ♥ ②

I(u0, v0) = JTJ =

fu · fu fu · fv

fv · fu fv · fv

x(u,v)

u

v

ΩRI2 RI

3

S

dudv

∂x∂u

∂x∂v

r ♥s♦tr♦♣② ♥ ♠♥tr② r s tr♥s♦r♠ ♥t♦ ♥ ♠♥tr② ♣s

1st ♥♠♥t ♦r♠ I(u0, v0) s s♦ rrr t♦ s t ♠tr t♥s♦r ♦ f s♥ t ♠s t ♣♦ss t♦ ♠sr ♦ st♥s ♥ ♥s r tr♥s♦r♠ ♥ t♥♦r♦♦ ♦ (u0, v0) sqr ♥♦r♠ ♦ t ♠ w′ ♦ t♦r w s ♥② ||w′||2 = wT Iw ♥ t ♦t ♣r♦t w′T

1 w′2 = wT

1 Iw2 tr♠♥s ♦ t ♥t♥ w1 ♥ w2 s tr♥s♦r♠ ♥①t st♦♥ s ♦♠tr ♥tr♣rtt♦♥ ♦t 1st ♥♠♥t ♦r♠ ♥ ts ♥s

♣r♦s st♦♥ s st ♦ ♥ ♠♥tr② s♣♠♥t r♦♠ ♣r♠trs♣ ♦t♦♥ (u0, v0) s tr♥s♦r♠ tr♦ t ♣r♠tr③t♦♥ f s s♦♥ ♥r ♦r ♦ s ♥♦ t♦ tr♠♥ t ♥ ♠♥tr② r ♦♠s

t s ♦♥sr t t♦ ♥s λ1, λ2 ♦ G ♥ t♦ ss♦t ♥t ♥t♦rsw1,w2 ♦t tt s♥ I s s②♠♠tr w1 ♥ w2 r ♦rt♦♦♥ ♥ rtrr② ♥tt♦r w ♥ rtt♥ s w = cos(θ)w1 + sin(θ)w2 sqr ♥♦r♠ ♦ w′ = Jw st♥ ♥ ②

||w′||2 = wT Iw

= (cos(θ)w1 + sin(θ)w2)T I(cos(θ).w1 + sin(θ).w2)

= cos2(θ)||w1||2λ1 + sin2(θ)||w1||2λ2+sin(θ) cos(θ)(λ1w

T2 w1 + λ2w

T1 w2)

= cos2(θ).λ1 + sin2(θ).λ2

♥ qt♦♥ t r♦ss tr♠s wT1 w2 ♥ wT

2 w1 ♥s s♥ w1 ♥ w2 r♦rt♦♦♥ t s s ♥♦ t r t ①tr♠ ♦ ||w′||2 ♥ ♥t♦♥ ♦ θ

∂||w′(θ)||2∂θ

= 2 sin(θ) cos(θ)(λ2 − λ1)= sin(2θ)(λ2 − λ1)

①tr♠ ♦ ||w′(θ)||2 r t♥ ♦t♥ ♦r θ ∈ 0, π/2, π, 3π/2 ♦r w = w1

♦r w = w2 r♦r t ♠①♠♠ ♥ ♠♥♠♠ s ♦ ||w′(θ)||2 r λ1 ♥ λ2

♥

• ①s ♦ t ♥s♦tr♦♣② ♣s r Jw1 ♥ Jw2

• ♥ts ♦ t ①s r√

λ1 ♥√

λ2

♦t s ♠♥t♦♥ ♥ ♣tr t ♥ts ♦ t ①s√

λ1 ♥√

λ2 s♦ ♦rrs♣♦♥ t♦ t s♥r s ♦ t ♠tr① J ♦♠tr ♥tr♣rtt♦♥ ♦ t ❱ ss♦ ①♣♥ ♥ tt ♣tr ❲ r♠♥ tt t s♥r ♦♠♣♦st♦♥ ❱♦ ♠tr① J rts

J = UΣVT = U

σ1 0

0 σ2

0 0

VT

r U : 3× 3 ♥ V : 2× 2 r s tt tr ♦♠♥ t♦rs ♦r♠ ♥ ♦rt♦♥♦r♠ss s♦ s② tt t② r ♥t ♠trs ♥ Σ s ♠tr① s tt ♦♥② ts♦♥ ♠♥tsσ1, σ2 r ♥♦♥③r♦ srs σ1, σ2 r t s♥r s ♦J ♥ ♦r s ② ssttt♥ t ♦♥ ♠tr① J t ts ❱ ♦t♥

I = JTJ

= (UΣVT )T (UΣVT )

= VΣTUTUΣVT = VΣTΣVT

= V

[σ2

1 00 σ2

2

]

VT

♥U s ♥t ♠tr① t ♥tr tr♠UTU ♦ t tr ♥ s q t♦ t ♥tt②♠tr① ♥ ♥ss ❲ t♥ ♦t♥ ❱ ♦ t ♠tr① I tt s s♦ ♦♥③t♦♥♦ I ② ♥t② ♦ t ❱ t rt♦♥ t♥ t ♥s λ1, λ2 ♦ G♥ t s♥r s σ1, σ2 ♦ J λ1 = σ2

1 ♥ λ2 = σ22

❲ ♥ ♥♦ t ①♣rss♦♥ ♦ t ♥ts ♦ t ♥s♦tr♦♣② ♣s σ1 ♥ σ2 ❲rst r t ①♣rss♦♥ ♦ t ♦♥ ♠tr① s ♥t♦♥ ♦ t r♥t t♦rs

I = JTJ =

(E FF G

)

t

E = f 2u

F = fu · fv

G = f 2v

♥ts ♦ t ①s ♦ t ♥s♦tr♦♣② ♣s ♦rrs♣♦♥ t♦ t ♥s ♦I r ①♣rss♦♥ ♥ ♦♥ ② ♦♠♣t♥ t sqr r♦♦ts ♦ t ③r♦s ♦ trtrst ♣♦②♥♦♠ |I− σId| tt s s♦♥ ♦rr qt♦♥ ♥ σ

σ1 =√

1/2(E + G) +√

(E −G)2 + 4F 2

σ2 =√

1/2(E + G)−√

(E −G)2 + 4F 2

r ♦ X, Y ss ♥ tr♥

♦r♠t♦♥ ♥②ss ♦r tr♥t srs

♦r♠t♦♥ ♥②ss ♥tr♦ ♥ t ♣r♦s st♦♥ ♥♦s t ♦♠♣tt♦♥ ♦t r♥ts ♦ t ♣r♠tr③t♦♥ s ♥t♦♥ ♦ t ♣r♠trs u ♥ v ♥ t s♦ tr♥t sr t ♣r♠tr③t♦♥ s ♣s ♥r ♥t♦♥ r♦rt r♥ts r ♦♥st♥t ♥ tr♥

♦r st②♥ t ♦♠♣tt♦♥ ♦ ts r♥ts ♥ t♦ ♠♥t♦♥ tt ♦rstt♥ s st② r♥t r♦♠ t ♣r♦s st♦♥ ♥ ♦r s s ♣r♦s② ♠♥t♦♥ t sr s ♥ ♥ ♦r ♦ s t♦ ♦♥strt t ♣r♠tr③t♦♥ ♥ tsstt♥ t s♠s ♠♦r ♥tr t♦ rtr③ t ♥rs ♦ t ♣r♠tr③t♦♥ t ♥t♦♥ tt ♦s r♦♠ t sr ♥♦♥ t♦ t ♣r♠tr s♣ ♥♥♦♥s ♥t♦♥ s s♦ ♣s ♥r ♦ ♣♦rt ♦r♠t♦♥ ♥②ss t♦ ts stt♥ t s♣♦ss t♦ ♣r♦ tr♥ t ♥ ♦rt♦♥♦r♠ ss X, Y s s♦♥ ♥ r ♥ ♥ s ♦♥ ♦ t rts pi ♦ t tr♥ s t ♦r♥ ♥ ts ss ♥ st② t ♥rs ♦ t ♣r♠tr③t♦♥ tt s t♦ s② t ♥t♦♥ tt ♠♣s ♣♦♥t (X, Y ) ♦ t tr♥ t♦ ♣♦♥t (u, v) ♥ ♣r♠tr s♣ s ♥t♦♥ rts

u(X, Y ) = λ1ui + λ2uj + λ3uk

v(X, Y ) = λ1vi + λ2vj + λ3vk

r (λ1, λ2, λ3) ♥♦t t r②♥tr ♦♦r♥ts t t ♣♦♥t (x, y) ♥ t tr♥♦♠♣t s ♦r

λi

λj

λk

=1

2|T |X,Y

Yj − Yk Xk −Xj XjYk −XkYj

Yk − Yi Xi −Xk XkYi −XiYk

Yi − Yj Xj −Xi XiYj −XjYi

XY1

r 2|T |X,Y = (XiYj − YiXj) + (XjYk − YjXk) + (XkYi − YkXi) ♥♦ts t ♦r ♦ t tr♥ ♥ s♣ ts t♠

r s♦ rs ♥ ss♦t r♥ts

② ssttt♥ t s ♦ λ1 λ2 ♥ λ3 ♥ u = λ1ui + λ2uj + λ3uk rs♣ v ♦t♥

(

∂u/∂X

∂u/∂Y

)

= MT

ui

uj

uk

=1

2|T |X,Y

(Yj − Yk Yk − Yi Yi − Yj

Xk −Xj Xi −Xk Xj −Xi

)

ui

uj

uk

r t ♠tr① MT s♦② ♣♥s ♦♥ t ♦♠tr② ♦ t tr♥ T s s♦♥ ♥ r ts r♥ts r r♥t t str♦♥② rt t t

r♥ts ♦ t ♥rs ♥t♦♥ ♦♠♣t ♥ t ♣r♦s st♦♥ r♥t ♦ u rs♣ v ♥trsts t s♦us rs♣ t s♦vs t rt ♥ ♥st ♦ ♥t♥♥t t♦ t♠ ♥ ts ♥♦r♠ s t ♥rs ♦ t ♦♥ ♦♠♣t ♥ t ♣r♦s st♦♥

s r♥ts ♥ t♥ s t♦ t ①♣rss♦♥ ♦ t ♦♥ ♠tr①JT s ♦♦s

JT =

(∂u∂X

∂v∂X

∂u∂Y

∂v∂Y

)

= 12|T |x,y

(0 −11 0

)(X3 −X2 X1 −X3 X2 −X1

Y3 − Y2 Y1 − Y3 Y2 − Y1

)

u1 v1

u2 v2

u3 v3

= 12|T |x,y

(0 −11 0

)(X1 X2 X3

Y1 Y2 Y3

)

0 1 −1−1 0 1

1 −1 0

u1 v1

u2 v2

u3 v3

s♦♥ ♥ ♦ ts qt♦♥ s♦s ♦ t ♦♥ ♠tr① ♦♠♥s t t♦rs ♦ t s ♦ t tr♥ t t (u, v) ♦♦r♥ts tr ♥ s ♠♦rs②♠♠tr ①♣rss♦♥ tt ss t ♦♦r♥ts ♦ t ♣♦♥ts rt② sqr ♠tr① s♣s t ♦♦r♥ts ♦ t tr♥ s s♦s ♥ ♥trst♥ s♠rt② t t♠♥r② ♥♠r i ♥ t ts s r♣rs♥tt♦♥ ♦ t ♠♥r② ♥♠r i ❲ ♦rt ♠♦r ♦♥ t ♥s t ♦♠♣① ♥②ss ♥ t♦♥ tt s t♦♥♦r♠ ♠t♦s

Pr♠tr③t♦♥ ♠t♦s s ♦♥ ♦r♠t♦♥ ♥②ss

s st♦♥ rs ts ♠t♦s s♥ t ♦r♠s♠ ♥tr♦ ♥ ♣tr ♥t♦♥ ♦r ♦r ♦♥ rtr ♥ t♦ r♥ t rr ♦t ♣♦sss♦r ♦ ♦♥s♦♥

• ♦ t ♠t♦s st② t ♥t♦♥ tt ♦s r♦♠ t sr t♦ t ♣r♠trs♣ s ♥ t ♣r♦s t♦♥ s s st ② t t tt t (u, v)♦♦r♥ts r ♥♥♦♥ r♦r t s ♠♦r ♥tr t♦ ♦ r♦♠ t ♥♦♥♦r t sr t♦ t ♥♥♦♥ ♦r t ♣r♠tr s♣

• t ♦tr ♦ t ♠t♦s s t ♥rs ♦♥♥t♦♥ ♥ st② t ♥t♦♥tt ♦s r♦♠ ♣r♠tr s♣ t♦ t sr s ♥ t♦♥ s s st ② t t tt t ♠s t ♦r♠s♠ ♦♠♣t t ss r♥t♦♠tr② ♦♦s ❬♦ r♠♦ ❪ tt s ts ♦♥♥t♦♥

♥ ♦t ♦♥♥t♦♥s r st ♦t r s ② r♥t t♦rs ♦rs s♦♥ ♦r♠t♦♥ ♥②ss ♦r ♦♥ ♦♥♥t♦♥ ♥ s② r♦♠t ♦tr ♦♥ r♦r ①♣t t rs ♦ ♦♥s♦♥ ts ♦s ♥♦t ♥tr♦ ♠t② t♦ ♥rst♥ ts ♠t♦s ♦ r t ♥♦tt♦♥s s ♥ ts st♦♥ rrt t t ♥t♦♥ tt ♠♣s (X, Y ) ♦ ♦♦r♥ts ♥ t tr♥ t♦ (u, v)♦♦r♥ts ♥ ♣r♠tr s♣ Ω

p

qq

q

12

3 p

q...q...

q...

q...

q...

r t t♦ ♦ tr♥ ♣s rt① p s ♦♥str♥ t♦ r♠♥ ♥ tr♥ ♦ t ♣♦②♦♥ ♥ ② ts ♥♦rs qi t t r♥ ♦ ♣♦②♦♥ ts ♥ ② t ♥trst♦♥ ♦ t ♣♥s ♥ ② t s♣♣♦rt ♥s ♦ ts ss

♦tt♦♥s ♦r t ♦♥♥t♦♥ (u, v) 7→ (x, y, z)

s②♠♦s J′, I′, σ′1, σ

′2 r rt t t ♥t♦♥ tt ♦s r♦♠ t ♣r♠tr

s♣ t♦ t sr ♥ ♥♦t rs♣t② t ♦♥ ♠tr① t rst ♥♠♥t♦r♠ ♥ t ♥ts ♦ t t♦ ①s ♦ t ♥s♦tr♦♣② ♣s

❲ ♥♦ st② t rt♦♥s t♥ ts s ♥ t♦s ss♦t t t ♥rs♥t♦♥ ♦♥ ♠tr① ♦ t ♥rs ♦ ♥t♦♥ f s q t♦ t ♥rs ♦t ♦♥ ♠tr① ♦ f−1 r♦r J′ = J−1

♦r♦r t s s② t♦ tt t ❱ ♦ J rtsUΣVT tΣ = (σ1, σ2)♥ t U,V t♦ ♥t ♠trs UTU = Id ♥ VTV = Id t♥ t ❱ ♦ J′

rts VΣ−1UT ♦♥ ♥ tt t ♣r♦t ♦ t t♦ ♠trs s t ♥tt②♠tr① r♦r t ♥ts ♦ t s♠st ♥ rst ①s ♦ t ♥s♦tr♦♣② ♣s♦ t ♥rs ♥t♦♥ r ♥ ②

σ′1 =

1

σ2

; σ′2 =

1

σ1

r♠ t ts ♥t♦♥s ♥ ♥♦ ♣r♦ t♦ r sr ♠t♦s s♦♥ ♦r♠t♦♥ ♥②ss ♥ ①♣rss t♠ ♥ ♦♠♠♦♥ ♦r♠s♠ s ①♣♥ ♦r ♥ t♦ t r ♦ ♥t②♥ tr t sr → ♣r♠trs♣ ♥t♦♥ ♦r♣r♠trs♣→ sr ♥t♦♥ s s ♦r ♦♥ ts ♠t♦s t♦♠♦r ♣rs♦♥s

• t♦ ♦ tr♥ ♣s s♦♠ ♦ t ♠t♦s ♦♥str♥ rt① p t♦ r♠♥ ♥t r♥ ♦ t ♣♦②♦♥ ♥ ② ts ♥♦rs qi s ♥♦t♦♥ s strt ♥r ♦ ♦♠♣t t r♥ ♦ ♣♦②♦♥ t s ♦r ♥st♥ ♣♦ss t♦ ♣♣②tr♥ ♥ ♦♠♥s r♥tr♥t ♣♦②♦♥ ♣♣♥ ♦rt♠ t♦ t ♣♦②♦♥

r Pr♠tr③t♦♥ ♠t♦s s ♦♥ ♦r♠t♦♥ ♥②ss ♦t♥ ♥ t♦ ♠♥♠③ ♥♦♥♥r ♦t ♥t♦♥s ♦ rt t ♦♠♣tt♦♥s ts ♠t♦s♦t♥ s ♠trs♦t♦♥ ♣♣r♦ s ♦♥ ♦♣♣ ts Pr♦rss s tstrtr

♣♣ ② ts ♦rt♠ s sr ♥ ♠♦st ♥r ♦♠♣tr r♣s♦♦s ❬♦② t ❪

• s♥ t② r s ♦♥ t ♥s ♦ t rst ♥♠♥t ♦r♠ t ♦t ♥t♦♥s ♥♦ ♥ ♦r♠t♦♥ ♥②ss r ♦t♥ ♥♦♥♥r ♥ tr♦rt t♦ ♠♥♠③ ♥ ♥ ♥t ② ♦ rt t ♦♠♣tt♦♥s ♦♠♠♦♥② s t♥q ♦♥ssts ♥ r♣rs♥t♥ t sr ♥ ♠trs♦t♦♥♠♥♥r s ♦♥ ♦♣♣s Pr♦rss s t strtr ❬♦♣♣ ❪ ♦rt♠ strts ② ♦♣t♠③♥ s♠♣ rs♦♥ ♦ t ♦t t♥ ♥tr♦st t♦♥ rts ♥ ♦♣t♠③s t♠ ② trt r♥♠♥ts

♦ tt s♥ t ♥r ♥♦t♦♥s rt t ♦r♠t♦♥ ♥②ss ♥t ♣rtr s♣ts tt ♦♥r♥ t ♦♣t♠③t♦♥ ♦ ♦t ♥t♦♥s ♥♦♥ ♦r♠t♦♥ ♥②ss ♥ r sr ss ♠t♦s tt ♦♥ t♦ tst♦r②

r♥r♥ ♦r♠t♦♥ t♥s♦r

st♦r② t♦ ♠♥♠③ t ♦r♠t♦♥s ♦ ♣r♠tr③t♦♥ ♦♥ ♦ t rst ♠t♦ss ♦♣ ② ♦t t ❬❪ ♠♥ ♥ tr ♣♣r♦ ♦♥ssts♥ ♠♥♠③♥ ♠tr① ♥♦r♠ ♦ t r♥r♥ ♦r♠t♦♥ t♥s♦r s ♥♦t♦♥♦♠s r♦♠ ♠♥s ♥ ♠srs t ♦r♠t♦♥ ♦ ♠tr ♥tt② ♥♦ tt t ♠tr t♥s♦r G s q t♦ t ♥tt② ♠tr① t♥ ♥s♦♠tr ♣r♠tr③t♦♥ r♥r♥ ♦r♠t♦♥ t♥s♦r s ♥ ② G− Id♥ ♠srs t ♥♦♥s♦♠tr② ♦ t ♣r♠tr③t♦♥ ♦t t ♠♥♠③ tr♦♥s ♥♦r♠ ♦ ts ♠tr① ♥ ②

‖I− Id‖2F = (λ1 − 1)2 + (λ2 − 1)2

r λ1 ♥ λ2 ♥♦t t ♥s ♦ t rst ♥♠♥t ♦r♠ I

P

♦r♠♥♥ ♥ r♥rs P ♦st② s♦♠tr Pr♠tr③t♦♥ ♦ rs ♠t♦❬♦r♠♥♥ ♥ r♥r ❪ s t♦ ♦r ♥♦ t rst ♠s ♣r♠tr③t♦♥♠t♦ tt ♦♠♣ts ♥tr ♦♥r② s ♠t♦ s s ♦♥ t ♠♥♠③t♦♥ ♦t rt♦ t♥ t t♦ ♥ts ♦ t ①s ♦ t ♥s♦tr♦♣② ♣s s ♦rrs♣♦♥st♦ t ♥♦r♠ ♦ t ♦♥ ♠tr①

K2(JT ) = ‖JT‖2‖J−1T ‖2 = σ1/σ2

♥ ♠♥♠③♥ ts ♥r② s t ♥♠r ♣r♦♠s ♦r♠♥♥ ♥ r♥r r♣ t t♦ ♥♦r♠ ‖.‖2 ② t r♦♥s ‖.‖F tt s t♦ s② t sqr r♦♦t♦ t s♠ ♦ t sqr s♥r s

KF (JT ) = ‖JT‖F‖J−1T ‖F

=√

σ21 + σ2

2

√(

1σ1

)2

+(

1σ2

)2

=σ21+σ2

2

σ1σ2= tr(IT )

t(JT )

s ♥ s♥ ♦rt♥t ♥t♦♥s ♦ tr♠s ② s♠♣ ①♣rss♦♥ t t ♥ ♥ ①♣rss♦♥ ♦rrs♣♦♥s t♦ t rt♦ t♥ t tr ♦ t ♠tr t♥s♦r♥ t tr♠♥♥t ♦ t ♦♥ ♠tr① s ♥t ♥ t ♦r♥ rt ts ♥ s♦ ♥tr♣rt s t rt ♥r② ♣r ♣r♠trs♣ r t tr♠tr(I) ♦rrs♣♦♥s t♦ t rt ♥r② ♥ t ♦♥ t(J) t♦ t rt♦ t♥tr♥s r ♥ ♥ ♥ ♣r♠tr s♣

♥trst♥② s ①♣♥ ② t t♦rs KF (JT ) s♦ rts

KF (JT ) = σ1/σ2 + σ2/σ1 = K2(JT ) + K2(J−1T )

s tr♥t ①♣rss♦♥ rs tt t ♥s♦tr♦♣② ♥ ♦t rt♦♥s r♦♠ Ω t♦ S♥ r♦♠ S t♦ Ω s t♥ ♥t♦ ♦♥t

trt ♠♥♠③t♦♥

♦tt ② t①tr ♠♣♣♥ ♣♣t♦♥s ♥r t ❬❪ st t ② s♥st♦r ♥ ♣r♠tr s♣ s ♦r♠ ♥ t s t①tr♠♣♣ ♦♥t♦ t sr ②♣♣②♥ t ♣r♠tr③t♦♥ ♦r ts rs♦♥ tr ♦r♠s♠ ss t ♥rs ♥t♦♥ tt ♠♣s t ♣r♠tr s♣ ♦♥t♦ t sr r♦r s t ♥♦tt♦♥sJ ′ G′ σ′

1 σ′2 s ①♣♥ t t ♥♥♥ ♦ t st♦♥

♣♦ss ② ♦ rtr③♥ t ♦r♠t♦♥s ♦ t①tr s t♦ ♦♥sr ♣♦♥t♥ rt♦♥ ♥ ♣r♠tr s♣ ♥ ♥②③ ♦ t t①tr s ♦r♠ ♦♥ ttrt♦♥ ♥r t ts t strt s ①t② ♦rrs♣♦♥s t♦ t♥♦t♦♥ ♦ rt♦♥ rt tt ♥tr♦ ♥ t♦♥ ♦r tr♥ T t②

r ♦♠ rsts ♦♠♣t ② strt L2 ♠♥♠③t♦♥ ♣r♠tr③ ♠♦s♦rts② ♦ Pr♦ ♥r ♥ r

♥ t♦ ♥rs tt ♦rrs♣♦♥ t♦ t r ♦ t strt ♦r rt♦♥sstrt L2(T ) ♥ t♦ t ♠①♠♠ strt strt L∞(T )

L2(T ) =√

(σ′21 + σ′2

2 )/2 =√

((1/σ1)2 + (1/σ2)2) /2L∞(T ) = σ′

2 = 1/σ1

r t ①♣rss♦♥ ♦ σ′1 = 1/σ2 ♥ σ′

2 = 1/σ1 r ♥ t t ♥♥♥ ♦ tst♦♥ ♦ ♥rs ♦ tr♥ T r ♦♠♥ ♥t♦ ♦ ♥r② L2(S)♥ L∞(S) ♥ s ♦♦s

L2(S) =√

P

T |T |L2(T )P

T |T |

L∞(S) = maxT L∞(T )

r s♦s s♦♠ rsts ♦♠♣t t ts ♣♣r♦ s ♦r♠s♠ s ♣rtr② st t♦ t①tr ♠♣♣♥ ♣♣t♦♥s s♥ t ♠♥♠③s t ♦r♠t♦♥stt r rs♣♦♥s ♦ t s rtts tt ts t②♣ ♦ ♣♣t♦♥ ♥ts t♦ ♦♦r♦r s♠♣ ♠♦t♦♥ ♦ ts ♠t♦ ♦s t ♦♥t♥ts ♦ t t①tr t♦ t♥ ♥t♦ ♦♥t ♥ tr♦r t♦ ♥ s♥♣t ♣r♠tr③t♦♥ ❬♥rt ❪

②♠♠tr ♥r②

s♠r ♠t♦ s ♣r♦♣♦s ♥ ❬♦r♥ t ❪ s ♦♥ t r♠r ttsr♥♥ ♥ strt♥ s♦ trt t s♠ t② r♣ t L2 ♥ L∞ ♥r②t t ♦♦♥ ♦♥ ♠♦r s②♠♠tr t ts rs♣t

DT = max(σ′2, 1/σ

′1) = max(1/σ1, σ2)

♦♠♥ ♥r②

♦ ♥tr♦ ♠♦r ①t② ♥ ts ♠t♦s ♥r t ❬❪ ♣r♦♣♦s t♦ s ♦♠♥ ♥r② t tr♠ σ′

1σ′2 tt ♣♥③s r ♦r♠t♦♥s ♥ tr♠

σ′1/σ

′2 tt ♣♥③s ♥r ♦r♠t♦♥s ♦ tt t ♥♠r ♦♣t♠③t♦♥ ♦

t ♦t ♥t♦♥ tr♠ x s r♣ ② t ①♣rss♦♥ x + 1/x s

Ecombined = Eangle × (Earea)θ

t

Earea = σ′1σ

′2 + 1

σ′

1σ′

2

= t(J′T ) + 1

t(J′

T )

Eangle =σ′

1

σ′

2

+σ′

2

σ′

1

= σ2

σ1+ σ1

σ2= EMIPS

r t ♣r♠tr θ ♠s t ♣♦ss t♦ ♦♦s t rt ♠♣♦rt♥ ♦ ♦t tr♠s♥ ♥ ♥♦t tt t ♥r tr♠ ♦rrs♣♦♥s t♦ t ♥r② ♠♥♠③ ② t P♠t♦ s ♦

♦♥♦r♠ ♠t♦s

♦♥♦r♠ ♠t♦s r rt t t ♦r♠s♠ ♦ ♦♠♣① ♥②ss ♥♦♦♥♦r♠t② ♦♥t♦♥ ♥s rtr♦♥ t s♥t rt② t♦ ♦r ♦♦ ①tr♣♦t♦♥ ♣ts tt ♥ ♦♠♣t ♥tr ♦♥rs rr ♥trst tts ♦r♠s♠ ♠② r t ①♥t ♦♦ ② ♠ ❬❪

s s♥ ♥ ts st♦♥ ♦r♠t♦♥ ♥②ss ♥tr♦ ♥ t♦♥ ♣②s ♥trr♦ ♥ t ♥t♦♥ ♦ ♥♦♥st♦rt ♣r♠tr③t♦♥ ♠t♦s ❲ ♥♦ ♦s ♦♥ ♣rtr ♠② ♦ ♠t♦s ♦r t ♥s♦tr♦♣② ♣s s r ♦r ♣♦♥t ♦t sr s s♦♥ ♥ r ts s♦ ♠♥s tt t t♦ r♥t t♦rs fu

♥ fv r ♦rt♦♦♥ ♥ t s♠ ♥♦r♠ ♦♥t♦♥ ♥ s♦ rtt♥ sfv = n × fu r n ♥♦ts t ♥♦r♠ t♦r ♥trst♥② ♣r♠tr③t♦♥s ♦♥♦r♠ ts s s♦ t s ♦ t ♥rs ♥t♦♥ s♥ t ♦♥ ♠tr① ♦t ♥rs s q t♦ t ♥rs ♦ t ♦♥ ♠tr① rt♦♥ ♥ s♦ s♥ ♥ r t s♦ rs r ♦rt♦♦♥ t s s♦ t s ♦ tr ♥♦r♠t♦rs ♥② ♦♥♦r♠t② s♦ ♠♥s tt t ♦♥ ♠tr① s ♦♠♣♦s ♦ r♦tt♦♥ ♥ s♥ ♥ ♦tr ♦rs s♠rt② r♦r ♦♥♦r♠ ♣♣t♦♥s♦② ♦rrs♣♦♥ t♦ s♠rts ♥ tr♠s ♦ r♥t ♦♠tr② ♣tr ts♠♥s tt t ♥ts ♦ t ♥s♦tr♦♣② ♣s r t s♠ σ1 = σ2 ❲ ♥♦ rr♥t ♠t♦s tt ♦♠♣t ♦♥♦r♠ ♣r♠tr③t♦♥

s ♦rrs♣♦♥s t♦ t ♦♥ ♦ t ♣r♠tr③t♦♥ t tr♠♥♥t ♦ t ♦♥♠tr① tt ♥s t r♥t ♠♥t ♦r rs

u

v

Ω

S

du

dv

∂x∂u

∂x∂v

r ♦♥♦r♠ ♣r♠tr③t♦♥ tr♥s♦r♠s ♥ ♠♥tr② r ♥t♦ ♥ ♠♥tr② r

X

Y

i

j

ku

v

r ♥ tr♥ ♣r♦ t ♦ (X, Y ) ss t s s② t♦ ①♣rss t♦♥t♦♥ tt rtr③s ♦♥♦r♠ ♠♣s

♥ ♦♥trst t t ①♣♦st♦♥ ♦ t ♥t ♣♣r ❬é② t ❪ ♣rs♥tt ♠t♦ ♥ tr♠s ♦ s♠♣ ♦♠tr rt♦♥s t♥ t r♥ts ❲ t♥♦rt t t ♦♠♣① ♥②ss ♦r♠s♠ ♥ sts t rt♦♥ t ♦tr♠t♦s

♠t♦ st qrs ♦♥♦r♠ ♣s s♠♣② ①♣rsss t ♦♥♦r♠t② ♦♥t♦♥ ♦ t ♥t♦♥s tt ♠♣s t sr t♦ ♣r♠tr s♣ ❲ ♥♦♦♥sr ♦♥ ♦ t tr♥s ♦ t sr ♣r♦ t ♥ ♦rt♦♥♦r♠ ss (X, Y )♦ ts s♣♣♦rt ♣♥ r ♥ ts ♦♥t①t ♦♥♦r♠t② rts

∇v = r♦t90(∇u) =

(0 −11 0

)

∇u

r r♦t90 ♥♦ts t ♥t♦s r♦tt♦♥ ♦ rs❯s♥ t ①♣rss♦♥ ♦ t r♥t ♥ tr♥ r t t ♥ ♦ t♦♥

qt♦♥ tt rtr③s ♣s ♥r ♦♥♦r♠ ♠♣s rrts

MT

vi

vj

vk

−(

0 −11 0

)

MT

ui

uj

uk

=

(00

)

r t ♠tr① MT s ♥ ② qt♦♥ ♥ t ♦♥t♥♦s stt♥ ♠♥♥ ♣r♦ tt ♥② sr ♠ts ♦♥♦r♠

♣r♠tr③t♦♥ ♦r ♥ ♦r s♣ s ♦ ♣s ♥r ♥t♦♥s ♦♥② ♦♣ srs ♠t ♦♥♦r♠ ♣r♠tr③t♦♥ ♦r ♥r ♥♦♥♦♣sr ♠♥♠③ ♥ ♥r② ELSCM tt ♦rrs♣♦♥s t♦ t ♥♦♥♦♥♦r♠t② ♦t ♣♣t♦♥ ♥ t srt ♦♥♦r♠ ♥r②

ELSCM =∑

T=(i,j,k)

|T |

∥∥∥∥∥∥

MT

vi

vj

vk

−(

0 −11 0

)

MT

ui

uj

uk

∥∥∥∥∥∥

2

♦t tt ♦♥♦r♠t② s ♥r♥t tr♦ s♠rts ♣♣ ♥ ♣r♠tr s♣♦r ts rs♦♥ t qrt ♦r♠ ELSCM s ♥♦t ♥t ♣♦st ♥ ts ♠tr① ss♥r ♦r t s ♣♦ss t♦ ♥ t rs ♦ r♦♠ ♦ t s♠rt②② r♠♦♥ t♦ rts r♦♠ t st ♦ rs s♥ t t♥q ♣rs♥t ♥t♦♥ s♦♥ qrt ♦r♠ s ♣r♦② ♣♦st ♥t

❲ ♦♥sr ♦♥♦r♠ ♠♣s r♦♠ t ♣♦♥t ♦ ♦ t r♥ts ♥ t♥①t st♦♥ ①t rt♦♥s t♥ ♦♥♦r♠ ♠♣s ♥ r♠♦♥ ♥t♦♥s ss♦ s♦s s♦♠ ♦♥♥t♦♥s t ♦trs r②♥tr ♠♣♣♥ ♠t♦ ♥ t ts♠♦r r♥t ♥r③t♦♥s

♦♥♦r♠ ♠♣s ♥ r♠♦♥ ♠♣s

♦♥♦r♠ ♠♣s ♣② ♣rtr r♦ ♥ ♦♠♣① ♥②ss ♥ ♠♥♥♥ ♦♠tr② ♦♦♥ s②st♠ ♦ qt♦♥s rtr③s ♦♥♦r♠ ♠♣s

∂v∂x

= −∂u∂y

∂v∂y

= ∂u∂x

s s②st♠ ♦ qt♦♥s s ♥♦♥ s ②♠♥♥ qt♦♥s ② ♣② ♥trr♦ ♥ ♦♠♣① ♥②ss s♥ t② rtr③ r♥t ♦♠♣① ♥t♦♥s s♦ ♥②t ♥t♦♥s ❲ ♦s ♦♥ t ♦♠♣① ♥t♦♥ U(X) = u(X) + iv(X)t X = x + iy t s t♥ ♣♦ss t♦ ♣r♦ tt t rt U ′(X) ♦ U t ♣♦♥t X♥ ②

U ′(X) = limA→0

U(X)− U(A)

X − A

①sts ♥ ♦♥② ②♠♥♥ qt♦♥s r sts♥♦tr ② ♦ ♥rst♥♥ ts rt♦♥ s ♥ ② t ♦rr ②♦r ①♣♥s♦♥

♥ t s ♦ ♥t♦♥ f r♦♠ R t♦ R t ②♦r ①♣♥s♦♥ ♦ f rts f(x0 + a) ≃f(x0) + f ′(x0)a s r③s ♦ ♥r ♣♣r♦①♠t♦♥ ♦ t ♥t♦♥ f r♦♥ t♣♦♥t x0

♥♦ ♦♥sr ♦♠♣① ♥t♦♥ U(X) ♥ t ♦♠♣① rt s ♥t ②♦r ①♣♥s♦♥ rts U(X0 +A) ≃ U(X0)+U ′(X0)A ♦♥sr t ♦♠♣①♥t♦♥ U s ♦♠tr tr♥s♦r♠ ♦ t ♣♥ ♥ ② rt♥ U ′(X0) = Reiθ ♥♣♦r ♦r♠ t U(X0 + A) ≃ U(X0) + ReiθA ♥ ♦tr ♦rs r♥t ♦♠♣①♥t♦♥s ♦② s♠rt② ♦ t ♣♥ ♦♠♣♦s ♦ tr♥st♦♥ ♦ t♦rU(X0) r♦tt♦♥ ♦ ♥ θ ♥ s♥ ♦ t♦r R s s ♥♦tr ①♣♥t♦♥♦r t ②♠♥♥ qt♦♥s t r♦tt♦♥ ♥ t s♥ ♣♣ t♦ t t♦r♥t t♦rs ♥ t♦ ♠t

♦t tt t ♦♠♣① ♥♠rs t ♥r② ♠♥♠③ ② ts s♠♣♦r♠ tt rs t s②♠♠tr r♦s ♣② ② t ♦♦r♥ts (Xi, Yi) ♦♥ t sr♥ t ♦♦r♥ts (ui, vi) ♥ ♣r♠trs♣

ELSCM =1

2

∣∣∣∣∣∣

(P1 P2 P3

)

0 1 −1−1 0 1

1 −1 0

U1

U2

U3

∣∣∣∣∣∣

2

r U1 = (u1 + iv1) rs♣ U2 U3 ♥ r P1 = (X1 + iY1) rs♣ P2 P3♥♦tr ♥trst♥ ♣r♦♣rt② ♦ ♦♠♣① r♥t ♥t♦♥s s tt tr ♦rr

r♥tt② ♠s t♠ r♥t t ♥② ♦rr ❲ ♥ t♥ s t ②♠♥♥ qt♦♥s t♦ ♦♠♣t t ♦rr rts ♦ u ♥ v

s stss ♥ ♥trst♥ rt♦♥ t t ♣♥ ♦♣rt♦r ∆

∂2u∂x2 = ∂2v

∂x∂y

∂2u∂y2 = − ∂2v

∂x∂y

♦r ∆u = ∂2u

∂x2 + ∂2u∂y2 = 0

∆v = ∂2v∂x2 + ∂2v

∂y2 = 0

♥ ♦tr ♦rs t r ♣rt ♥ t ♠♥r② ♣rt ♦ ♦♥♦r♠ ♠♣ r t♦r♠♦♥ ♥t♦♥s t♦ ♥t♦♥s t ③r♦ ♣♥ s s t ♣♦♥t ♦ ♦♣t ② sr♥ t ❬❪ t♦ ♦♣ tr ♦♥♦r♠ ♣r♠tr③t♦♥ ♠t♦♥r② q♥t t♦ s t② ♦♠♣t t♦ r♠♦♥ ♥t♦♥s tt♥t ♦rr ♦ ♥ t ♦rr st ♦ ♦♥str♥ts ♥♦r t ♦♥♦r♠t② ♦ t♣r♠tr③t♦♥ ♥ ♥tr♦ ♦♣♥ tr♠ t♥ t us ♥ t vs

♥♦tr ② ♦ ♦♥sr♥ ♦t ♣♣r♦s ♠♥t♦♥ ② P♥ ♥ P♦tr❬❪ ♥ ♣r♦② t t ♦r♥ ♦ sr♥ ②r ♥ ③s ♥tt♦♥ s ♥② Pts ♣r♦♠ ❬Pt s ❪ ♥ ♦s r ts ♣r♦♠♦♥r♥s t ①st♥ ♦ sr t ♠♥♠♠ r s tt ts ♦rr ♠ts t♦s r ♦ ♠♥♠③ t r ♦ sr ♦s ❬❪ ♥ ó ❬❪ ♥tr ♦r♥t ❬❪ ♦♥sr rts ♥r② t ♥tr ♦ t sqr ♥♦r♠♦ t r♥ts sr t♦ ♠♥♣t srt③t♦♥ ♦ ts ♥r② s ♣r♦♣♦s② P♥ ♥ P♦tr ❬❪ t t ♠ ♦ ♥ ♣rt s♦t♦♥ t♦ Pts♣r♦♠ ♥ t srt s rts ♥r② rs r♦♠ t r ♦ t sr r♥ s tr♠ tt ♣♥s ♦♥ t ♣r♠tr③t♦♥ t ♦♥♦r♠ ♥r② ♦♥♦r♠ ♥r② s q t♦ ③r♦ t ♣r♠tr③t♦♥ s ♦♥♦r♠ rt♦♥t♥ ts tr q♥tts s ①♣♥ ♦

∫

S

t(J)ds

︸ ︷︷ ︸

r ♦ t sr

=1

2

∫

S

‖fu‖2 + ‖fv‖2ds

︸ ︷︷ ︸

rts ♥r②

− 1

2

∫

S

‖fv − r♦t90(fuX)‖2︸ ︷︷ ︸

♦♥♦r♠ ♥r②

s rt♦♥ s s② t♦ ♣r♦ ② ①♣♥♥ t ♥trt tr♠s ♥ ♦ ssX, Y ♦ t t♥♥t ♣♥

∂X

∂u

∂Y

∂v− ∂X

∂v

∂Y

∂u︸ ︷︷ ︸

t(J)

=1

2

∥∥∥∥∥

(∂X∂u

∂Y∂u

)∥∥∥∥∥

2

+

∥∥∥∥∥

(∂X∂v

∂Y∂v

)∥∥∥∥∥

2

︸ ︷︷ ︸

‖fu‖2+‖fv‖2

− 1

2

∥∥∥∥∥

(∂X∂v

+ ∂Y∂u

∂Y∂v− ∂X

∂u

)∥∥∥∥∥

2

︸ ︷︷ ︸

‖fv−r♦t90(fu)‖2

r♦r ♠♥♠③s t ♦♥♦r♠ ♥r② ♥ sr♥ t s ♠t♦ ♠♥♠③rts ♥r② ♥ t r♥ t♥ ts t♦ q♥tts ♦rrs♣♦♥s t♦ t♦♥st♥t r ♦ t sr ♦t ♠t♦s r q♥t

t ♠t♦s ♠♥t♦♥ ♦ r s ♦♥ rt♦♥s t♥ t r♥ts t♦♥ ♦r t rst ♥♠♥t ♦r♠ ♦ t ♣r♠tr③t♦♥ ♦r ts rs♦♥ t②♥ q s ♥②t ♠t♦s ♥ t ♥①t ♣tr ♦s ♦♥ ♦♠tr♠t♦s tt ♦♥sr t s♣ ♦ t tr♥s ♥ ♠♦r s♣② tr ♥s

♣tr

♥♣ t♦s

♥st ♦ ♥♥ ♣♥r ♣r♠tr③t♦♥ ♥ tr♠s ♦ rt① ♦♦r♥ts ♦t t ♠t♦ ❬r ♥ trr r t ❪ ♥ tr♣ttr♥s ♦rt♠ ❬r② t ❪ ♥ t ♥ tr♠s ♦ t ♥s ♦t ♣♥r tr♥s r ♣r♦s ♦♠♣rs♦♥ t♥ ♥s♣ ♥ rt♦♥♦r♠ ♠t♦s s ♠♦♥strt ♥s♣ ♠t♦s ♥tr♦ s♥♥t② ssstrt ♥t♦ t ♣r♠tr③t♦♥ ♦♥ ♠♦s tt r♦♥s ♦ ss♥ rtr

♠t♦ ♥ s tt♥♥ ❬r ♥ trr ❪s s ♦♥ t ♦♦♥ ♦srt♦♥ ♣♥r tr♥t♦♥ s ♥q② ♥ ②t ♦r♥r ♥s ♦ ts tr♥s ♠♦♦ s♠rt② tr♥s♦r♠t♦♥ s ♦♥ ts♦srt♦♥ t t♦rs r♦r♠t t ♣r♠tr③t♦♥ ♣r♦♠ ♥♥ (ui, vi) ♦♦r♥ts ♥ tr♠s ♦ ♥s tt s t♦ s② ♥♥ t ♥s αt

k r αti ♥♦ts t

♥ t t ♦r♥r ♦ tr♥ t ♥♥t t♦ rt① k♦ ♥sr tt t ♥s ♥ tr♥t♦♥ st ♦ ♦♥str♥ts ♥s

t♦ sts

• r♥ t② ♦r tr♥ t

∀t ∈ T, αt1 + αt

2 + αt3 − π = 0;

• P♥rt② ♦r ♥tr♥ rt① v

∀v ∈ Vint,∑

(t,k)∈v∗

αtk − 2π = 0,

r Vint ♥♦ts t st ♦ ♥tr♥ rts ♥ r v∗ ♥♦ts t st ♦♥s ♥♥t t♦ rt① v

• P♦stt② αtk > 0 ♦r ♥s ❲ ♥♦t tt ts ♦♥str♥t ♥ ♥♦r ♥

♠♦st ♣rt st♣s ❬r t ❪ s♠♣②♥ t s♦t♦♥ ♣r♦ss

• ♦♥strt♦♥ ♦r ♥tr♥ rt① ts ♦♥str♥ts ♥srs tt ssr ② ♣rs ♦ tr♥s s t s♠ ♥t

∀v ∈ Vint,∏

(t,k)∈v∗

sin αtk⊕1 −

∏

(t,k)∈v∗

sin αtk⊖1 = 0.

♥s k ⊕ 1 ♥ k ⊖ 1 ♥♦t t ♥①t ♥ ♣r♦s ♥ ♥ t tr♥♥tt② ♥♦t tt t ♣r♦t sin αt

k⊕1 sin αtk⊖1 ♦rrs♣♦♥s t♦ t ♣r♦t ♦

t rt♦ t♥ t ♥ts ♦ t♦ ♦♥st s r♦♥ rt① k t② ♦♥♦t ♠t t s t ♣♦ss t♦ tr♥ r♦♥ rt① k t♦t ♥♥ ♦♥ tstrt♥ ♣♦♥t

② sr ♦r ♥s tt r s ♦s s ♣♦ss t♦ t ♦r♥ ♠s ♥sβt

k ♥ sts② t♦s ♦♥str♥ts ♦♥str♥ ♥♠r ♦♣t♠③t♦♥ ♣r♦♠ s s♦ s♥ t r♥ ♠

t♣rs ♠t♦ s st♦♥ stt♦♥r② ♣♦♥t ♦ t r♥♥ s ♦♠♣ts♥ t♦♥s ♠t♦ s t♦♥ st♣ rqrs t♦ s♦ ♥r s②st♠♦ s③ 2nv + 4nf r nv ♥♦ts t ♥♠r ♦ ♥tr♦r ♠s rts ♥ nf t♥♠r ♦ ts ② t♥ ♦♥rt t s♦t♦♥ ♥s ♥t♦ t (u, v) rt① ♦♦r♥ts s♥ ♣r♦♣t♦♥ ♣r♦r rst♥ ♣r♠tr③t♦♥s r r♥tt♦ ♥♦ ♣♣ tr♥s ♦② t t ♥ ♦♥t♥ ♦ ♦r♣s t♦rs ♣r♦ ♠♥s♠ ♦r rs♦♥ s ♦r♣s t t s ♥♦ r♥ts♦ ♦♥r♥ ♦r♥ ♠t♦ s rt② s♦ ♥ srs r♦♠ stt②♣r♦♠s ♥ t ♥t♦ ♦♥rs♦♥ st ♦r r ♠ss

s ♠♥t t♦ ② ❬r t ❪ t♥q rss♥♦t ♣r♦♠s ♥tr♦s st ♥t♦ ♦♥rs♦♥ s♥ t ♠t♦ t♦ ♦t♥ (u, v) ♦♦r♥ts r♦♠ t st ♦ ♥s r t ❬❪ s♦rst② s♣s ♣ t s♦t♦♥ ② ♥tr♦♥ ♦t rt ♥ rr s♦t♦♥♣♣r♦s ♦r t rt s♦r t② st r♦♠ t♦♥ t♦ sst♦♥ s♦t♦♥st♣ s♠♣②♥ t strtr ♦ t ♥r s②st♠ s♦ t st♣ ② t♥ st strtr ♦ t ss♥ ♠tr① t♦ s♦ t ♥r s②st♠ t♦t ①♣t② ♥rt♥t ♥tr ss♥ s t② ♠♥ t♦ r t s③ ♦ t ①♣t② ♥rt ♠tr①t st♣ ② t♦r ♦ t♦ ♦t 2nv ♥♦t tt ♦♥ ♠♥♦ ♠s 2nv ≈ nfs s t ♦♥ ♠② ①♣t ♦r ♣r♠tr③t♦♥ ♣r♦♠ s♥ ts ♦rrs♣♦♥st♦ t ♥♠r ♦ rs ♦ r♦♠ ❲ ♠ ♥ ♥trst♥ ♦srt♦♥ ♦t t♦♥r♥ ♦r ♦ tr♦♦t t sst♦♥ ♣r♦ss ❲ t ts t♦ t♥ trt♦♥s t♦ r t ♠♥♠③t♦♥ rr♦r t r♥t ♦ t r♥♥♦ 1e−6 ♣rt② t ♥s r♦♠ trt♦♥ t♦ ♥ ♦♥ ♦r ♥ t r♥♠t♣rs ♥ tr ♦♥ trt♦♥ r t ♦♠♣t ♥s r r② ♦st♦ t ♥ ♦♥s ♥ t② ♦♠♣t② ♦♥r t♥ ♦♣ ♦ trt♦♥s r

r ♠♦t♦♥s ♦ t ♦r♠t♦♥ r ♣r♦♣♦s ♦r t st ②rs ♦r♥st♥ ❩②r t ❬❪ ♥tr♦ t♦♥ ♦♥str♥ts ♦♥ t ♥s ♥♦r♥t ♣r♠tr ♦♠♥ t♦ ♦♥① ♦♥rs ts r♥t♥ ♦ tt②

r② t ❬❪ s r ♣ttr♥s ♣♣r♦ r r ♦rrs♣♦♥st♦ ♠s ♥ ♦♥trst t♦ ss r ♣♥ t② s ♥trst♥ rs t♣rsr ♥trst♦♥ ♥s θe r ♥ ts ♥s t r r ♦♦s t ♥q ♠♥♠③r ♦ ♦♥① ♥r② ♠t♦ rst ♦♠♣ts t ♥trst♦♥♥s s♥ ♥♦♥♥r ♦♥str♥ ♦♣t♠③t♦♥ ♥ t♥ ♥s t r r s♥♥♦♥str♥ ♠♥♠③t♦♥ ♦ ♥ t ♥trst♦♥ ♥s t rst ♦♠♣ts st ♦s tr♥ ♥s αk

ij r ♦s t♦ t ♥s βkij ♥ sts② t ♦♦♥

r ♦♥r♥ ♦ ♣r♠tr③t♦♥ rst tr ♦♥ trt♦♥rr♦r ♠trs ♦ Lstretch

2 = 1.10634, Lshear2 = 4.69e−4, AngD = 4.19e−4 ♠

Lstretch2 = 1.09616, Lshear

2 = 5.21e − 3, AngD = 5.09e−3 t♦ trt♦♥s rr♦r ♠trs ♦ Lstretch

2 = 1.10641, Lshear2 = 4.64e−4, AngD = 3.81e−4 ♠

Lstretch2 = 1.09645, Lshear

2 = 5.15e − 3, AngD = 5.04e−3 t♥ trt♦♥s rr♦r ♠trs ♦ Lstretch

2 = 1.10638, Lshear2 = 4.64e−4, AngD = 3.81e−4 ♠

Lstretch2 = 1.09639, Lshear

2 = 5.15e−3, AngD = 5.04e−3 ♦r♠s ♦r t strtsr ♥ ♥r st♦rt♦♥ r t♥ r♦♠ ❬r t ❪

r r ♣ttr♥s ♥♦tt♦♥s

♦♥str♥ts

• r♥ t② ♦r tr♥ t

∀tijk ∈ T, αtij + αt

ik + αtkj − π = 0;

• P♥rt② ♦r ♥tr♥ rt① v

∀v ∈ Vint,∑

(t,ij)∈v∗

αtij − 2π = 0,

r Vint ♥♦ts t st ♦ ♥tr♥ rts ♥ r v∗ ♥♦ts t st ♦♥s ♥♥t t♦ rt① v

• P♦stt② αtij > 0 ♦r ♥s

• ♦ ♥② ♣r♦♣rt② ♦r eij

∀eij ∈ E, αkij + αl

ij < π

❲ ♥♦t tt tr ♦ ts ♦♥str♥ts r s♠r t♦ t♦s ♠♣♦s ② t st♣ t t ♥♦♥♥r r♦♥strt♦♥ ♦♥str♥t ♦♥ ♥tr♦r rts r♣ ② t♥qt② ♦ ♥② ♦♥str♥t ♣r ♥trst♦♥ ♥s r ♦♠♣tr♦♠ t s tr♥ ♥s s♥ s♠♣ ♦r♠ ♥ t s♦t♦♥ ♦r t♥trst♦♥ ♥s s ♦♥♦r♠ ♦♥② ♦r ♥② tr♥t♦♥ t t♦rs ♠♣♦② ♣r♣r♦ss♥ st tt ♥♦s ♥tr♥s ♥② tr♥t♦♥ t t ♥ st♦ t ♠t♦ t ♦♠♣t r r ♦♥rt t♦ t ♦♦r♥ts ♠t♦s♣♣♦rts qt② ♥ ♥qt② ♦♥str♥ts ♦♥ t ♥s ♦♥ t ♦♥r② ♦ t♣♥r ♣r♠tr ♦♠♥ ♠r t♦ t ♣r♠tr③t♦♥ s ♦② tt ♥ ♦♥t♥ ♦ ♦r♣s ♠♦♥t ♦ st♦rt♦♥ ♥tr♦ ② t ♠t♦ s♦♠♣r t tt ♦ t♥qs

r② t ♣r♦♣♦s ♥ ①t♥s♦♥ ♦ t ♠t♦ t♦ ♦ ♣r♠tr③t♦♥ ♦♠ss ② ♥tr♦♥ ♦♥ s♥rts ② ♦sr tt ♥ ♥ s♣ ♦r♠t♦♥

r ① ♥ r♦♥r② ♣r♠tr③t♦♥s ♦♠♣t s♥ r ♣ttr♥s

t ♦♥② r♥ t♥ ♣r♠tr③♥ t ♠s ♦♥r② ♥ ts ♥tr♦r ♠s st ♦♥str♥ts ♠♣♦s ♦♥ t ♥tr♦r rts r ♥♦t ♠♣♦s ♦♥ ♦♥r② ♦♥s♥ tt t s ♣♦ss t♦ ♥ ♦ ♣r♠tr③t♦♥ ② s♣②♥ ♥ ♥♦♥♥tsst ♦ ♠s rts s ♦♥r② rts s t② rst ♦♠♣t s♦t♦♥ ♥♥ s♣ t st ♦ ♦♥ s♥rt② rts s♣ ② t sr s ♦♥r② ♦r♣♥r ♣r♠tr③t♦♥ t♦ ♣r♦r♠ t ♥t♦ ♦♥rs♦♥ t② tr ♦♠♣t ♣ts t♥ ts rts ♦t♥ ♣r♠tr③t♦♥ s ♦② ♦♥t♥♦s ♣t♦ tr♥st♦♥ ♥ r♦tt♦♥ r②r ①♣t t t ♦♥ s♥rts ♣r♦♣♦s♣♣r♦ ♥ rt② ♣♣ t♦ ♦tr ♥s♣ ♠t♦s s s

r ♦ ♣r♠tr③t♦♥ t ♦♥ s♥rts

♣tr

♠♥tt♦♥ ♥ ♦♥str♥ts

♦r ♣rt ♣r♣♦ss ♣♥r ♠s ♣r♠tr③t♦♥ s t♦ rss t♦ t♦♥ ♥s s♠♥t♥ ♦s ♦r ♥s ♠ss t♦ ♥ ♣♥r ♠♥ ♥ ♥♦r♥ s♣ ♣♦♥tt♦♣♦♥t ♦rrs♣♦♥♥s r♥ ♣r♠tr③t♦♥

♠♥tt♦♥

P♥r ♣r♠tr③t♦♥ s ♦♥② ♣♣ t♦ srs t s t♦♣♦♦② ♥ ♦ssrs ♥ srs t ♥s rtr t♥ ③r♦ t♦ t ♣r♦r t♦ ♣♥r ♣r♠tr③t♦♥ s ♣r♦s② ♥♦t rtr sr ♦♠♣①t② s② ♥rss ♣r♠tr③t♦♥ st♦rt♦♥ ♥♣♥♥t ♦ t ♣r♠tr③t♦♥ t♥q s ♦ ♦♣r♠tr③t♦♥s t ♦ st♦rt♦♥ t srs ♠st t t♦ r t ♦♠♣①t② ♥ ts ♥tr♦ s♦♥t♥ts ♥t♦ t ♣r♠tr③t♦♥ t ♥t♥ t ♦♥t♥ ♦s ♦ s♠ st♦rt♦♥ ♥ s♦rt ts s t♦ t s♣♦ss t♦ s ♦♥str♥ ♣r♠tr③t♦♥ t♥qs t♦ r r♦sst s♦♥t♥ts

tt♥ ♥ rt ♥rt♦♥ r ♠♦st ♦♠♠♦♥② s ♥ ♦♠♣t♥ ♣r♠tr③t♦♥s ♦r ♠♣♣♥ ♦ t①trs ♥ ♦tr s♥s ♦♥t♦ t sr ② r s♦s ♦r ♣♣t♦♥s s s ♦♠♣rss♦♥ ♥ r♠s♥ t♥qs ♦r tt♥srs ♥ r♦② ♥t♦ t♦ t♦rs s♠♥tt♦♥ t♥qs ♣rtt♦♥ t sr ♥t♦ ♠t♣ rts t♦♥ ♥ s♠ ♥rt♦♥ t♥qs ♥tr♦ ts ♥t♦ t sr t ♣ t s s♥ rt t♦♥ t♣rts rt ② s♠♥tt♦♥ t②♣② ♦♥r ♦♥rs t♥ t♦s rt ②s♠ tt♥ ♦r t② ♥ ♦t♥ ♠♦r ♥t② ♣ ♥t♦ ♦♠♣t ♣♥r♦♠♥

trt ♠♥tt♦♥s

♣♥♥ ♦♥ t ♣♣t♦♥ ♠s s♠♥tt♦♥ t♥qs s r♥t rtr ♦rrt♥ rts ♦r ♣r♠tr③t♦♥ srs r r♦♥ ♥t♦ sr rts s ttt ♣r♠tr st♦rt♦♥ ♥ ♣r♠tr③♥ rt s s♥t② ♦ t

s ♣tr s t♥ r♦♠ ❬r t ❪ t ♠♥♦r ♥s

r trt s♠♥tt♦♥s s♦rts ❬❩♦ t ❪ rts ❬é② t ❪ rts rts ❬s t ❪ rts ♠trt ♦♠tr② ♠s ❬♥r t ❪ rts ♥ ❬❩♥ t ❪ rts

♥♠r ♦ rts r♠♥s s♠ ♥ tr ♦♥rs r ♣t s s♦rt s ♣♦ss ♥♣♥s r ♦♣ ② ♥t♦♥ ♦♥ ♣♦ss ♣♣r♦ s t♦ s♠♥t t sr♥t♦ ♥r② ♣♥r rts ❬♦t t r♥ t ♥r t ♦♥t♥r t ❪

♥t② ♦t ♥r t ❬❪ ♥ ♦♥t♥r t ❬❪ ♥tr♦ ♣♥rs♠♥tt♦♥ ♠t♦s ♥s♣r ② ♦② q♥t③t♦♥ r ♦rt♠s trt t♥rt r♦♥ ♥ rs♥ sts tr rt r♦♥ trt♦♥ t ♠t♦sst t st ss t rs♣t t♦ rt ♥ r♣ts t ♣r♦ss t♦rs♠♦♥strt tt ② trt♥ t② ♦t♥ ttr rsts t♥ s♥ ♣ss ♠t♦s

P♥s r s♣ t②♣ ♦ ♦♣ srs s ♦r ♣r♠tr③t♦♥ ♣r♣♦ss♣♥r s♠♥tt♦♥ s ♦rrstrt ♥ s② ♥rts ♠♦r rts t♥ ♥ssr②r r♥t ♣♣r♦s ♦s ♦♥ ♦♣ s♠♥tt♦♥ ♥st ❬é② t ❩♦ t s t ❪ é② t ❬❪ ♣r♦♣♦s t♦ tt ♠♥rtr r♦♥s ♦♥ t ♠s ♥ t♥ ♥rt rts strt♥ r♦♠ ss rrtst r♦♠ t♦s r♦♥s s ♣♣r♦ t♥s t♦ ♣tr ♠♥② ♦♣ r♦♥st ♥ s♦ ♥tr♦ rts r r r♦♠ ♦♣

❩♦ t ❬❪ ♣r♦♣♦s s♠♥tt♦♥ ♠t♦ s ♦♥ s♣tr ♥②ss ♦ tsr ② ♦♠♣t ♠tr① ♦ ♦s ♠s st♥s ♥ t♥ ♣r♦r♠ str♥ ② r♦♥ rts r♦♥ t ♥ rtst ♣♦♥ts ♥ s♣ ♥ ② t♦♠♥♥t ♥t♦rs ♦ t ♠tr①

❩♥ t ❬❪ s♠♥t t sr ♥t♦ tr r♦♥s ② ♥♥ s♦♦♥t♦rs r t tr r ♥rss s♥♥t② t♥ t trs r sss ♥r ♣s♦s t ♣s♦s ♦r s♣rs ♥ t s♥ ♥ts rr ♦rrs♣t② s s♠ t rst♥ ♣s r ♦s t♦ ♥ ♦♣♥ ♥ ♣r♠tr③ t tt st♦rt♦♥

♠r t♦ ❬♥r t ❪ s t ❬❪ s ♦② trt♦♥s ♦ r♦♥♥ rs♥ t ♥st ♦ ♦♦♥ ♦r ♣♥r r♦♥s t② sr ♦r rr sst♦ ♦♣ srs t s♦ ♦♣ srs ♦ ♦♥st♥t s♦♣ rrtr③ ② ♥ ♦♥st♥t ♥ t♥ t ♥♦r♠ t♦ t sr ♥ s♦♠①s t♦r ② ♣r♦ ♠tr t♦ ♠sr rt ♦s② ♣♣r♦①♠ts s sr ♥ s ts ♠tr ♥ t rt r♦♥ ♥ rs♥ sts

rt ♣♥ rtt♦♥ t♥qs rs ♥ t♦♥ ♣♦st♣r♦ss♥ ♥ ♦♦♥ t ♣r♠tr③t♦♥ ♦ ♥ rt t♦s rts ♥ t♦ ♣ ♦r ♣ ♥ ♦♠♠♦♥ ♣r♠tr ♦♠♥ ♦r ♥t st♦r ♦ t ♣r♠tr③ ♠ss t ♣♥ s t♦ s ♦♠♣t s ♣♦ss ♦♣t♠ ♣♥♣r♦♠ s Pr ts ♦♥② rst ♦r ♣♣r♦①♠t ♣♥ ♦rt♠s ①st trs ♦rt♠ ❬é② t ❪ ♥tr♦s rts ♦♥ ② ♦♥ sr♥ ♦r t stt ♦♥ t tr♦♥t ♦ t rts ♣ s♦ r

♠ tt♥

t s ♣♦ss t♦ r t ♣r♠tr③t♦♥ st♦rt♦♥ t♦t tt♥ t sr ♥t♦s♣rt ♣ts ② ♥tr♦♥ ♠t♣ ♣rt ts ♦r s♠s ♥s s♥ ♣ts t②♣② s t♦ s♦rtr ts t♥ t♦s rt ② s♠♥tt♦♥ P♣♦♥ ♥

r ♠ tt♥ ❬♦r♥ t ❪ ❬ t ❪ ❬r ♥rt ❪ t ♦♦r s③s t sr st② t ♠♦r s r♦♥s s♦♥♥ r♥ ♥ ss s ♥ r

♦rs♦ ❬❪ ♥rt s ts ♠♥② s♥ ♥t♦r ♦ s♦r♥ t ❬❪ ♣r♦r♠ ♣r♠tr③t♦♥ ♥ tt♥ s♠t♥♦s② ②

♥♦ t ♠s rts ♦♥t♦ t ♣♥ ♦♥ tr t ♦tr ♦♣t♠③♥ t ♦ ♠♣♣♥❲♥r t st♦rt♦♥ ♦ t ♠♣♣♥ rs trs♦ t② t t ♠s t♦ rt s rst t② r ♦♥ ♦♥ t st♦rt♦♥ t ♥ ♥ ♣ t ♦♥ ♥♦♠♣t ♦♥rs ♦ ♠sr st♦rt♦♥ t② s t s♥r s ♦ t♣rtr♥ ♠♣♣♥

t ❬❪ s ♣r♠tr③t♦♥ rsts t♦ tt t tt♥ ♣r♦ss t♦rs rst ♣r♠tr③ t sr s♥ s♣ ♣rsr♥ ♣r♠tr③t♦♥ ❬♦tr❪ ② t♥ ♥ t ♣♦♥t ♦ ♠①♠ ♣r♠tr st♦rt♦♥ ♦♥ t ♠♣♣♥ ♥♥rt t s♦rtst t r♦♠ t sr ♦♥r② t♦ tt ♣♦♥t ② r♣t t♣r♦ss ♥t t st♦rt♦♥ s ♦ rt♥ trs♦

♠str ♦rt♠ ❬r ♥ rt ❪ ♦♥srs t r♥t ♦♠tr②♣r♦♣rts ♦ t sr ♥♣♥♥t ♦ ♣rtr ♣r♠tr③t♦♥ t♥q t rst♥s r♦♥s ♦ ss♥ rtr ♦♥ t sr ♥ t♥ ss ♠♥♠ s♣♥♥♥tr ♦ t ♠s s t♦ ♦♥♥t t♦s ♥② t ts t ♠s ♦♥ t tr sr ♥ rt ❬❪ s t ♥t ② st② ♠tr ♥ ♦♠♣t♥ t♠♥♠ s♣♥♥♥ tr s ② t② r t♦ tr t ts tr♦ t ss s♣rts ♦ t sr ♥ t ♣♦t♥t r♦sst s♦♥t♥ts ♥ t①tr ♦r ♦tr♠♣s ♦♥ t sr

♥s rt♦♥ ♠ tt♥ ♠t♦s rqr ♥ ①♣t ♣r♣r♦ss♥ st t♦♦♥rt srs t ♥s ♥t♦ t♦♣♦♦ ss ♥rt♦♥ ♦ ♠♥♠ ♥tts tt ♦♥rt ♥s sr ♥t♦ t♦♣♦♦ s s Pr ❬rs♦♥ ♥rP ❪ s ♣r♦♠ r ♦t ♦ tt♥t♦♥ ♦♥ t ♦♠♣tt♦♥♦♠tr② ♦♠♠♥t② t ♥♠r ♦ ♠t♦s ♣r♦♣♦s ♦♣rt t r♥s ♦ sss r② ♣rt ♣♣r♦ s t♥ ② t ❬❪ ♦ tr s♣♥♥♥ r♣ ♦ t s ♥ t ♠s ♥ t♥ ♣r♥ ts r♣ ♦t♥♥ ♥sr♥ t

r ♦♥str♥t ♥♦r♠♥t t t♠r ❬r♦② t ❪ ①trt♦ ♣♦t♦s t tr ♣♦♥ts s♣ ♥♣t ♠♦ t ♦rrs♣♦♥♥ rtst rst♥ t①tr

r ♦♠♥♥ ♠t♣ t①trs ❬❩♦ t ❪

r ♦ ①♠♣s ♦ ♦♥str♥ t①tr ♠♣♣♥ ❬é② ❪

♦♥str♥ts

♦♠t♠s ♣r♠tr③t♦♥ ♥s t♦ ♦♠♠♦t sr ♦♥str♥ts s♣②♥ ♦rrs♣♦♥♥s t♥ rts ♦ t ♠s ♠♦st ♠♣♦rt♥t ♣♣t♦♥ ♦ ♦♥str♥♣r♠tr③t♦♥ s t①tr ♠♣♣♥ ♠♦s r♦♠ ♣♦t♦r♣s ❬é② r♦②t ❪ s rs ♥ ❩♦ t ❬❪ s ♠♦r ♦♠♣① ♦♥str♥ts t♦♦ t sr t♦ ♦♠♥ sr ♠s t♦ ♣r♦ ♦♠♣t t①tr ♦r ♠s ♦♥str♥ ♣r♠tr③t♦♥ ♥ s♦ s t♦ r♦sss♠ s♦♥t♥ts ❬r♦②t ❩♦ t ❪

♠t♦s ♦r ♥♦r♥ ♦♥str♥ts ♥ s♣t ♥t♦ t♦ t②♣s t♦s tt ♥♦rs♦t ♦r ♣♣r♦①♠t ♦♥str♥ts ♥ tt tt ♥♦r r ♦♥str♥ts t♦s s♦♥ ♥r② ♠♥♠③t♦♥ ♥ ♦♠♠♦t s♦t ♦♥str♥ts ② ♥ qrt tr♠t♦ t ♥r② ♥t♦♥ ♠sr♥ t st♥ t♥ t ♦♥str♥t trs ♥ trr♥t ♦♥rt♦♥ ♥ tr sr ♦t♦♥ ❬é② ❪ ♦♥str♥ts ♦rrs♦♥② ♥ ♣rt ♥ ♥ s♦ ♥t② s♥ t② ♦♥② ♥rtr♠s t♦ t ♥r② t s♦♠t♠s r t♦rt r♥ts ♦t t ♦r♥♣r♠tr③t♦♥ ♠t♦ s s tt② r ♦ ♦♥str♥t ♣♣r♦①♠t♦♥t②♣② rss t t st♥ t♥ t ♦♥str♥ ♥ ♥♦♥str♥ rt①♦t♦♥s ♦♠ ♣♣t♦♥s s s ♥ t①tr s♦♥t♥ts ♦♥ s♠s ♥ t♣r♠tr③t♦♥ ❬r♦② t ❩♦ t ❪ rqr r ♦♥str♥ts t♦ ♣rt ♥♠♥t ♦ t t①tr ♦♥ t s♠s

♥ ② t♦ ♥♦r r ♦♥str♥ts s ② ♥ t♠ ♥t♦ rr ♣r♠tr③t♦♥ ♦r♠t♦♥ s♥ r♥ ♠t♣rs ❬sr♥ t ❪ s ♣♣r♦ ♦s♦♥str♥ts t♦ ♥ ♦♥ ♣♦♥ts ♥s tr♥s ♥ ♦♥ rtrr② ♥ s♠♥ts trts ♦ t tr♥t♦♥ ♥ ♦♥str♥ ♠♦r s② ② t♥ t ♦rrs♣♦♥♥rs ♦t ♦ t s②st♠ ♦r t s s② t♦ s♦ tt ♦r ♥ ♠s ♦♥♥tt② ♥♦t r② st ♦ ♦♥str♥ts ♥ sts s ♠t♦s ts tt ♣rsrt ♠s ♦♥♥tt② t♦ ♥rt t ♣r♠tr③t♦♥s ♦r ♠♥② ♥♣ts

t♦s tt ♥♦r r♦♥str♥ts r♦st② ♥tr♦ t♦♥ rts ♥t♦t ♠s s t② ♦ ♦♥ t♦ ♥sr tt ♦♥str♥ s♦t♦♥ ①sts st♥ t ❬❪ ♥♦r r ♦♥str♥ts ② ♦r♠♥ ♥ ①st♥ ♠♥ ♥ ♥rts ♥ ♥ssr② ♦rt② ts ♠t♦ ♥ ♥ r sts ♦ ♦♥str♥tst s ①tr♠② ♦♠♣t

t♠r ♦rt♠ ❬r♦② t ❪ ♦♠♣t t ♣r♠tr③t♦♥ ②sts♥ ♦rs ♣t ♦rrs♣♦♥♥s t♥ t ♥♣t ♥ t ♣r♠tr ♦♠♥ ♣r♦ tr ♣♦♥ts ♦♥ t ♥♣t ♠♦ ♥ t ♣r♠tr ♦♠♥ r♦♥♥t s♥ ♥t♦r ♦ rs tt ♣rtt♦♥ t sr ♥t♦ ♣ts tt rt♥ ♣r♠tr③ tr②♥ t♦ ♠♥t♥ ♦♥t♥t② ♥ s♠♦♦t♥ss t♥ t♠ r tr♥ ♣r♦ss s ② st ♦ t♦♣♦♦ rs tt ♥sr tt trst♥ ♣ts ♦♥sst♥t t♥ t ♦ts ♥ ♣r♠tr③ ♥ t♦♠♥ ② ♦♠♣t t tr♥t♦♥s ♦ t ♥♣t ♥ t ♣r♠tr ♦♠♥ s♠t♥♦s② ♦♥t♥t② ♥ s♠♦♦t♥ss t♥ ♣ts ♥ ♦t♥ ② r①♥t ♣r♠tr③t♦♥

❩♦ t ❬❪ ♦ t sr t♦ ♦♠♥ sr ♠s t♦ ♣r♦ t①tr ♦r

♠s ② ss♥♥ s♦♠ sr ♣ts t♦ r♥t ♠s s s s♥ ♥♣♥t♥t♥qs t♦ rt t①tr ♦r ♥② ♥ss♥ tr♥st♦♥ ♣ts t♥ t♠ ♥t♦♥ t♦ ♦♠tr s♠♦♦t♥ss ♦ t ♠♣ t② t ♥t♦ ♦♥t t ♦♥t♥t②♦ t t①tr s♥ ♥ ♣♣ s♥ t ♠② ♦♠ r♦♠ r♥t s♦rs ♦r t♦♥♦r♥ ♣ts

♣tr

♦♠♣rs♦♥ ♦ P♥r t♦s

r♥t ♠t♦s r ♠♥♠③ r♥t t②♣s ♦ st♦rt♦♥ ♠trs ② ♠♦st ♣r♠tr③t♦♥ ♣♣t♦♥s ♦r st ♦♥ ③r♦ st♦rt♦♥ ♣r♠tr③t♦♥st♦ ♠♦st r t♦r♥t t♦ s♦♠ ♠♦♥t ♦ st♦rt♦♥ s♦♠ ♥ ♠♦r t♦r♥t t♦sr ♥ ♦trs t♦ strt ♥ ♥r ♣♣t♦♥s tt ♣♥ ♦♥ rr rs ♦rs♠♣♥ s s r♥t t②♣s ♦ t ♠♣♣♥ ♥ s②♥tss s s ♦♠♣rss♦♥♥ rr rs♠♣♥ s♠s ♦♠tr② ♠s ❬ t ❪ t♥ t♦ ♣r♦r♠ ttr ♦♥ strt ♠♥♠③♥ ♣r♠tr③t♦♥s s♥ strt s rt② rt t♦♥rs♠♣♥ ♥ ♦♥trst ♣♣t♦♥s s ♦♥ rrr s♠♣♥ s s r♠s♥ ❬sr♥ t ❪ r r② s♥st t♦ sr♥ t ♥ ♥ qt s♥♥tstrt ❲♥ ♣t s ♦ sr ♦r strt r ♥♦t tt♥ s srs t♦♦ ♦♠♣① t sr ♥s t♦ t ♣r♦r t♦ ♣r♠tr③t♦♥ ♥ ♦rr t♦ ♣t st♦rt♦♥

♥ t♦♥ t♦ st♦rt♦♥ sr ♦tr t♦rs s♦ ♦♥sr ♥ ♦♦s♥ ♣r♠tr③t♦♥ ♠t♦ ♦r ♥ ♣♣t♦♥ t ♥

• r rss ① ♦♥r② ♥② ♠t♦s ss♠ t ♦♥r② ♦ t ♣♥r♦♠♥ s ♣r♥ ♥ ♦♥① ① ♦♥r② ♠t♦s t②♣② s r②s♠♣ ♦r♠t♦♥s ♥ r r② st ♠t♦s r st ♦r s♦♠♣♣t♦♥s ♦r ♥st♥ t♦s tt t③ s ♠s ♣r♠tr③t♦♥ st♦♥ r♦♥r② t♥qs tr♠♥ t ♦♥r② s ♣rt ♦t s♦t♦♥ r ♦t♥ s♦r t t②♣② ♥tr♦ s♥♥t② ss st♦rt♦♥

• ♦st♥ss ♦st ♣♣t♦♥s ♦ ♣r♠tr③t♦♥ rqr t t♦ t ♦rs♦♠ ♣♣t♦♥s ♦ tt② ♥♦ tr♥ ♣s s s♥t ♦trs rqr ♦ tt② ♦♥t♦♥s t ♦♥r② ♦s ♥♦t s♥trst ♥② sst ♦ t ♣r♠tr③t♦♥ ♠t♦s ♥ r♥t ♦ ♦r ♦ tt②♦♠ ♦ t ♦trs ♥ r♥t tt② t ♥♣t ♠ss sts② s♣♦♥t♦♥s

• ♠r ♦♠♣①t② ①st♥ ♠t♦s ♥ r♦② ss ♦r♥t♦ t ♦♣t♠③t♦♥ ♠♥s♠ t② s ♥t♦ ♥r ♥ ♥♦♥♥r ♠t♦s ♥r♠t♦s r t②♣② s♥♥t② str ♥ s♠♣r t♦ ♠♣♠♥t ♦r s①♣t t s♠♣t② s② ♦♠s t t ♦st ♦ ♥rs st♦rt♦♥

s♠♠r③s t ♠♦r ♦♠♠♦♥② s ♠t♦s ♥ tr♠s ♦ ts ♣r♦♣rts sts t r♥t♠s ♦r s♦♠ ♦ t ♠♦r ♦♠♠♦♥② s ♠t♦s ❲ s t② t ❬❪ ♠♦♥ s②st♠ t♦ t♠ t ① ♦♥r② ♠t♦s ♥ ♦r t ♦tr ♠t♦s t t♠♥s r ♣r♦ ② t t♦rs s ①♣t♥r t♥qs r ♦t ♦♥ ♦rr ♦ ♠♥t str t♥ t ♥♦♥♥r ♦♥srtss ♥ t ♥♦♥♥r ♠t♦s r r② st t♥ ss t♥ t♦ ♠♥tst♦ ♣r♦ss r s③ ♠♦s rs t♦ s♦ s♦♠ t②♣ ♣r♠tr③t♦♥rsts

r rs t ♥r② ♦♥① ♦♥rs ♣r♠tr③ t ♥r ♠t♦s♠s ♠ t r♣t tt♣♦rrs♦tr

r rs t ♥♦♥♦♥① ♦♥rs ♣r♠tr③ t ♥r ♠t♦s♠s ♠ t r♣t tt♣♦rrs♦tr

r rs t ♥♦♥♦♥① ♦♥rs ♣r♠tr③ t ♥♦♥♥r ♠t♦s♠s ♠ t r♣t tt♣♦rrs♦tr

♣tr

tr♥t s ♦♠♥s

♦♠ ♣♣t♦♥s r qt s♥st t♦ s♦♥t♥ts ♥ t ♣r♠tr③t♦♥ ♦r ♥♥♦tt♦rt t♠ t ♥ s ss ♥ t ♦t t♦ ♣r♠tr③ s ♥♦t t♦♣♦♦ s t s ♦rt t♦ s r♥t s ♦♠♥ ♦r t ♣r♠tr③t♦♥①♠♣s ♦ s ♦♠♥s tt ♥ ♥stt ♥ s♠♣ ♦♠♣①ss♣rs ♥ ♣r♦ ♣♥r r♦♥s t tr♥st♦♥ rs

♥ t♦♥ ♥♠r♦s ♣♣t♦♥s ♦ ♣r♠tr③t♦♥ rqr r♦ss♣r♠tr③t♦♥ ♦r ♥trsr ♠♣♣♥ t♥ ♠t♣ ♠♦s Prs ♠♣♣♥ t♥ ♠♦s ♥ s ♦r t tr♥sr ♦ r♥t ♣r♦♣rts t♥ t ♠♦s ♥♥strt♦rr ♦♥s s s t①tr ♥ ss ♦♦s ♦♥s s s ♦r♠t♦♥ ♥♥♠t♦♥ t ♥ s♦ s ♦r ♥♥ ♥ ♠♦r♣♥ s s ♠s ♦♠♣t♦♥♥ r♣r ♠♦st ♦♠♠♦♥ ♣♣r♦ ♦r ♣rs ♠♣♣♥ s t♦ ♣r♠tr③ ♦t♠♦s ♦♥ ♦♠♠♦♥ s ♦♠♥ r♦♥r② ♣♥r ♣r♠tr③t♦♥ s r②♥st ♦r ts ♣r♣♦s ♥st tr♥t ♦♠♥s s s s♠♣ ♦♠♣① ♦r s♣r r ♦♠♠♦♥② s

❯♥t ♣r

♥t ♦ t s♣r ♦♠♥ ♦r t ♣♥r ♦♥ s tt t ♦s ♦rs♠ss ♦♥t♥♦s ♣r♠tr③t♦♥ ♦ ♥s ♠♦s ♥ tr r r ♥♠r♦ s ♠♦s ♥ s s t s♣r ♦♠♥ s r ♠ tt♥t♦♥ ♥ tst ②rs t sr ♣♣rs ♣s ♦t ts t♦♣ ♦♠ r♦r♦s t♦r② s♥ ♦♣ tt♥ ♦s t♦ t ♦ ♥rst♥♥ ♦ ♣♥r ♣r♠tr③t♦♥s s ♥♦ts ♦r ♦r ♠♥ t②♣s ♦ s♣r ♣r♠tr③t♦♥ ♣♣r♦sss trt ①t♥s♦♥ ♦ ♣♥r r②♥tr ♠t♦s str♦r♣ ♣r♦t♦♥s♣r ♥r③t♦♥ ♦ r②♥tr ♦♦r♥ts ♥ ♠trs♦t♦♥ ♠♥

♥ ttrt ♣♣r♦ ♦r s♣r ♣r♠tr③t♦♥ s t♦ ①t♥ t r②♥tr♦♥① ♦♥r② ♣♥r ♠t♦s t♦ t s♣r r ♠t♦s ❬① ♥❨ ♦t t ❪ s ss trt♦♥s t♦ ♦t♥ s ♣r♠tr③t♦♥ ② strt ② ♦♠♣t♥ ♥ ♥t ss ♥ t♥ ♠♦♥ t rts ♦♥ t t♠ rst ♦♠♣t♥ ♣♦st♦♥ ♦r t rt① s♥ r②♥tr ♦r♠t♦♥ ❬t ❪ ♥ t♥ ♣r♦t♥ t rt① t♦ t ♥t s♣r s♥r t ❬❪

r ♣r Pr♠tr③t♦♥

s♣t t ♠s ♥ t♦ ♠♣ t t ♦♥t♦ rt r ♥ ♠ ♠s ♦♥t♦ ♠s♣r s♥ ♠♦ tt ♣r♦r rtt② s ♣r♦♥ ② t ❬❪ ♣r♦t ss trt♦♥s rs t rs ♦r ♦♥② ♥t ♥♠r ♦trt♦♥s s t rst ♣♣r♦s t s♦t♦♥ t s♠ t♠t② ♦♠s♥st t rs ♥rss ♥ t s②st♠ ♦♣ss t♦ ♥rt s♦t♦♥ t ❬❪ ♥♦t tt ts ♦r s ♥♣♥♥t ♦ st♣ s③

r t ❬❪ ♦♠♣t ♣♥r ♣r♠tr③t♦♥ ♦ t ♠s rst s♥ ♦♥♦ t tr♥s s ♦♥r② ② t♥ s t str♦r♣ ♣r♦t♦♥ t♦ ♦t♥t s♣r ♠♣♣♥ rst ♣♥s qt ② ♦♥ t ♦ ♦ t ♦♥r②tr♥ s ♣♣r♦ ♦rs qt ♥ ♣rt ♦r t ♦s♥t ♦r ♥② t♦rt r♥ts s♥ t str♦r♣ ♣r♦t♦♥ s t ♦♥② ♦r t ♦♥t♥♦ss ♥ ♥ ♣r♦ tr♥ ♣s ♥ t srt s s♠♣ ♣r♦♦ ② ①♠♣ ♦ts stt♠♥t ♥ ♦t♥ ② ♠♥♥ t rt r s♣♣♦rt♥ t ♦ ♠♣♣ s♣r tr♥ ♦♥t♥♦s str♦r♣ ♣r♦t♦♥ ♠♣sts rt r t♦ r ♥ t ♦r♥ ♣♥ tr rt① ♥ ♣rtr♥ t ♣♥ t♦ r♦ss r♦♠ t ♥tr♦r t♦ t ①tr♦r ♦ t r t♦t ♥♥t tr♥ ♦r♥tt♦♥ s♣r tr♥ ♣ ♦r s rst ♦ ts♣rtrt♦♥ s t ♠ ♦ ♦♥ t s♣r r♦ss r♦♠ ♦♥ s t♦ t ♦tr ♦t s♣r

♦ts♠♥ t ❬❪ s♦ ♦ t♦ ♦rrt② ♥r③ t ♠t♦ ♦ r②♥tr♦♦r♥ts t ts ♥ts t♦ t s♣r ♥r③t♦♥ s s ♦♥ rstsr♦♠ s♣tr r♣ t♦r② t♦ ❱rèr ❬❪ ♥ ①t♥s♦♥s t♦ ♦ás③ ♥rr ❬❪ ② ♣r♦ qrt s②st♠ ♦ qt♦♥s s s♣rq♥t ♦ t r②♥tr ♦r♠t♦♥ t♦rs ♦ ♥♦t ♣r♦ ♥ ♥t ②t♦ s♦ t rst♥ s②st♠ ♥ ts tr ♠t♦ s ♠t t♦ r② s♠ ♠ss t ❬❪ ♥tr♦ ♠t♦ ♦r ♥t② s♦♥ t s②st♠ ② ♣r♦♥ ♦♦ ♥t ss ♥ s♥ r♦st s♦r rst s♠r t♦ ❬s♥r t ❪t② ♣rtt♦♥ t ♠s ♥ t♦ ♥ ♠ ♦♥ ♠s♣r s♥ ♣♥r♣r♠tr③t♦♥ ♦♦ ② str♦r♣ ♣r♦t♦♥ ② t♥ s ♥♠rs♦t♦♥ ♠♥s♠ ♦♠♥s ss trt♦♥ t ♥♦♥♥r ♠♥♠③t♦♥t♦ ♦t♥ t ♥ s♦t♦♥

♥ ♥t ♥ t tr♥t s sst ② ♠trs♦t♦♥ t♥qss ♠t♦s ♦t♥ ♥ ♥t ss ② s♠♣②♥ t ♠♦ ♥t t ♦♠s ttrr♦♥ ♦r t st ♦♥① tr② ♠ t ♦♥ t s♣r ♥ t♥ ♣r♦rss② t rts ❬♣r♦ ♥ Pr♥ ♥ ♦♣♣ ❪ ♣r♦ ♥ ❬❪ ♦♠♣t t ♠♥ s♥ ♣r② t♦♣♦♦ ♦♣rt♦♥s ♥ ♦ ♥♦t tt♠♣t t♦♠♥♠③ ♥② t②♣ ♦ st♦rt♦♥ Pr♥ ♥ ♦♣♣ ❬❪ ♦t♥ s♣r ♣r♠tr③t♦♥ ② tr♥t♥ r♥♥ st♣s tt rts r♦♠ ♠trs♦t♦♥ ♦♠♣♦st♦♥ ♦ t ♦t t r①t♦♥ ♦ s♥ rt① ♦t♦♥s ♥s tr ♥♦r♦♦s r①t♦♥ s ♠ t♦ ♠♥♠③ t strt ♠tr ♦ t ♣r♠tr③t♦♥ ♥ sr♥t t♦ ♠♥t♥ ♠♥

r Pr♠tr③t♦♥ ♠t♦s ♦r st♦♣♦♦② ♦♠♥ t s♠♥tt♦♥ ♦rt♠s ♥ rt t①tr ts r♦♠ s♣ ♦ rtrr② t♦♣♦♦② ♦rt r ♥♠r ♦ s♦♥t♥ts ♥ ♣r♦♠t ♦r t ♣♣t♦♥s ♦♣r♠tr③t♦♥ ♦rt♠s ♦ ♥♦t sr r♦♠ ts ♣r♦♠ t ♦rts② ♦ t t

♥♦ Pr♦t t♥♦r

♠♣ ♥ qrtr ♦♠♣①s

s s♥ ♥ ♣trs ♥ ♣r♠tr③t♦♥ ♠t♦s ♥ ♣t s♣ t st♦♣♦♦② ♥ ♦♥t♦♦♥ ♦rrs♣♦♥♥ t ♦♠♥ ♦r s♣ t rtrr②t♦♣♦♦② t s ♣♦ss t♦ ♦♠♣♦s t s♣ ♥t♦ st ♦ rts s♥ s♠♥tt♦♥♦rt♠ ❱ ❬♦♥t♥r t ❪ rt s t♥ ♣r♠tr③ sr ♥ ts s♦t♦♥ ♦rs t s ♥♦t ♦♠♣t② stst♦r② ② ♦♥s♦ ♠ t sr st t♦ ♥ ♦♦r♥t s②st♠ ♦♥ t r♦♠ t ♣♣t♦♥ ♣♦♥t ♦ rt ♦♥rs r t t♦ ♥ ♥ r♠s♥ ♦rt♠s♥ ♥tr♦ rtts ♥ t①tr ♠♣♣♥ ♣♣t♦♥s ♦r ts rs♦♥ ♦s ♥ts st♦♥ ♦♥ ♦ ♣r♠tr③t♦♥ ♦rt♠s tt ♦ ♥♦t rqr s♠♥t♥ tsr r

♦ ♦♠♣t s ♦ ♣r♠tr③t♦♥ t ♦♠tr② ♣r♦ss♥ ♦♠♠♥t② rst♦♣ ♠t♦s tt ♦♣rt ② s♠♥t♥ ♣r♠tr③♥ ♥ rs♠♣♥ t♦t ♦ ♦r ♥♦ ts s rst ♦♣ ♥ t P ♠t♦ ❬ t ❪ trs♦t♦♥ ♣t Pr♠tr③t♦♥ ♦ rs s s♦♥ ♥ r ts ♠t♦ strts ② ♣rtt♦♥♥ t ♥t ♦t r ♥t♦ st ♦ tr♥r rts t s ♦♠♣① r ♥ ♣r♠tr③t♦♥ ♦ rt s ♦♠♣t ♥ t ♦t s rr② rs♠♣ ♥ ♣r♠tr s♣ r rtr r♥♠♥ts ♦ t ♠t♦ ♠♣r♦ t ♥trrt ♦♥t♥t② ❬♦

r P ♠t♦ ♥ ts rts ♦♠♣t ♦ ♣r♠tr③t♦♥② ♦♠♣♦s♥ t ♥t sr ♥t♦ st ♦ tr♥r rts ♥ rr②rs♠♣s t ♦♠tr② ♥ t ♣r♠tr s♣ ♦ ts rts

♦s② t ❪ ♦r♠③ ② t ♥♦t♦♥ ♦ tr♥st♦♥ ♥t♦♥ ①♣♥ rtr♥ ts st♦♥ s r♣rs♥tt♦♥ tts ♥♥ rr r♣rs♥tt♦♥s ♥♠♣♠♥t♥ ♠trs♦t♦♥ ♣r♦ss♥ t♦♦s ♦♥ t♦♣ ♦ t ❬s♦ t ❪

st♦r② t ♠♦st ♣♦♣r ♥♦♥♣♥r s ♦♠♥ s ♥ s♠♣ ♦♠♣①❬ t s♦ t t Pr♥ t ♦♦s② t Pr♥♦♠♦ t r♥r t r♦② ♥r ❪ s♠♣ ♦♠♣① ♥ ♦♥sr s st t ♦♥♥tt② ♣rt ♦ trt♦♥ tr♥ ♠s t sts ♦ rts s ♥ s ♦st ♣♣t♦♥st②♣② s s♠♣ ♦♠♣①s r♣rs♥t♥ ♠♥♦s t ♦♥r② ♥ ♥ ♦♥② ♥t t♦ ♦r s t s♠ ♥♠r ♦ ♠♥ts ♥ ♠t♦ ♦r♦t♥♥ s ♦♠♣①s s t♦ s♠♣② ♥ ♦r♥ ♠s ♥ st s ♠ss ♥ ♦s♥ t ♦r♥ ♠s s ♣r♠tr③ ② ss♥♥ ♦ ts rts t♦ s♠♣① ♦ t s ♦♠♥ rt① ♦r ♦♥ t r②♥tr ♦♦r♥ts♥s t

r② ♠t♦s t♦♦ t♦st♣ ♣♣r♦ t♦ ♦♠♣t♥ ♣r♠tr③t♦♥ ♥ trst st♣ ♠♥ts ♦ t ♥ ♠s r ss♥ t♦ s ♦ t s s♠♣ ♦♠♣① t s♦♥ st♣ ♦ ♦♠♣t r②♥tr ♦♦r♥ts ♦r ts ♠♥ts s②s♥ ♦♥ ♦ t ① ♦♥r② ♣r♠tr③t♦♥ ♠t♦s sss rr s st♣s♦ r♣t t t②♣② ♥♦t ♠① ♦r r♥t ♠t♦s s s ❬♦♦s②t ❪ tr② t♦ ♣r♦r♠ ♦t st♣s t t s♠ t♠

♦♠♣t♥ s ♦♠♣①s

♦ ♦t♥ t s♠♣ ♦♠♣① ❬ t ❪ r♦ ❱♦r♦♥♦ r♦♥s ♦ s r♦♠s ♣♦♥ts ♥ t♥ s t tr♥t♦♥ s ♣♦♥ts r ♥t② ♥s♥ s♦rtst ♣ts r♦ss ♠s s tt ♣r♦ t ♥t ♦♥rs ♦ t ♣ts♦rrs♣♦♥♥ t♦ s ♦♠♥ s ♦ strt♥ ♦ ts ♣ts t t♦ ♥t♣ts r ♣r♠tr③ t♦ sqr ♣t ♥ qst♦♥ s t♥ r♣ t t

♦♥ ♦ t sqr ♠♣♣ ♦♥t♦ t ♠s sr❬ t ❪ s♠♣② t ♦r♥ ♠s ♣♥ tr ♦ ♦rrs♣♦♥♥s t♥

t ♦r♥ rts ♥ t s ♦ t s♠♣ ♠s trs ❬s♦ t ❪ r ♥ ❬♦♦s② t ❪ s str♥ t♥qs t♦ ♥rtt ♣t ♦♥♥tt② ♥ r t s♠s r♦♠ t

♦♥strt♦♥ ♦♠s ♠♦r ♥♥ ♥ ♠t♣ ♠♦s ♥ t♦ ♣r♠tr③ ♦♥ t s♠ ♦♠♣① ❬Pr♥ t r♥r t r♦②♥ r ❪ Pr♥ t ❬❪ ♣rtt♦♥ ♠s ♥t♦ tr♥r ♣ts ♦rrs♣♦♥ t♦ t s ♦ sr ♥ s♠♣ ♦♠♣① ② r♥ ♥t♦r ♦ ♣tst♥ srs♣♣ tr rts tt ♦rrs♣♦♥ t♦ t rts ♦ t s ♠s

r♥r t ❬❪ ♥ r♦② ♥ r ❬❪ ①t♥ t ♠t♦s ♦ Pr♥t ❬❪ ♥ r♦② t ❬❪ t♦ ♦♥strt t s♠♣ ♦♠♣① t♦♠t②♥ ♣r t♦ t ♣t ♦r♠t♦♥ ♥♣t t♦ ♦t ♠t♦s ♥s st ♦ ♦rrs♣♦♥♥s t♥ tr rts ♦♥ t t♦ ♥♣t ♠♦s ♠t♦s s t♦s st rts ♦ t s ♦♠♣① ② s♠t♥♦s② tr ♣ts ♦♥ t ♥♣t ♠sst♥ ♦rrs♣♦♥♥ ♣rs ♦ rts s♣tt♥ ①st♥ ♠s s ♥ssr② r♥t ❬❪ r t rst t♦ ♦r ♥♦ t♦ s qrtr s ♦♠♥ ♦♠♥ s ♠ ♠♦r st ♦r qrtr r♠s♥ ♦ t ♥♣t sr ♥ ♦rs♣♥ tt♥ r♥ t ❬❪ ♥rt t s ♦♠♥ ♠♥②

♣♣♥ t♦ t s ♠s

♥ t srt ss♥♠♥t t♦ s ♦♠♥ s s ♥ ♦♥ t r②♥tr ♦♦r♥ts ♥ ♦♠♣t s♥ ①♦♥r② ♣♥r ♣r♠tr③t♦♥ rr ♠t♦s♦♠♣t t r②♥tr ♦♦r♥ts ♦♥ s ♦♥ t ♥t ss♥♠♥t ♦ t rts t♦ t s tr♥s ♦r r♥t ♠t♦s ❬♦♦s② t r♦②♥ r r♥ t ❪ s ♥ trt ♣r♦ss r rts ♥ rss♥ t♥ s s

♦♦s② t ❬❪ ♣r♦r♠ t rt①t♦♣t ss♥♠♥t ♥ ♦♦r♥t r①t♦♥ ♥ s♥ ♣r♦r ② tt♥ rts r♦ss ♣t ♦♥rs s♥ tr♥st♦♥♥t♦♥s tr♥st♦♥ ♥t♦♥ ①♣rsss t r②♥tr ♦♦r♥ts ♦ rt① trs♣t t♦ s ♦♠♥ s r②♥tr ♦♦r♥ts ♦r ♥♦r♥ s ♦♠♥ ♦r ts ♣r♦r ♦♥② t ♠s ♦ t s ♦♠♥ rts ♥s t♦ ①rtr t♥ t s s s ♥ t ♣r♦s ♠t♦s t♦rs r① t s♦♠♥ rts s♣rt② ♣r♦♠♣t♥ ♥ r♥ ♦ t ♠♥ r①t♦♥ ♥ ♣rt ts② s r♣t ♦♥② r② t♠s ♠♣♠♥tt♦♥ s♦♠t♠s ♥s t♦ srs♦♠ r①t♦♥ rsts ♥ ♠s rts ♠♦ r♦♥ s ♦♠♥ rts ♥ ♣t r②♥tr ♦♦r♥ts tt r ♥ ♦r t s ♦♠♥ s r♦♥ ttrt①

r♥ t ❬❪ ♥ r♦② ♥ r ❬ ❪ ① t ♦♥r② ♦ r♦♣♦ s ♠s s ♣t t r②♥tr ♦♦r♥ts ♥ t ♥tr♦r ♥ t♥ ♣♦ss②rss♥ s♦♠ rts t♦ r♥t s ♥s t r♦♣ ♠t♦s r ♥ tr♦♣♥ t② s ♥ t ♦ ♦ ♣r♠tr③t♦♥ t♥q s ♦r t r②♥tr♦♦r♥ts ♦♠♣tt♦♥

r ♦ ♣r♠tr③t♦♥ s♦ r♥t ♠♥♦ ♥ t ♠t♠ts trtr s st ♦ ♣r♠tr③ rts (ϕ, ϕ′, . . .) ♦♥♥t ② r♥ttr♥st♦♥ ♥t♦♥s (τϕ→ϕ′ . . .)

♦ Pr♠tr③t♦♥

s ♠② ♦ ♠t♦s s st ♦ tr♥r rts t♦ ♥ t s ♦♠♣① ♦r s♦♠♣♣t♦♥s s s t①tr ♠♣♣♥ ♦r sr ♣♣r♦①♠t♦♥ t t♥s♦r♣r♦ts♣♥s t s ♣rrr t♦ s s ♦♠♣① ♦♠♣♦s ♦ qrtrs r♥s♠s st t rst st t t♦♠t② ♦♥strt♥ ♦♦ qrtr s♦♠♣① s st ♥ ♦♣♥ ♣r♦♠ r♥t ♦ t P ❬♦r♠♥♥ ♥ r♥r ❪♠t♦ ♣♣ t♦ qrtr s ♦♠♣① s ♣r♦♣♦s ❬r♥ t ❪ ♠t♦ ts t sr ♥trt② ♥ t s ♦♠♣① ♦r r♥t② ♥s t♦t♦♠t t ♣r♦ss r ♠ s s♦♥ rtr ♥ ts st♦♥

♥①t sst♦♥ ♥tr♦s t ♥♠♥t ♥♦t♦♥s ♥ r r♥t♥s ♥ ts r

r♥t ♠♥♦

❲ strt ② ♥♥ t ♥♦t♦♥ ♦ r♥t ♠♥♦ tt ♦r♠③s t ♥♦t♦♥♦ ♦ ♣r♠tr③t♦♥ ♦t♥ t t ♥♥♥ ♦ t st♦♥ s ♠s t♣♦ss t♦ ♥ ♦② s♠♦♦t ♣r♠tr③t♦♥ ♦♥ sr ♦ rtrr② ♥s ②♦♥♥t♥ ♠t♣ ♦ ♣r♠tr③t♦♥s ♦ ♦r ♥♦ t ♥♦t♦♥ ♦ r♥t♠♥♦ s rst ♥tr♦ t♦ t ♦♠tr② ♣r♦ss♥ ♦♠♠♥t② ② r♠♠ ♥s ❬r♠♠ ♥ s ❪ ♦r r♥t② ♦♥strt s ♣r♦♣♦s t♦ ♥srs ♦ ss C∞ ❬❨♥ ♥ ❩♦r♥ ❪

♥ sr S ♦♥sr st ♦ ♣♦ss② ♦r♣♣♥ t♦♣♦♦ ss C rts ♥ st ♦ ♥t♦♥s ϕ tt ♣t rt C ♥ ♦rrs♣♦♥♥ t

♦♠♥ Ω r ♦♦r♥ts ♥♦t ② θ, φ st♦ ♥t♦♥s ϕ ♥s r♥t ♠♥♦ t ♦♦♥ ♦♥t♦♥ s sts ♥ t♦ rts C ♥ C′ tr ♥trst♦♥ C∩C′ s ♦♠♦♠♦r♣ t♦ s t♥t t♦ ♠s ♦ tr ♥trst♦♥ C ∩ C′ ♥ ♣r♠tr s♣ tr♦ ϕ ♥ ϕ′ r♥ ② r♥t ♦♠tr tr♥s♦r♠ τϕ→ϕ′

∀p ∈ C ∩C′, ϕ′(p) = τϕ→ϕ′ (ϕ(p))

♥t♦♥s τϕ→ϕ′ r tr♥st♦♥ ♥t♦♥s ❬♦♦s② t ❪ ♠♥♦ s s t♦ ♥ t tr♥st♦♥ ♥t♦♥s r tr♥st♦♥s ♦♠♣①♠♥♦s ♠t ♥② ♦♥t♦♦♥ ♦♥♦r♠ ♥t♦♥ s tr♥st♦♥ ♥t♦♥ t♦♥ ♦r t ♥♦t♦♥ ♦ ♦♥♦r♠t② ♥ ♣rtr ts ♥s s♠rts ♥t♦♥s ♦♠♣♦s ♦ tr♥st♦♥s r♦tt♦♥s ♥ s♥s

①tr♦r s

② s♥ r♣rs♥tt♦♥ ♦ ♠♥♦ s st ♦ rts t t ss♦t tr♥st♦♥♥t♦♥s t s ♣♦ss t♦ ♠♥♣t ♥t♦♥s ♥ ♦r t ♠♥♦ ♦♠♣t trr♥ts ♥ ♥trt t♠ ♦r ssts ♦ t ♠♥♦ ♦r ts ♦♠♣tt♦♥s♦t♥ ♥♦ t ♦♥ ♠tr① ♦ ♦ ♣r♠tr③t♦♥ ♥ ♥♦♥tr ♠♥♥r♥ ♠ t ♦♠♣tt♦♥s qt ♥♦ s♣t t ♣♦sst② ♦ s♥ ♦r♠t♦♦s ♣ ts ② ♦ ♦♥t♥ ♦♠♣tt♦♥s r♠♥s rstrt♥ r♦♠ ♥♥tt ♣♦♥t ♦

①tr♦r s s ♥ ♥t tr♥t t♦ ts ♣r♦♠ s s ♦♠tr s t♦t ♦♦r♥ts tt ♦♥srs t ♥t♦♥s s strt ♠t♠t ♦ts♦♠♥ t ♦♣rt♦rs t t ♥ t♦ s t ♥t♦♥s ♦♥ st ♥s t♦ ♥st♥ttt♠ ♥ ♦ ♦♦r♥t r♠ ♦r ② ♣♦st♣♦♥♥ t ♦r r♣rs♥tt♦♥ ♥ ♦♦r♥ts t t ♥ ♦ t ♠t♠t rs♦♥♥ ♠♦st ♦♠♣tt♦♥s rs♠♣ ② r ♦♥srt♦♥s ② t r strtr ♥ ②t ♥t♦♥s ♥ t ♦♣rt♦rs

①tr♦r s ♥r③s t ♥♠♥t t♦r♠ ♦ s tt ♥s t♥tr ♦ ♥t♦♥

∫ b

a

f(x)dx = F (b)− F (a)

r F ♥♦ts t ♣r♠t ♦ f ♥r③ t♦r♠ rts

∫

Ω

dω =

∫

∂Ω

ω

r ω s r♥t ♦r♠ ∂Ω ♥♦ts t ♦rr ♦ Ω ♥ d ♥♦ts t ①tr♦rrt ♦r t ♠♦♠♥t ♦♥ ♠② t♥ ♦t ♦r♠ s s♦♠t♥ tt ♥tst♦ ♥trt ♦r ♠♥s♦♥ ♦♠♥ r♦r ♦r♠s ♦rrs♣♦♥ t♦ ss♥t♦♥s ♦r♠ t♦ t♦r s tt ♦♥ ♥ ♥trt ♦♥ r ♥ ♦r♠st♦ ♥t♦♥s tt ♦♥ ♥ts t♦ ♥trt ♦r sr ♦♠♥ ♦t t ①tr♦r

rt ts s ♥r③t♦♥ ♦ t r♥t ♦♣rt♦r ①tr♦r rt ss♦ t ♥♦t ♣r♦♣rt② ♦ ♥s♥ ♥ ♣♣ t ddω = 0 ①♣♥② rtr

r♥ t♦r♠ str♦rs②ss s ♣rtr s

∫

ol

Kdv =

∫

∂♦K ·NdS

r N ♥♦ts t ♥♦r♠ t♦ t ♦rr ♦ t ♦♠ ♦t tt ♥ t ①tr♦rs rs♦♥ t sr ♣r♦t t t ♥♦r♠ s r② t♥ ♥t♦ ♦♥t ② t♥trt♦♥ ♦r ∂Ω ts s t ② ①tr♦r s ♦♥srs t ♦rr ♦ sst♦r ♥r② ①tr♦r s ①ts s②♠♠tr② t♥ t ♥trt ♥tts r♥t ♦r♠s ♥ t ♥trt♦♥ ♦♠♥s ♥s ♥trt♦♥♦ ♦r♠ ω ♦r ♥ Ω s t♥ rt♥ t②♣ ♦ ♥♥r ♣r♦t ♦r ♦t ♣r♦t< ω, Ω > ♦rr ♦♣rt♦r ∂ ♦♠♣ts t ♦rr ∂Ω ♦ ♥ Ω ♥ rtr♥s♥♦tr ♥ ♦ ♦r ♠♥s♦♥ ❲t ts ♥♦tt♦♥ t♦s t♦r♠ rts

< dω, Ω >=< ω, ∂Ω >

♥ ♦tr ♦rs r♦♠ ♥ ♥tt ♣♦♥t ♦ t ①tr♦r rt d ♦♠s t♦rr ♦♣rt♦r ∂ ♥ ♦♥ sts r♦♠ t ♥trt ♦r♠ ω t♦ t ♥trt♦♥♦♠♥ Ω s ①ts t t② t♥ t ①tr♦r rt d ♥ t ♦rr♦♣rt♦r ∂ ♦r ts rs♦♥ r♥t ♦r♠s r s♦ ♦♥s ♥ t ①tr♦rrt s s♦ t ♦♦rr ♦♣rt♦r

♥② ♦♥ ♥ ♥♦t tt t ♥♠♥t t♦r♠ ♦ s s rtr s s♣ s ∫

(a,b)

f(x)dx =

∫

a,b

dF =

∫

a−∪b+F = F (b)− F (a)

s s t ♦r♥tt♦♥ ♦ t ♦rr ∂(a, b) = a− ∪ b+ tt ♥tr♦s t s♥ ♥r♦♥t ♦ F (a)

♦ ♦r ♥♦ ♥♦t♦♥s ♦ ①tr♦r s rst ♣♣r ♥ t ♦♠♠♥t② ♥P♥ ♥ P♦trs ♣♣r ❬P♥ ♥ P♦tr ❪ r t② s ♦ t② t♦ ♦♠♣t ♠♥♠ srs ♥ t ❬ ♥ ❨ ❪ s t♥♠♥t ♥♦t♦♥s ♥♦ ♥ P♦♥rés ♦♥tr t♦ ♦♠♣t ♦ ♣r♠tr③t♦♥s ♥ r♥ ♦♣ ♥ s P tss srt ♦♥tr♣rt ♦ ①tr♦r s ❬r♥ ❪ s♣② st t♦ ♦♠tr② ♣r♦ss♥ t ♠ss ♦rr♥t② ♥ rör ♥ sr♥ ♦rs t P ♦t s ttr ♦rs ♦♥trt t♦ ♠ ts ♦♠♣① strt ♥♦t♦♥s ss t♦ r ♦♠♠♥t② ♥ rrt t ♥trst rr t♦ tr ♦rs ♥♦ts ♦r ♠♦rts ♦♥ ts s♥t♥ t♦♣ ♥①t sst♦♥ q② rs ♠t♦s s ♦♥♦♦♠♦♦② ♥ ♥tr♦s t rt ♥♦t♦♥s ♥ t ♦♦s ♣ t ♥r♦♥t♥♦s stt♥ ♦r t ①♣♥t♦♥s

r ♦♠♦♦② ss ♦ t♦rs ♥ ♦ t♦rs

♦♠♦♦② ♥ ♦♦♠♦♦②

s s♥ ♥ t ♣r♦s sst♦♥ t ♥♦t♦♥ ♦ ♦rr ♦ ♥ ♥trt♦♥ ♦♠♥ ♣②s ♥tr r♦ ♥ ①tr♦r s s♥ t ♥♠♥t t♦r♠ t♦s♦♥♥ts ♥trs ♦r t ♦♠♥ t ♥trs ♦♥ t ♦rr s ①ts ♥♠♥t ♦♥♥t♦♥s t♥ t t♦r② ♦ ♥trt♦♥ ♥ t t♦♣♦♦② ♦ t srrtr③ ② ts ♦♠♦♦② r♦♣ ♦r ♥st♥ t♦ ♦s rs C1 ♥ C2 r♦♠♦♦② q♥t ♦♥ ♥ tr♥s♦r♠ ♥t♦ t ♦tr ♦♥ ② ♦♥t♥♦s♦r♠t♦♥ ♦r ♥r② t♦ ♦s rs r ♦♠♦♦②q♥t tr ♥♦♥♦rrs♣♦♥s t♦ t ♦rr ♦ sst ♦ t ♦t s ♥♦t♦♥ ♠s t ♣♦ss t♦♦♠♣♦s ♥② r ♥ t sr ♥t♦ s♠ ♦ ♥♠♥t rs ♥ ♦♠♦♦② ss r s♦s ♥ ①♠♣ ♦ ♦♠♦♦② ss ♦r t♦rs ♥ ♦t♦rs ♦r ♥r② ♥ ♦♠♦♦② ss ♦ ♥ ♦t ♦ ♥s g ♥ ♦tr ♦r ♦♣ t g ♥s s 2g ♠♥ts s ♥ s♥ ♥ t r ♥ s 2♥♠♥t rs ♦♥ ♦ t♠ ♥s r♦♥ t ♥ ♥ t ♦tr ♦♥ ♥s ♥t ♣r♣♥r rt♦♥

❲ ♠♥t♦♥ ♥ t ♣r♦s st♦♥ t t② t♥ ♥trt♦♥ ♦♠♥s♥ r♥t ♦r♠s s t② s♦ ♣♣s t♦ t ♥♦t♦♥ ♦ ♦♠♦♦② tt rtst♦ rs ♦r ♥s r♦♠ ♦♦♠♦♦② tt rts t♦ ♦♥s ♦r r♥t♦r♠s ♥ r ❲ rst ♥ t♦ t♦ ♠♦r ♥t♦♥s

• ♦r♠ ω s ♦s dω = 0

• ♦r♠ ω s ①t t s q t♦ t ①tr♦r rt ♦ ♥♦tr ♦r♠ ∃σ/ω =dσ

♦t tt ①t ♦r♠s r ♦s s♥ ddω = 0∀ω s ♦♥ t t② t♥♦r♠s t♥s t♦ ♥trt ♥ ♥s ♦♠♥s ♦ ♥trt♦♥ ♥ ♥♦ ttr

♥ ①tr♦r s t ♦ t♥ s t ♦t♥❲ s ♥ tt t ♥♦t♦♥ ♦ ♦rr ♥♦ ♥ t ♥♠♥t t♦r♠ ♣②s ♥tr r♦

♥rst♥ t tr♠♥♦♦② ♥ ♥♦tt♦♥s t ♦♥r② ♦♣rt♦r ∂ s t♦ t①tr♦r rt d ♦s ♦r♠ ω s tt dω = 0 s t♦ ♦s ♥ Ω s tt ∂Ω = 0 ♦r♦r t ♦♥r② ♦ ♥ s ②s ♦s ∂∂Ω = ⊘)ts ①♣♥s ② t ①tr♦r rt ♥ss ♥ ♣♣ t (ddω = 0)

❲ ♥ ♥♦ ♦rt ♦♥ ♦♦♠♦♦② ♦♥ sr ♦♥ ♥ ss♦t t♦ ♥♠♥t r ♦ t ♦♠♦♦② ss st ♦ ♦s ♦r♠s t♦r s ttr q♥t t rs♣t t♦ ♦♦♠♦♦② ♦r ♣rs② t♦ ♦r♠s ω1 ♥ ω2 rq♥t tr ♥tr ♦♥ t rs ♦ t ♦♠♦♦② ss ♠t ♦t ttts s t s tr r♥ ω2 − ω1 s ①t s♥ t ♥trt♦♥ ♦ t r♥♦♥ ♥ ♠♥t Ω ♦ t ♦♠♦♦② ss s

∃σ/ω2 − ω1 = dσ ⇒< Ω, dσ >=< ∂Ω, σ >=< ⊘, σ >= 0

r♠♥ tt t ♠♥ts ♦ t ♦♠♦♦② ss r ♦s rs ❲ ♥ ♥♦ t ♥r ♥t♦♥ ♦ ♦♦♠♦♦② tt s t♦ s② t q♦t♥t s♣ ♦ ♦s♦r♠s ♦♥ ①t ♦r♠s t♦ ♦s ♦r♠s r q♥t tr r♥ s ♥ ①t♦r♠ s rt♦♥ s t♦ ♦♠♦♦② t♦ ♦s rs r q♥t t② rt ♦rr ♦ sst

s s♦♥ ♥ r ♥ ❨ s ts ♥♦t♦♥s t♦ ♦♠♣t ♦ ♦♥♦r♠ ♣r♠tr③t♦♥ ♦♥ sr ♦ rtrr② ♥s ❬ ♥ ❨ ❪ ♦♦ s♦ t② ♦♠♣t ♦♦♠♦r♣ ♥t♦♥ t ♥r③t♦♥ ♦ ♦♥♦r♠ ♥t♦♥s ♠♥t♦♥ ♥ t♦♥ s ♦♥ ♥ ♠♣♦rt♥t t♦r♠ tt stts tt ♦♦♠♦♦② ss ♦♥t♥s ♥q r♠♦♥ ♦♥♦r♠ ❲ r② s♥ tt t♦♦r♥ts ♦ ♦♥♦r♠ ♠♣ r t♦ r♠♦♥ ♥t♦♥s ♠r② ♦♦♠♦r♣♥t♦♥ s ♦♠♣♦s ♦ t♦ ♦♥t ♦rt♦♦♥ r♠♦♥ ♦♥♦r♠s ♥ tr♠t♦ ♦♣rts s ♦♦s t② rst ♦♠♣t ♦♠♦♦② ss ♦ t sr s♥♦r ♥st♥ rs♦♥s ♠t♦ ❬rs♦♥ ♥ ❲tts② ❪ ♦r ③rss ♦♥ ❬③rst ❪ t♥ t② ♦♦♠♦♦② ss ♥ ♥ ♣r ♦ ♦♥tr♠♦♥ ♦♥♦r♠s ♥② t② ♥trt t ♦♥♦r♠s t♦ ♥ t ♣r♠tr③t♦♥ ♦t tt ♦r ♥ ♦t ♦ ♥s g t ♣r♠tr③t♦♥ s ♥ssr②2g s♥rts ② t ❬② t ❪ ♣r♦♣♦s ② ♦ t♦♠t♥ t♣♠♥t ♦ ts s♥rts ♥ ♥r③ t♠ t♦ rt♦♥ ♥s s r

♥ ♣rt t srt rs♦♥ ♦ t s r♠♦r ♦♣rts ② s♦♥ ♥rs②st♠s t tr t②♣s ♦ ♦♥str♥ts r♠♦♥t② ♦s♥ss ♥ t② r♠♦♥t② qt♦♥s r s♠r t♦ t ♦♥s s ② ♣♥r ♠t♦s t s♠ ♦t♦rs ♦r s ♥♥t t♦ rt① ② t ♦t♥♥t r♠♦♥ ts s③r♦ ♦s♥ss qt♦♥s stt tt t s♠ ♦ t t♦rs ♦r t s ♦ s ③r♦ r♥t s ♥♦ r♥ t② ♦♥t♦♥s r♣ t♦♥r② ♦♥t♦♥s ♥ trt♦♥ ♣r♠tr③t♦♥s t② ♠♣♦s ① s ♦r t♥tr ♦ t ♦♥ t ♦s ♦♦♣s ♦r♠♥ t ♦♠♦♦② ss ♦ t sr t 2grs tt t ♦♣♥ t ♥s ♦ t ♠s r s t ♥s ♦ t sr r

♥ t ♦rr ♦ sst s t♦ ①t♥ss ♦r ♦r♠s

r ♠t♦ ♦♣ ② ♥ ❨ t♦ ♦♥strt r♥t ♠♥♦rst ♦♠♣ts ♦♠♦♦② ss t♥ s ♦♦♠♦♦② ss ♥ ♥s t♥q r♠♦♥ ♦♥ ♦r♠ ♥ ♦♦♠♦♦② ss ♥② t u ♥ v ♣♦t♥ts r ♦t♥ ② ♥trt♥ ts r♠♦♥ ♦♥♦r♠s tt t♦tr ♥ ♦♦♠♦r♣ ♥t♦♥ t ♦rts② ♦ t♥♦r ♣r♠tr③t♦♥ ♦rts② ♦ ❳

r t s ♥♦♥ r♦♠ ♠t♠t♥s ♥ rtt♥ ♦ts ♠rs tt t♠s ♦ ♥ ♦t ♦ ♥s g s 2g s♥rts ♦r ♣♦s s s♣r g = 1s t♦ ♣♦s t s ♣♦ss t♦ s ♣♦ ♥t♦ t♦ ♣♦s ♦r ♦rqrtr♣♦s ♦rr ♦ ♠t♣t② ♦ ♣♦ ♦r t r♦tt♦♥ ♥ tt t♠s ♥r♦s ♥ tr♥♥ r♦♥ t ♣♦ s t ♥① ♦ t ♣♦

s②st♠s r s♦ ♥ s②st♠ t ♥tr r♦ss ♦♥ ♦ t rs s ♦♥str♥t♦ ♥ t ♦trs t♦

tr t♦rs ♣r♦♣♦s ♠t♦s tt rt② ♦♠♣t t ♣r♠tr③t♦♥s ♦♥ ♠♦ ♦tr ♦♥t♦♥s ♦r ♥st♥ t♥r ♥ sr ♣r♦♣♦s t♦♠ t ♦t q♥t t♦ s s♥ t r♣ ♥ ♥srt tr♥st♦♥ t♦rs ♥tts ♦♥t♦♥s rt t♦ t rts ♦t ♦♥ t t r♣ ❬t♥r ♥ sr❪ ♦♥ t ♦♣ ♥♣♥♥t② t s♠ ❬♦♥ t ❪ s♥ t♦r♠s♠ ♦ ①tr♦r s ♠♥t♦♥ ♦ ♥ ♥ s♥rts ♦ rt♦♥♥①

♦ s♠♠r③ ts ♠t♦s ❬ ♥ ❨ r② t ♦♥ t ❪ ♠♣t② rt ♣r♠tr③t♦♥ ② s♦♥ ♦r r♥t ♦♥♦r♠s ♥st♦ ss♦t♥ (u, v) ♦♦r♥ts t rts s ♥ t②♣ ♣♥r ♣r♠tr③t♦♥ t②ss♦t ♣♥r t♦rs (du, dv) t t s ♦ t ♠s ♦♠♣t♥ r♥t ♦r ♦♥♦r♠ s ♣r♠tr③t♦♥s ♥ ♦♥rt t♦ s♣ ♦r♠ ♦ ♣♥r♣r♠tr③t♦♥ ② ①♥ rt① t♥ ♥ r♦♥ t♦ ♦tr rts ♥ ♥

r r♦♠ tr♥t ♠s t t Pr♦ ♦ Pr♠tr③t♦♥♠t♦ strts ② s♠♦♦t♥ t♦r ♥tr ♥ t♥ ♦♠♣ts ♣r♠tr③t♦♥ s tt t r♥t t♦rs r ♥ t t t♦r rt

t♦rs ♠t♦ r♥ts tt ♥s t♦♣♦♦ s r♦♥ t s♠♦♦r♥ts r ♦t♥ ♦r rt① rrss ♦ t ♣t s t♦ t♦ t r♦♠t s♦r rt① ♥s ♥ ♠♣♣ t♦ ♥ ♥♥t ♣♥ s♥ ♠t♣r♦♥t♦♥ ② t♥ t ♣♥ s♥ ♠♦r ♣r♦r♠ ♥♠♥t ♦♠♥♦♥ ② rs ♣r t♦ ② t♦ ♦r r ♥s ♣♣②♥ ts ♠t♦rt② ♦r t ♣♥ ♠t♣ t♠s ♣r♠tr③t♦♥s r r♥t t♦ ♦♥t♥♦s r②r ①♣t t s♠ ♥♠r ♦ s♥r ♣♦♥ts ♥ r tr♦rs♦♠t♠s rrr t♦ s ♦② ♦♥t♥♦s ♣r♠tr③t♦♥s

Pr♦ ♦ Pr♠tr③t♦♥

Pr♦ ♦ Pr♠tr③t♦♥ ♠t♦ ❬② t ❪ s♦♥ ♥ r ♠s t tt♥ t s♥rts ♥tr② ♠r r♦♠ t ♦♣t♠③t♦♥ ♦ t ♣r♠tr③t♦♥ s ♦♥sq♥ t s ♥♦t ♣♦ss t♦ tr♠♥ t ♦♠♦♦② ss ♥♥ s ♥ ③ t s ♥s♦tr♦♣ r♠s♥ ♠t♦ ❬③ t ❪ t♠t♦ rst ♦♠♣ts ♥ t♦r ② s♠♦♦t♥ t ♣r♥♣ rt♦♥s ♦rtr ♥ t t② s t♦ ♦ t ♦♦r♥ts t♦ ♥ r♦♥ t trs♦ t ♦t ♦ ♦ s♦ t ♠t♦ ss t ♥tr ♣r♦t② ♦ t s♥ ♥ ♦s♥ ♥t♦♥ ♦♠♣t ♦♣t♠③t♦♥ ♣r♦♠ s rstt ♥ tr♠s ♦ ♥ rstt ♦rrs♣♦♥ t♦ t s♥ ♥ ♦s♥ ♦ t t ♦♦r♥ts ♠♥ t②s tt t ♠t♦ ♥rts ♥ rts s ♥ tr♥s r♦♥ t s♥rts r♦r t rqrs ♣♦st♣r♦ss♥ st♣ t ts ♣♦♥t ♥ tr s♠t♦s s ♦♥ ♦♦♠♦♦② t t② rqr ♠♥ ♥tr♥t♦♥ ♦r PP t trqrs ♥ ♥♥t ♣♦st♣r♦ss♥ t♦ ① t s♥rts ♦r ♥♦tr t♦r②♦ ♠t♦s s ♦♥ ♥t♦rs ♦♠♣tt♦♥s s♠ ♣r♦♠s♥ rsr ♥ t♦♥ ♦t t♦♠t ♥ s♠♣ ♠t♦s

♣tr ♠t♦s

♣tr ♠t♦s st② t ♥♥t♦♥s ♦ ♦♣rt♦rs ♦r ♥t♦rs ♦ ♠trs♥ t srt stt♥ r rs ♦♥ ts t♦♣ r ❬♦ts♠♥ ② ❪ ♥ s♦ s♦♠ rr♥s ♦♥ t ♦♦♥ ♣ tt♣

♦rr♣t♦♥s❲ ♦s ♦♥ t ♣ ♦♣rt♦r tt ♣②s ♥♠♥t r♦ ♥ ♦♥♦r♠ ♠s

♣r♠tr③t♦♥ s s♦♥ ♥ t♦♥ ♥♥t♦♥s f ♦ t ♣ ♦♣rt♦r ∆ r rtr③ ②

∆f = λf

♦r ♦rt♥ ♦♥ t ♥♥t♦♥s ♠♦r t ♦t t ♣♥ ♥ ts ♥r③t♦♥s ♣♥ ♣②s ♥♠♥t r♦ ♥ ♣②ss ♥♠t♠ts ♥ R

n t s ♥ s t r♥ ♦ t r♥t

∆ = r = ∇.∇ =∑

i

∂2

∂x2i

♥tt② t ♣♥ ♥r③s t s♦♥ ♦rr rt t♦ r ♠♥s♦♥s♥ s rtrst ♦ t rrrt② ♦ ♥t♦♥ s ∆f(P ) ♠srs t r♥t♥ f(P ) ♥ ts r ♥ s♠ ♥♦r♦♦ ♦ P ♥r③♥ t ♣♥ t♦ r srs rqr ♦♠♣① t♦♥s tt ♥ rt② s♠♣ ② ♠t♠t t♦♦ ①tr♦r s st♦♥ ♦r Ptr r♦rsr♣ ♦rs ♦r ♠♦r ts ♦t ①tr♦r s ♥r③s t r♥t ② d ♥ r♥ ② δ = ∗d∗ r ∗ ♥♦ts t ♦ str r t♥♣♥♥t② ♦ ♥② ♦♦r♥t r♠ ❯s♥ t ♥t♦♥ ♦ t ♣♥ ♥ ♥r③ t♦ ♥t♦♥s ♥ ♦r ♠♥♦ S t ♠tr g ♥ s t♥ t ♣tr♠ ♦♣rt♦r

∆ = r = δd =∑

i

1√

|g|∂

∂xi

√

|g| ∂

∂xi

r |g| ♥♦ts t tr♠♥♥t ♦ g t♦♥ tr♠√

|g| ♥ ♥tr♣rt s

♦ s t♦r s♥ t ♦ r ♠♥t dA ♦♥ S s ♥ ② dA =√

|g|dx1∧dx2 ♥♥t♦♥s ♥ ♥s ♦ t ♣♥ ♦♥ ♠♥♦ sr S r

t ♣rs (Hk, λk) tt sts② −∆Hk = λkHk − s♥ s r rqr ♦r

t ♥s t♦ ♣♦st ♥ ♦s r t ♥♥t♦♥s ♦ t ♣♦♣rt♦r ♥ t ♥t♦♥ ss s♥s ♥ ♦s♥s ♦ ♦rr ♥②ss ♥ sqrt② ♦rrs♣♦♥ t♦ t ♥t♦♥ ss ♦ t srt ♦s♥ r♥s♦r♠ s♦r ♥st♥ ② t P ♠ ♦r♠t ♥② t ♥♥t♦♥s ♦ t ♣tr♠ ♦♣rt♦r ♦♥ s♣r ♥ t ♣r r♠♦♥s ss r s♦s♦ t② ♦♦ ♦r ♠♦r ♥r ♦t ♥ t ♥r③s s♣r r♠♦♥s

s s s② t♦ ② ♥♦t♥ tt ♥ t ♣ ♦♣rt♦r ♦rrs♣♦♥s t♦ t st♥r s♦♥♦rr rt ♥♥t♦♥s r s♠♣② sin ωt rs♣ cos ss♦t t t ♥s −ω2

r ♦♠ ♦ t ♥♥t♦♥s ♦ t ♣ ♦♣rt♦rs

t♦ rtrr② ♠♥♦s ts ♥t♦♥ ss s ♥tr② t ♥♦ r♠♦♥sss ❬❱t ♥ ② ❪ s tr♣♦rt s tr ♦r♠ ♥t♦♥ s♥t ♥t ♠♥t ♦r♠s♠ ♥ ①♣♥s ♦ t♦ ♥t② ♦♠♣t t ♥ tsrt stt♥ ♦♥ ♥ ♣♣r♦①♠t t ♥♥t♦♥s ② ♦♠♣t♥ t ♥t♦rs♦ srt ♣♥ ♥② r♥t ♣♣t♦♥s ♥ s ts ♥♥t♦♥s s♦♠♦ t♠ r♣ t srt ♣♥ t ♦♠♥t♦r ♣♥ tt ♦♥② tst ♠s ♦♥♥t♦♥s ♥t♦ ♦♥t

• ♠tr① r♦rr♥ ❬r ❪ ♥ ♠s r♦rr♥ ❬s♥r ♥ ♥str♦♠❪

• ♠s ♦♠♣rss♦♥ ❬r♥ ♥ ♦ts♠♥ ❪

• s♣ sst♦♥ ❬tr t ❪

• r♣ ♠♥ ❬♦r♥ ❪

r♣ ♠♥ ♣r♦♠ ♦ t♦ ♥② r r♣ ② ♣♥ ts rts ♦♥ t sr♥ s♦s str♦♥ ♦♥♥t♦♥s t t ♣r♠tr③t♦♥ ♣r♦♠ ♦ttt tts ♣♣r r s rt t♦r♠ s ♣r♦ s ♥tt ♦ t♦ r r♣ ❬tt ❪ s ♥r ♣r♦♠ s s♦ ♥♦♥ ② t t♦♠t r♥♥rsr ♦♠♠♥t② s ♥♦ r♥♥ ♣r♦♠ s♦ ♠♥s♦♥ rt♦♥ s♦r ♥st♥ rt♥ s ♣ tt♣s♠s⑦♠♥♦ ♥♦tr ♦rs ♥ r♣ ♠ ♥ R

n ♦♥ trs t♦ st♠t ts ♥tr♥s ♠♥s♦♥s ts r sr ♦♠ ♦r ♥ ♦t ♦ ♥ r♠♥s♦♥ ♥ ♦ts ♠t♦s P ❬♥♥♠ t ❪ ♦♠♣ts t ♦s st♥s t♥ ♣r ♦ rt① ♥ t r♣ ♥ t♥ ss ♠t♠♥s♦♥s♥ ❬❨♦♥ ❪ t♦ ♦♠♣t ♥ ♠♥ tt st ♣♣r♦①♠ts ts st♥st♠♥s♦♥ s♥ s♠♣② ♠♥♠③s ♥ ♦t ♥t♦♥ tt ♠srs t t♦♥ t♥ t ♦s st♥s ♥ t ♥t s♣ ♥ t ♥ st♥s ♥t ♠♥ s♣ ♦r ♦s st♥ t♦♥ ② ♦♠♣t♥ t ♥t♦rs ♦ t ♠tr① D = (di,j) r di,j ♥♦ts t ♦s st♥ t♥ rt①i ♥ rt① j

s♦♠♣s ♥ t♠♥s♦♥ s♥ r s t♦ ♥ ♣r♠tr③t♦♥ ♦rt♠s ♥ ❬❩♠♥ t ❪ ♥ ♠♦r r♥t② ♥ t rts ♠t♦ ❬❩♦

r ♣tr r r♥t♦♥ rst ♦♠♣ts ♣ ♥♥t♦♥ t♥ ①trts ts ♦rs ♦♠♣① s♠♦♦ts t ♥ ss t t♦ ♣rtt♦♥ t♠s ♥t♦ qs tt ♥ ♣r♠tr③

t ❪ s ♥ r♦s♦ts rt❳ ♦♠♥ t t ♣♥ ♦rt♠ ♣rs♥t ♥ ❬é② t ❪ ♥trst♥② t rts ♠t♦ ss ♦r ♦ts♠♥t♥ t ♠♦ ♥ ♣r♠tr③♥ t rts s ♣r♦s ♥ ♥ ♦r♥tt♦rt r♠♦r tt ♥ rt② s② tr♥st ♥t♦ ♥t ♠♣♠♥tt♦♥s

♦r t s♣tr ♣r♠tr③t♦♥ ♠t♦s st ♦ st ♥ t♦ ♣rtt♦♥t ♠s ♥t♦ rts ♦r r♥t② ♦♥ t s t ♣♥ t♦ ♦♠♣♦s ♠s ♥t♦ qrtrs ❬♦♥ t ❪ ♥ ② tt tts ♦♥strt♥ ♦② s♠♦♦t ♣r♠tr③t♦♥ s s♦♥ ♥ r tr ♠t♦ rst ♦♠♣ts♦♥ ♥♥t♦♥ ♦ t ♣♥ t 38th ♥ ts ①♠♣ t♥ ①trt t ♦rs♦♠♣① ♦ ts ♥t♦♥ trs ♥ s♠♦♦ts t ♦rs ♦♠♣① ♥ ss t t♦ ♣rtt♦♥t ♠s ♥t♦ qs s qs r ♣r♠tr③ ♥ ♥trrt s♠♦♦t♥ss ♥ rtr ♦♣t♠③ s♥ ♦ r①t♦♥ ❬♦♦s② t r♥ t ❪

♣tr

r♦ssPr♠tr③t♦♥ ♥trsr ♣♣♥

♥ r♦ss♣r♠tr③t♦♥ ♦r ♥trsr♠♣♣♥ st♣ t ♣r♠tr ♦♠♥ ♦r♦♥ ♠s s ♥♦tr sr ♠s r♦ss♣r♠tr③t♦♥ s s t♦ ♠♦r♣ ♦r ♥t♥ ♠ss ♥ t♦ tr♥sr ♣r♦♣rts t♥ t♠ ♦r ♠♦r♣♥ ♥ t♦♥ t♦♦t♥♥ ♠♣♣♥ t s ♥ssr② t♦ ♦t♥ ♦♠♠♦♥ ♦♠♣t ♦♥♥tt② ♦r tt♦ ♠ss ♠♦st ♦♠♠♦♥ ♣♣r♦ ♦r ♣rs ♠♣♣♥ s t♦ ♣r♠tr③ ♦t♠♦s ♦♥ ♦♠♠♦♥ s ♦♠♥ r♦♥r② ♣♥r ♣r♠tr③t♦♥ s r②♥st ♦r ts ♣r♣♦s ♥st tr♥t ♦♠♥s s s s♠♣ ♦♠♣① ♦r s♣r r ♦♠♠♦♥② s

s ♦♠♣① t♦s

♥② ♠t♦s s ♠♣♣♥ t♦ ♦♠♠♦♥ s♦♠♣① t♦ ♦t♥ r♦ss♣r♠tr③t♦♥ ♥ t s ♠st sr t ♦♥strt♦♥ s s♥♥t② ♠♦r ♥♥t♥ ♥ ♣r♠tr③♥ s♥ ♠s Pr♥ t ❬❪ ♣rtt♦♥ ♠s ♥t♦tr♥r ♣ts ♦rrs♣♦♥ t♦ t s ♦ sr ♥ s♠♣ ♦♠♣① ②r♥ ♥t♦r ♦ ♣ts t♥ srs♣♣ tr rts tt ♦rrs♣♦♥ t♦t rts ♦ t s ♠s

r♥r t ❬❪ ♥ r♦② ♥ r ❬❪ ①t♥ t ♠t♦s ♦ Pr♥t ❬❪ ♥ r♦② t ❬❪ t♦ ♦♥strt t s♠♣ ♦♠♣① t♦♠t②♥ ♣r t♦ t ♣t ♦r♠t♦♥ ♥♣t t♦ ♦t ♠t♦s ♥s st ♦ ♦rrs♣♦♥♥s t♥ tr rts ♦♥ t t♦ ♥♣t ♠♦s ♠t♦s s t♦s st rts ♦ t s ♦♠♣① ② s♠t♥♦s② tr ♣ts ♦♥ t ♥♣t ♠sst♥ ♦rrs♣♦♥♥ ♣rs ♦ rts s♣tt♥ ①st♥ ♠s s ♥ssr②

♥ t s s rt t ♠ss ♥ ♠♣♣ t♦ t s s♥ t t♥qsr ♥ t ♣tr ♦♥ tr♥t ♦♠♥s r♥r t ❬❪ s ♥ tr♥t♣♣r♦ ② ♥r ♦♠♣t ♥ ①♣t ♠♣ t♥ t rs♦t♦♥ ♦ts♥ t s ♦♠♥ ♥st t② tr♥t t r♦ ♦ s ♦♠♥ t♥ tt♦ ♠ss t r♦s ♦♠♣①t② s ♥ ♠trs♦t♦♥ r♣rs♥tt♦♥ ②♣r♦rss② r♥ ♠s ② ♥ ♥ rts ♥ r①♥ tr ♦t♦♥ s♥ strts ♠tr ♠sr ♦♥ t♠♣♦rr② ♣♥r ♥♦♥ ♦ tr ♥♦r♦♦s

r s ♦♠♣① ♦♥strt♦♥

♥r② r♥ ♠t♦s

r ♠t♦s s ♥ ♥r②r♥ ♣♣r♦ ♦r r♦ss♣r♠tr③t♦♥ r ♦♥♠s s rt② ttrt t♦rs t ♦tr ❬♥ t ♠♥r ♥ P♦♣♦❪ ttrt♦♥ ♥r② ♥t♦♥ ♦♥ssts ♦ ♦♠♣♦♥♥ts tt ♣ t rts ♦♦♥ ♠s t♦rs ♥rst ♦t♦♥s ♦♥ t ♦tr tr②♥ t♦ ♣rsr t s♣ ♦ t♠s s ♠ s ♣♦ss ♦ tt ♦rrs♣♦♥♥ t② rqr t sr t♦ s♣②sr ♦③♥ ♣♦♥tt♦♣♦♥t ♦rrs♣♦♥♥s t♥ t ♥♣t ♠♦s ♠t♦s♦r ♥ t ♠ss r r② s♠r ♠♥s ♥ t s♠ ♣♦s ❬♥ t ❪ t t♥ t♦ ♦♥r t♦ ♣♦♦r ♦ ♠♥♠♠ t ♥rs ♥ s♣ r♥s ♠t♦s r qt s♥st t♦ t ts s ♥s t ♥r② ♥t♦♥ t♦♦♥t ♦r t r♥t ♦♠♣♦♥♥ts ♥ ♥t ♦ ts ♣♣r♦s s tt t②♥ ♥ ♠♣♣♥s t♥ ♠♦s ♦ r♥t t♦♣♦♦② ♥s t t♦ ts ♠♣sr ♥♦ ♦♥r t

♦♠♣t ♠s♥

♦r ♣♣t♦♥s s s ♠♦r♣♥ t s ♥♦t ♥♦ t♦ ♦t♥ r♦ss♣r♠tr③t♦♥t♥ t t♦ ♠♦s ♦r ts ♣♣t♦♥s t t ♥ ♦ t ♣r♦ss t t♦ ♠♦sr t②♣② rqr t♦ t s♠ ♦♥♥tt② r r tr ♠♥ ♣♣r♦s♦r ♥rt♥ s ♦♠♠♦♥ ♦♥♥tt②

• s ♠s ss♦♥ r ♠t♦s ♥♥ ❬Pr♥ t ❪ s ts ♠s ♦♥♥tt② ♥ r♥ t s♥ t ♦♥t♦♦r ss♦♥ ♣ttr♥♥tr♦♥ s ♠♥② s ♦ ss♦♥ s ♥ssr② t♦ ♣tr t ♦♠tr②♦ ♦t ♠♦s ♥t ♦ t ♠t♦ s s♠♣t② ts r s t♣♥♥ ♦♥ t s♣ ♦ s ♠s tr♥s ♠t♦ s♦ t♥s t♦ rqrr tr♥ ♦♥t t♦ ♣t r② r♦② t♦r ♦♠♣rt♦ ♥♣t ♠s s③s

• r② ♥♦tr ♣♣r♦ ♦r ♥rt♥ ♦♠♠♦♥ ♦♥♥tt② ❬① r♥r t ❪ s t♦ ♥trst t t♦ ♥♣t ♠ss ♥ t ♣r♠tr ♦♠♥♦♠♥♥ tr rts ♥ ♥rt♥ ♥ rts t ♥trst♦♥s ♠t♦ ♣rsrs ①t② t ♥♣t ♦♠trs t s ♥♦ttr t♦ ♠♣♠♥t♥ ss♦♥ ♥rss t tr♥ ♦♥t ② r♦② t♦r ♦

• ♠s♥ ♥ ❬r♦② ♥ r ❪ t t♦rs ♣r♦♣♦s ♥ tr♥tr t② s t ♦♥♥tt② ♦ ♦♥ ♦ t ♥♣t ♠ss s ss ♥ t♥r♥ t s ♥ssr② s ♦♥ ♥ ♣♣r♦①♠t♦♥ rr♦r t rs♣t t♦ t s♦♥♠s rst♥ ♠ss s♥♥t② ♦r tr♥ ♦♥t t♥ s♥ t♦tr t♦ ♣♣r♦s rst ② ♣♥s ♦♥ ♦ t ♥♣ts s sts t s♦r ♦r ♦♠♠♦♥ ♦♥♥tt② ❯♥ ♥ t ♦r② ♣♣r♦ s♦♠ ♥trs ♦ t s♦♥ ♠s r ♦♥② ♣♣r♦①♠t ♥ t s t t♦ t②r♣r♦ sr♣ trs

♣tr

♠r ♣t♠③t♦♥

♥ t ♣r♠tr③t♦♥ ♠t♦s st ♥ t st♦♥ ♦ ♦trs ♣r♠tr③t♦♥ ♠t♦ r♠♦♥ ♣r♠tr③t♦♥ ♦♥♦r♠ ♣r♠tr③t♦♥ ♥ trt♦ s♦ ♥r s②st♠ ♦r t♦ ♠♥♠③ qrt ♦t ♥t♦♥ s♣t② ♦t ♥♠r ♣r♦♠s ② ♥ ♠s ♣r♠tr③t♦♥ s tt t ♥♦ ♠trsr s② s♣rs s ♦♥sq♥ rst ①♣♥ ♦ t♦ ♥t② ♠♣♠♥t t strtr ♦r s♣rs ♠trs ♦ ♠ srs sr ♣r♦♣♦s ♥ ♠♣♠♥tt♦♥ tt ♥ ②♥♠② r♦ ♥ ♦♥ts r t♦ t ♠tr① rs♦♥ ♦ ts ♠♣♠♥tt♦♥ s ♥ ♥ t ♦♠♣♥♦♥ ♠tr s ♦♥ tstt♦r t strtr s t strtr s ♥ ♦r r♣t s♦tr❲ s♦ ♣r♦ rs♦♥ ♥ ♣♥ tt♣♦rrs♦tr ♦t ttts t strtr ♥ s② ①t♥ r ♥ ♠♣♠♥tt♦♥s s♦ ♣r♦ t ♦♦♥ trs ♥♦t t ♥ t ♠♣♠♥tt♦♥ ♥ ♥ t ♣♣♥① t♦♣ ts ♥t rs♦♥

st♦r ♦ t ♦♥ tr♠

st♦r ♦ ♦t s♣rs r♦s ♥ ♦♠♥s

s②♠♠tr st♦r ♦ ♥♦t st♦r t ♣♣r tr♥ ♦r s②♠♠tr ♠trs

♥♦♥sqr ♠trs

♠tr① × ♠tr① ♠t♣②

trs ♥ r ♥trst♥ ♦r ♠♣♠♥t♥ ♦ ♣r♦♥t♦♥r rqrs ♦r ♣r♦♥t♦♥r rqrs tr s♣s ♣ t ♦♥tr♥t ♦rt♠ t rqrs s♦♠ ♠♦t♦♥ ♥ t ♠tr① × t♦r r♦t♥trs ♥ ♥ s ② ♠♦r s♦♣stt ♥♦♥♥r s♦rs s s P♥

♠r ♠t♦s r ♣rtr② ♥t ♦r♠s♠ ♦r ♦♠tr ♠♦♥♣r♦♠s tt ♥♦ r ♠ss ♣♣r♦ ♦♥ssts ♥ ♦r♠③♥ t ♣r♦♠s ♥t♦♥ ♦ ♠♥② rs ♥ ♦♣t♠③♥ t rs ♦rrs♣♦♥ t♦ s ♦♦r♥ts ♦♦rs t①tr ♦♦r♥ts t tt t♦ t rts ♦ t ♠s

1

2 3 4

5 6

7 8 9

0

1

2

3

0 1 2 3j

i 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8

0 1 1 1 2 2 3 3 3

a

i

9=NNZ

1 0 2 3 1 3 1 2 3j

r ①♠♣ ♦ s♣rs ♠tr① t ♦r♠t

s ♥♠r ♦♣t♠③t♦♥ ♣r♦♠s ♥♦ r ♠trs ♥ r♦♠ t ♠s♥ t s tt t♦ t rts ♥♦r t s ♥♦r t ts ♦ t ♠s ♥♠♦st ss ts ♠trs r s♣rs ♥ strtr ♥t ♦r str♦♥② rtt t ♥② ♠tr① ♦ t r♣ ♦r♠ ② t ♠s ❲ strt ts st♦♥ ②r♥ s♦♠ ♥t t strtrs t♦ r♣rs♥t s♣rs ♠trs

t strtrs ♦r s♣rs ♠trs

♦rt♠ t strtr ♦r s♣rs ♠trsstrt tr① ④

♥t ♠♥s♦♥ ♦ t ♠tr①

♥t ❩ ♥♠r ♦ ♥♦♥③r♦ ♦♥ts

♦ ♥♦♥③r♦ ♦♥ts rr② ♦ s③ ❩

♥t r♦ ♥s rr② ♦ s③ ❩

♥t ♦♠♥ ♥s rr② ♦ s③ ❩

⑥

♦rt♠ tr① × t♦r ♣r♦t t t t strtr♦ ♠t♦ ② tr① ♦ ① ④

♦r♥t ④

②❬❪

⑥

♦r♥t ❩ ④

②❬❬❪❪ ❬❪ ①❬❬❪❪

⑥

⑥

1

2 3 4

5 6

7 8 9

0

1

2

3

0 1 2 3j

i 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8

1 0 2 3 1 3 1 2 3

0 1 4 6 9

a

colind

rowptr

9=NNZ

r ①♠♣ ♦ s♣rs ♠tr① r♣rs♥t ♥ t ♦r♠t

t strtr

♦ ♦♣t♠③ t st♦r ♦ s♣rs ♠trs t rst tt ♦♠s ♥tr② t♦ ♠♥♦♥ssts ♥ st♦r♥ ♦♥② t ♥♦♥③r♦ ♦♥ts t t ss♦t (i, j) ♥s r♦r ♥st ♦ st♦r♥ n2 ♦♥ts ♦♥② ♥ t♦ st♦r ❩ ♦♥ts ❩♠r ♦ ♦♥❩r♦ ♦♥ts ♥ t s♠ ♥♠r ♦ (i, j) ♥s s tstrtr s rrr t♦ s t ♦r♠t ♣t ♥ r ❲ s♦ s♦ ♥♠♣♠♥tt♦♥ ♥ ♦rt♠ ♥t♦♥ tt ♦♠♣ts t ♣r♦t t♥ s♣rs ♠tr① ♥ ♦r♠t ♥ t♦r s ♥ ② ♦rt♠ ♦r ♥st♥ ts♥t♦♥ s ♣rtr② s t♦ ♠♣♠♥t s♦r s ♦♥ t ♦♥t r♥t♠t♦ s ♦

s♣t t ♦♦s ♥ r③ ② t ♦r♠t s ♦♠♣r t♦ rr rr② ts r♣rs♥tt♦♥ st s♦s r♥♥② ♦ t ♦♥ts i ttst♦r t r♦ ♥① ss♦t t♦ ♥tr② ♦r ts rs♦♥ ts ♦r♠t s s♦♠ s♥ ♥♠r rrs t s ♠t t♦ ♥♣t ♦r♠ts s♥ ts s♠♣t② ♦rs ts♥rst♥♥ ♥ s ♥ ♣rt t ♦♠♣rss ♦ t♦r t strtrs ♣rrr s♥ t s ♠♦r ♦♠♣t ♥ ♠♠♦r② ❲ t ts ♦r♠t ♥ t ♥①tsst♦♥

♦rt♠ ♦♠♣rss ♦ t♦r t strtr ♦r s♣rs ♠trsstrt tr① ④

♥t ♠♥s♦♥ ♦ t ♠tr①

♥t ❩ ♥♠r ♦ ♥♦♥③r♦ ♦♥ts

♦ ♥♦♥③r♦ ♦♥ts rr② ♦ s③ ❩

♥t ♦♥ ♦♠♥ ♥s rr② ♦ s③

♥t r♦♣tr r♦ ♣♦♥trs rr② ♦ s③

♦ ♦♥ ♠♥ts rr② ♦ s③

⑥

♦♠♣rss ♦ t♦r

♦r♠t ♦♠♣rss ♦ t♦r s ♦♥ ♦ t ♠♦st ♦♠♠♦♥② s tstrtr ♦r s♣rs ♠trs s ♥ r ♥♠r rrs tr♥s♣♦s r♥t ♦♠♣rss ♦♠♥ t♦r s s♦ s ♣♥♥ ♦♥ t ♦rt♠s ♥♠♣♠♥tt♦♥ ♦s ❲ ♣rs♥t r t ♦r♠t ♦rt♠ ♣rs♥ts t♠♣♠♥tt♦♥ ♥ ♥ r s♦s ♥ ①♠♣ t strtr sstr ♥s t♦ r♣rs♥t ♥♦♥③r♦ ♦♥ts ♥ tr ♥s s ♥ t ♦r♠t t rr② st♦rs t ♥♦♥③r♦ ♥trs rr② ♦♥ s ♦r ♥tr②t ♦rrs♣♦♥♥ ♦♠♥ ♥① r♦s r ♥♦ ♥ r♥t ♠♦r ♦♠♣t♠♥♥r ② t r♦♣tr rr② s rr② s ♦r r♦ ts strt ♥ ♥ ♥ trr②s ♥ ♦♥ ♦ tt t ♦♥ ♦ ♦rt♠ ♦♠♠♦♥ ♣rt ♦♥ssts♥ ♦♠♣t♥ t rr② ♦♥ ② ♥ t♦♥ ♥tr② tt ♣♦♥ts ♦♥ ♥tr② ♣st tst ♥tr② ♦ t ♠tr① s t♦♥ ♥tr② ♦ ♦♥ s s♥tr② ♥ ♦s tst ♥ t ♠tr① × t♦r ♣r♦t s s♦♥ ♦

♦rt♠ tr① × t♦r ♣r♦t t t ♦r♠t♦ ♠t♦ ② tr① ♦ ① ④

♦r♥t ④

②❬❪

♦r♥t r♦♣tr❬❪ r♦♣tr❬❪ ④

②❬❪ ❬❪ ①❬♦♥❬❪❪

⑥

⑥

⑥

♦rt♠ s♦s ♦ t♦ ♠♣♠♥t t ♣r♦t t♥ s♣rs ♠tr① st♦r♥ t ♦r♠t ♥ t♦r ① s♥tr② r♦♣tr❬❪ ❩ ♦s t♦ ♣rtr s ♦r t st r♦ ♦ t ♠tr①

r♥t rs♦♥s ♦ t t strtr ①st s t ♠tr① s s②♠♠trt s ♣♦ss t♦ s s♦♠ ♠♠♦r② ② ♦♥② st♦r♥ t ♦r tr♥r ♣rt ♦ t ♠tr①tr r♥ts st♦r t ♦♥ ♦♥ts ♥ s♣rt rr② t♦ tt t ♠♣♠♥tt♦♥ ♦ t ♦ ♣r♦♥t♦♥r s ♦ tr r♥ts st♦r t ♦♠♥s♥st ♦ t r♦s ♦♠♣rss ♦♠♥ t♦r ♥② t ♦♦♠♣rss ♦ t♦r ♣rtt♦♥s t ♠tr① ♥t♦ ① s③ ♦s ♥ st♦rs t♠♥ t ♦r♠t s ♦t ♦♣t♠③s ♠♠♦r② sss ♥ ♦rs s♥ ①t♥♥♥strt♦♥ sts ♦♥ ♥t Pr♦ss♦r

t strtr ♥ ts r♥ts r ♦t ♦♠♣t ♥ ♥t ♦r♥ ♦r ♣rtr ♦♥t①t ♦ ♥♠r ♣r♦♠s rt t ♦♠tr ♦♣t♠③t♦♥ ♦ ♠ss t t strtr s s♦♠t t♦♦ r s ①♣♥ ♦ ♥ ♦♠tr ♦♣t♠③t♦♥ ♣r♦♠ t s ♥tr t♦ ss♠ t ♠tr① ② trrs♥ tr♣ tt ♦rrs♣♦♥s t♦ t ♠s r♥ t trrs ♦♥ts r ♠t♥ t ♠tr① ♥rt ② ♠♥tr② ♥♦r♦♦s st♥s ♥ ♥t ♠♥ts♣r♥ ♥ t t strtr rqrs t♦ ♥ t ♥♠r ♦ ♥♦♥③r♦

1

2 3 4

5 6

7 8 9

0

1

2

3

0 1 2 3j

i

(a ,j)i,j

(1,1)

(2,0) (3,2) (4,3)

(5,1) (6,3)

(7,1) (8,2) (9,3)

row[0]

row[1]

row[2]

row[3]

r ②♥♠ s♣rs ♠tr① t strtr

♦♥ts ♥ ♥ ♦♥strt♥ ♠tr① rqrs t♦ trrss ♦ t r♣ rst trrs tr♠♥s t ♥♦♥③r♦ ♣ttr♥ t st ♦ (i, j) ♥① ♦♣ss tt ai,j s ♥♦♥③r♦ ♥ t t strtr s ♦t ♥ ♠♠♦r② ♥② s♦♥ trrs ♦♠♣ts t ♥♠r s ♦ t ai,j ♥trs ♥ ♣rt ts ②♦ ♣r♦♥ s ♥♦♠♦rt t♦ ♠♣♠♥t ♦r ts rs♦♥ ♣r♦♣♦s ②♥♠♠tr① t strtr ♦r t ♥♠r ♦ ♥♦♥③r♦ ♦♥ts ♥ r② ♠♣♠♥tt♦♥ s ♥ ♦r rr② ♣♥ ♥ ♥ t r♣t ♣t♦r♠ tt♣♦rrs♦tr

②♥♠ ♠trs

r ②♥♠ s♣rs ♠tr① s t r♦♥ t ss stt♦r ♦ t st♥r rr② ♦r t♥r ♠♣t rr② s ♦♥t♥r ss s♦ t ♥trst♥♣t② ♦ r♦♥ ♥ ♠♠♦r② ♠♥♠③♥ t ♥♠r ♦ rqr ♦♣s s♦r ②♥♠ s♣rs ♠tr① t strtr ♣t ♥ r ♥ t ♥♦rt♠ ♦♥ssts ♦ ♥ rr② r♦ r♦ s ♠♣♠♥t ② stt♦r ♦♥① ♣rs ♥ ♣r s r♣rs♥t ② ♥ ♥st♥ ♦ t ♦ strtr ♦ t strtr s♦s ♠♠♦r② st♦r rqr♠♥t ♦♠♣r t♦ t ♦r♠t ♦r♥ ttr ①t② s ①t② s ①♣♦s t♦ t sr ② t♥t♦♥ tt s t t♦ t ♦♥t ai,j

♠tr①t♦r ♣r♦t r♠♥s s♠♣ t♦ ♠♣♠♥t t ts strtr s♦r ♦ s ♥ ♥ ♦rt♠ ♦r t s ♠♣♦rt♥t t♦ ♥♦t tt ts ttr①t② s ♦t♥ t t ①♣♥s ♦ ♦r ♣r♦r♠♥s ♦ t ♠tr① × t♦r♣r♦t ♣♣r♦①♠t② ② t♦r ♦ ♥ ♦r ①♣r♠♥tt♦♥ s ♥ ①♣♥② t ♦ss ♦ t ♦t② tt rsts ♥ ss ♥t s s ♦♠♣r t♦t r♣rs♥tt♦♥ ♦r ts rs♦♥ ♥ t s r r ♥♠r ♦ ♠tr①

s s ♦♥ ② ♦♥ t s③ ♦ t ♠♠♦r② s♣ ♦t t♦ t ♦♥t♥r t♠ ♦♣②♦♣rt♦♥ s ♥ssr② s ♠s t ♥♠r ♦ ♦♣s rs s ♥t♦♥ ♦ t ♦ ♦ t ♥s③

♦rt♠ ②♥♠ s♣rs ♠tr① t strtrss ♣rstr① ④♣

strt ♦ ④♦ ④ ⑥♦♥t ♦ ♥① ④ ⑥♥t ♥① ♦

⑥

ss ♦ ♣ stt♦r♦ ④♣

♦ ♥t ♥① ♦ ④♦r♥t s③ ④

ts❬❪♥① ♥① ④ts❬❪ rtr♥

⑥⑥stt♦r♦♣s❴♦♥①

⑥⑥

♣rstr①♥t ♠ ♠♥s♦♥♠ ④r♦ ♥ ♦❬♠❪

⑥

⑦♣rstr① ④ t❬❪ r♦ ⑥

♦ ♥t ♥t ♦ ④

r♦❬❪ ⑥

♦ r ④

♦r♥t ♠♥s♦♥ ④r♦❬❪r

⑥⑥

♥♠r ♦ ♥♦♥③r♦ ♦♥ts♥t ♥♥③ ♦♥st ④

♥t rst ♦r♥t ♠♥s♦♥ ④

rst r♦❬❪s③ ⑥rtr♥ rst

⑥♥t ♠♥s♦♥ ♦ r♦

⑥

♦rt♠ tr① × t♦r ♣r♦t t t ②♥♠ t strtr ② ①

♦ ♠♦ ② ♦♥st ♣rstr① ♦♥st ♦ ① ④

♦r♥t ♠♥s♦♥ ④

②❬❪

♦♥st ♦ r♦❬❪

♦r♥t s③ ④

②❬❪ ❬❪ ①❬ ❬❪♥① ❪

⑥

⑥

⑥

♦rt♠ ♦♥rt♥ ②♥♠ s♣rs ♠tr① ♥t♦ t ♦r♠t♦ ♦♥rt❴t♦❴

♦♥st ♣rstr①

tr① ❴

♥t rr②❴s ♦r ♦r ♦rtr♥

④

❴ ♠♥s♦♥

❴❩ ♥♥③

❴ ♥ ♦❬❴❩❪

❴♦❴♥ ♥ ♥t❬❴❩❪

❴r♦❴♣tr ♥ ♥t❬❴❪

♥t ♦♥t

♦r♥t ❴ ④

♦♥st ♣rstr①♦ r♦❬❪

❴r♦❴♣tr❬❪ ♦♥t rr②❴s

♦r♥t s③ ④

❴❬♦♥t❪ ❬❪

❴♦❴♥❬♦♥t❪ ❬❪♥① rr②❴s

♦♥t

⑥

⑥

❴♦❴♣tr❬❴❪ ❴❩ rr②❴s

⑥

× t♦r ♣r♦ts ♥ t♦ ♦♠♣t ♥ ♦♥tr♥t s trts♦r sr tr t ♠② ♠♦r ♥t t♦ ♦♥rt t ②♥♠ ♠tr① ♥t♦ t ♦r♠t ♦♥rs♦♥ r♦t♥ ♥ s♦ s t♦ ♥tr t ②♥♠ ♠tr①t strtr t s♣rs rt s♦rs ♣r❯ ❯ ❯P t tt st ♦r♠t s♦r ♦ ♦ t ♦♥rs♦♥ r♦t♥ s ♥ ♥ ♦rt♠ s ♥t♦♥ s ♥♦t t t♦ ♠♣♠♥t ♦♥② t♥ tt ♦♥ ♥s t♦ t r♦ s t t♦♥ ♣r♠tr rr②❴s tt srs s♦♠ ①♣♥t♦♥s ♥ ♥s ♦ ♥♦t s t s♠ ♦♥♥t♦♥ ♦r rr② ♥①♥ rst ♥tr②♦ ♥ rr② s ♥① ② ♥ ♥ ② ♥ ♦r ts rs♦♥ t ♠tr① s ♠♥t t♦ s ② r♦t♥ ts ♣r♠tr ♥s t♦ st t♦

♦ tt s♥ t ♠♥tr② t strtrs ♦r s♣rs ♠trs t ♦rt♠ t♦ ♠♥♣t t♠ ♥ tr ♠♣♠♥tt♦♥ ♥ ♥ s s♦♠♣♦♣r ♦rt♠s t♦ s♦ ♥r s②st♠s

♦♥ ♥r ②st♠s

ss ♦ ♦♠tr ♠♦♥ ♣r♦♠s rqr t♦ s♦ r s♣rs ♥r s②st♠ ♥ qt♦♥ ♦ t ♦r♠ Ax = b t A tt ♥♦ts n × n ♥♦♥s♥r sqr♠tr① b t♦r ♦ ♠♥s♦♥ n ♥ x t ♥♥♦♥ t♦r ♦ ♠♥s♦♥ n ♥ tsst♦♥ sr sr ♠t♦s t♦ s♦ ts ♥r s②st♠s

①t♦♥

r①t♦♥ ♠t♦ s t ♠♦st s♠♣ ♦t r♦♠ t ♦♥♣t ♥ t ♠♣♠♥tt♦♥ ♣♦♥ts ♦ s ♠t♦ s ② s ♥ t s t♦ ♠♣♠♥t ♦♠tr②♣r♦ss♥ ♦rt♠ t s t♥ tr r♣ ② ♠♦r ♥t ♦rt♠s st♦

r①t♦♥ ♠t♦ ♥ s② ♥rst♦♦ ② ♦♥sr♥ t ♣r♦♠ Ax = bs ♥r s②st♠

a1,1x1 +a1,2x2 + . . . +a1,nxn = b1

ai,1x1 +ai,2x2 + . . . +ai,nxn = bi

an,1x1 +an,2x2 + . . . +an,nxn = bn

♠t♦ ♦♥ssts ♥ trrs♥ t qt♦♥s ♦♥ ② ♦♥ ♥ ♦♠♣t ♦r qt♦♥ i t ♦ xi ♦t♥ ② ♣rt♥♥ tt t ♦tr rs xj ♦r j 6= ir ♥♦♥ s s t ♦♦♥ ♣t s♠ ♦r t r xi

xi ←1

ai,i

(

bi −∑

j 6=i

ai,jxj

)

♦♠♣t ♦rt♠ ♦r t r①t♦♥ ♠t♦ t♥ rts

‖Ax− b‖ < ǫ♦r i r♦♠ 1 t♦ n

xi ← 1ai,i

(

bi −∑

j 6=i ai,jxj

)

♥//♦r♥//

t ǫ t ♣rs♦♥ s♣ ② t sr t s s♦ ♣♦ss t♦ s♣② ♠①♠♠♥♠r ♦ trt♦♥s s♦ tt t ♦rt♠ st♦♣s ♥ t ♦s ♥♦t ♦♥r

s ♥ s♥ ts ♦rt♠ ♥♥♦t ♣♣ t♦ ♠trs t ③r♦ s ♦♥ t♦♥ ♦r ♥r② t s ♣♦ss t♦ ♣r♦ s♥t ♦♥t♦♥ ♦r ♦♥r♥ t ♠tr① s ♦♥ ♦♠♥♥t

∀i,∀j 6= i, |ai,i| > |ai,j|t♥ t ♦rt♠ ♦♥rs

t s ♣♦ss t♦ s♣♣ ts ♦rt♠ ② s♥ t t tt t ♣t ♣♣t♦ r xi ♣♦♥ts t♦r t rt rt♦♥ ♥tt② ♦♥ t rtr♥ tt rt♦♥ ② ♠t♣②♥ t s♣♠♥t ② t♦r ω s ② t♦ rtt s♣ ♦ ♦♥r♥ s♦♠♦ ♣t s♠ rts

xprev ← xi

xi ← 1ai,i

(

bi −∑

j 6=i ai,jxj

)

xi ← xprev + ω(xi − xprev)

s ♣t s♠ rst ♦♠♣ts t ♣t ♦ xi s ♦r ♥ t♥ ♠t♣s tss♣♠♥t rt t♦ t ♣r♦s xprev ② t♦r ω t s ♣♦ss t♦ ♣r♦tt t ♦rt♠ ♦♥rs ♥r t s♠ ♦♥t♦♥s s r①t♦♥ ♦♥ ♦♠♥♥t♠tr① ♥ ♦r ω ∈ [1, 2[ s♦♠♦ ♦rt♠ s rrr t♦ s sss♦rr①t♦♥ ② rrt♥ t ♣t s♠ ♥ ♠♦r ♦♠♣t ♦r♠ t ♦♠♣t ♦rt♠ s ♥ ②

‖Ax− b‖ < ǫ♦r i r♦♠ 1 t♦ n

xi ← (1− ω)xi + ωai,i

(

bi −∑

j 6=i ai,jxj

)

♥//♦r♥//

♦♣t♠ ♦ ♦r t ♣r♠tr ω s tr♠♥ ② t ♥s ♦ t ♠tr①A ♥r② ♦♠♣t♥ t♦s ♥s s ♠ ♠♦r ①♣♥s t♥ s♦♥ ts②st♠ ♦r ts rs♦♥ tr t♦rt st② ♦ t ♥♠r ♣r♦♠ ♥ t ♦♣t♠ ω ♦r t s tr♠♥ ♥ ♥ ♠♣r ②

♠♥ ♥t ♦ t ♠t♦ s ts s♠♣t② r♦♠ ♦t t ♦♥♣t♥ ♠♣♠♥tt♦♥ ♣♦♥t ♦ s s♠♣t② ♦r ts s ♥ t ♦♠tr②♣r♦ss♥ ♦♠♠♥t② ❬♥ ❪ ♦r s s tr t ♦♠♠♥t② ♥♦ss ♠♦r ♥t ♠t♦s s s t ♦♥t r♥t ♥ s♣rs rt s♦rs

♦♥t r♥t ♠t♦

♦♥t r♥t ♠t♦ s ♠ ♠♦r ♥t ♥ ♦s ♥♦t rqr ♠ ♦rtst♦ ♠♣♠♥t s ♦rt♠ s s ♦♥ t q♥ t♥ s♦♥ t ♥rs②st♠ Ax = b ♥ ♠♥♠③♥ t qrt ♦r♠ F (x) = 1/2xtAx−btx ♦r ♣rs②t ♦rt♠ ♦♠♣ts ss ♦ t♦rs tt s ♦rt♦♦♥ ♥ t s♣ ♦ t ♠tr① ss ♦ ♦♥t t♦rs ② ♣♣②♥ t r♠♠t ♦rt♦♦♥③t♦♥♦rt♠ ♦rt♥t s♠♣t♦♥s ♥ t ♦♠♣tt♦♥s ♠ t ♣♦ss t♦ ♦t♥ tt♦rs ♦ t ss ♦♥ ② ♦♥ ② ♦♥② ♣♥ ♦♥ t♦r ♥ ♠♠♦r② ♥①t ♦♥ st♥ ♦t♥ s ♥r ♦♠♥t♦♥ ♦ t ♣r♦s ♦♥ ♥ t r♥t ∇F = Ax−bt t rr♥t ♣♦♥t x ♦♠♣t ♦rt♠ rts

♦rt♠ Pr♦♥t♦♥ ♦♥t r♥t

i← 0 r← b−Ax d←M−1rδnew ← rTd δ0 ← δnew i < imax t δnew > ǫ2δ0

q← Ad α← δnew

dT q

x← x + αd r← r− αqs←M−1r δold ← δnewδnew ← rT s β ← δnew

δold

d← r + βd i← i + 1♥

♥ ts ♦rt♠ t ♠tr① M ♣r♦♥t♦♥r ♦s t♦ s♣♣ ts♣ ♦ ♦♥r♥ ♦ t ♦rt♠ ♦r ♥st♥ t ♦ ♣r♦♥t♦♥r s s♠♣②q t♦ t ♦♥ tr♠s ♦ t ♠tr① A ♦r s♦♣stt ♣r♦♥t♦♥rs ①st♦r ♦r ①♣r♠♥ts s♦♥ tt rs♦♥② ♥t rsts r ♦t♥ tt ♦ ♣r♦♥t♦♥r

s ♥ s♥ ♥ ts ♦rt♠ t ♠♦st ♦♠♣t ♦♣rt♦♥ s ♠tr① ×t♦r ♣r♦t t ♦tr ♦♠♣tt♦♥s r s♠♣② ♦t ♣r♦ts t♥ t♦rs♥ ♥r ♦♠♥t♦♥s ♦ t♦rs

♦♥t r♥t ♠t♦ ♥♥♦t ♣♣ t♦ ♥♦♥s②♠♠tr ♠trs ♦r ♥♦♥s②♠♠tr ♠tr① t s ♣♦ss t♦ ♣♣② t ♦♥t r♥t ♠t♦ t♦ t♥♦r♠ qt♦♥ AtAx = Atb rst♥ ♠t♦ s ♦♥t r♥t sqr ♦r s♥ ts sqrs t ♦♥t♦♥ ♥♠r t ♦ss ♦ ♥♠r stt②♠s ts ♠t♦ ♥♦t st ♥ ♥r

♥♦tr ♦♥ssts ♥ r♥ r♦♠ t s②st♠ Ax = b ♥ q♥t s②♠♠trs②st♠ s ♦♦s (

Id AAt 0

)(0x

)

=

(b0

)

t s t♥ ♣♦ss t♦ ♣♣ t ♦♥t r♥t ♠t♦ t♦ ts s②st♠ s ♥st ♦rt♠ ♦♥t r♥t r♥t ♥♠ st③ rts t s♣ ♦ ♦♥r♥

♠ ♦t♦♥♣♥ tt♣♦rrs♦tr

♣r❯ tt♣r♦⑦①♦②♣r❯

❯ tt♣t⑦st♦♦ts

❯P tt♣r♥s②♦♥r❯P

❯P tt♣srsrs♣rs♠♣

♦♠ r② s♣rs rt s♦rs

rr ♥trst ♥ ♠♦r ts ♦t t ♥r②♥ t♦r② ♥ r t①♥t tt♦r ② ❬❪ tt s ♦♥ t

♣rs rt s♦rs

♥♦tr ♠t♦ t♦ s♦ ♥r s②st♠ ♦♥ssts ♥ t♦r③♥ t ♠tr① ♥t♦ ♣r♦t♦ ♠trs tt r s♠♣ t♦ ♥rt ♦r ♥st♥ t ❯ t♦r③t♦♥ ♦♥ssts ♥♥♥ t ♦r tr♥r ♠tr① L ♥ t ♣♣r tr♥r ♠tr① U s tt t♣r♦t LU s q t♦ t ♠tr① A ♦ t s②st♠ t ♠tr① s s②♠♠tr U♦rrs♣♦♥s t♦ t tr♥s♣♦s ♦ L ♥ ts t♦r③t♦♥ s ♦t♥ t s t♥ r②s② t♦ s♦ ♥r s②st♠s tt ♥♦ A = LU s ♦♦s

LUx = b →

Lx1 = bUx = x1

s ts ♦rt♠ s♦s t ♥r s②st♠ ② s♦♥ t♦ tr♥r s②st♠s ②sss ssttt♦♥s r♦♠ ♦♥♣t ♣♦♥t ♦ t ss ♣♦t ♠t♦tt ♥ ss ♦rrs♣♦♥s t♦ ts ♦rt♠ ♦r♠③ ♥ st② r♥t ②

♥ t s ♦ s♣rs ♠trs r♥t ♠t♦s ♥ ♦♠♣t t t♦rs L ♥ U s♦ r♣rs♥t ② s♣rs ♠trs ♦rrs♣♦♥♥ ♦rt♠s r qt t t♦♠♣♠♥t ♦rt♥t② r♥t rrs ♦♥ t ♥t ♣r♦♣♦s r② ♠♣♠♥tt♦♥s s r ♣♥ r♦♥t♥ ♣r♦s ♥ ♣♥ Pt♦ ♦♥strt s♣rs ♥r s②st♠ r♦ ② r♦ t♦ r♠♦ rs ♦ r♦♠ s ①♣♥♥ t♦♥ ♥ t♥ s♦s t ♥r s②st♠ tr t ts t♥ trt s♦r♦r t ♦♥ ♦ t ♥tr s♣rs rt s♦r

s r♠r ② ♦ts t ❬❪ ts rt s♦rs r ♣rtr② ♥t ♦r♦♠tr② ♣r♦ss♥ ♣r♦♠s t sr ♠ss ♠♦♥ ts s♦rs ❯ st ♥♦t ♥t♦♥t② ♦ ♦♣rt♥ ♥ ♥ t♦r ♠♥♥r ② st♦r♥ t ♥ ❯ t♦rs ♦♥ t s s ♠s t ♣♦ss t♦ s♦ ♥r s②st♠s

♦♥s♦♥s ♦♥ ♥r s♦rs

❲ ♦♥ ts sst♦♥ t s♦rt s♠♠r② ♦ ts ♥r s②st♠ s♦rs ♥ ♥ts ♠t♦s r ♦t s② t♦ ♥rst♥ ♥ t♦ ♠♣♠♥tt ♦ ♥♦t ♣r♦r♠ ♦r ♠♦r t♥ rs ♦♥tr♥t ♠t♦ r③s ♦♦ tr♦ t♥ t s ♦ ♠♣♠♥tt♦♥ ♥ ♥② rt s♣rs

♠t♦ ♣r♦s ♦♥sr①t♦♥ s② t♦ ♠♣♠♥t ♥♥t

♦ ♠♠♦r② ♦st♦♥t r♥t s② t♦ ♠♣♠♥t rs♦♥② ♥t

♦ ♠♠♦r② ♦strt ♠t♦s t ♠♦st ♥t ♠♠♦r② ♦st

♣ ♦s ♠♣♠♥tt♦♥s r ♦♠♣①

sss ♦ ♥r s♦r ♣r♦s ♥ ♦♥s

s♦rs r t ♠♦st ♥t t ♠② s♦♠t♠s ♦♥s♠s ♦♥sr ♠♦♥ts ♦ ♥② rt s♦rs t t♦r st♦r ❬sr t ❪ t t sr♥t r♦♠ t ♥② ♦ s♣rs rt s♦rs ♣♥ t ♦♥tr♦ ♦ ts ♠♦♥t ♦

♥t♦♥s ♦ ♠♥② rs

❲ ♥♦ ♦s ♦♥ ♥t♦♥s F : x 7→ F (x) r♦♠ Rn t♦ R ♦r ♥st♥ s ♥t♦♥

♥ ♦r♠③ t ♦♠tr rtr♦♥ tt s♦ ♠t ② sr ♥ ts ♦♥t①tt r♠♥ts ♦ t ♥t♦♥s s t♦r x tt trs t ♦♦r♥ts t trts ♦ t ♠s ♥t♦♥ rtr♥s r ♥♠r tt ♠srs t r♥ss ♦t s♦sr ♦♠tr② ♥ t s ♥t♦♥s ♦r s ♦ ts q♥tt② ♦rrs♣♦♥ t♦ ttr qt② ♦♣t♠③t♦♥ ♦rt♠ t♥ ss ♦r t ♣r♠trs ♦t ♥t♦♥ t ♦♦r♥ts t t rtstt ♠♥♠③ t ♥t♦♥ ♥ ts♦♥t①t s ♥t♦♥ s s♦ ♥ ♥r② ♦r ♥ ♦t ♥t♦♥ ❱rt♦♥♠t♦s ♠♥♠③ ♥ ♦t ♥t♦♥ ② st②♥ t rt♦♥s ♦ F (x) ♥ ♥t♦♥♦ t ♣r♠trs x = [x1, . . . xn] s rt♦♥s r ♦r♠③ ② t rts ♦F (x) ♦r s♣② ♦s ♦♥ ♠t♦s tt ♦♣rt t ♦rr s rqrst♦ ♥ t ♥♦t♦♥ tt ♦rrs♣♦♥s t♦ t rst ♦rr rt ♦r ♠trt ♥t♦♥s t r♥t ♥ t ♦♥ tt ♦rrs♣♦♥s t♦ t s♦♥ ♦rr rt t ss♥ ❲ t♥ st② s♦♠ t♦r♠s tt tt t ♦♠♣tt♦♥s♥ ①♣rss♦♥s tt ♦♠♥ t rts ♦ F

r♥t

r♥t ♥♦r♠t♦♥ t ♦rr s r♣rs♥t ② t t♦r ♦ rtsrt t♦ rs t r♥t ♦ F ♥ ♥♦t ② ∇F

∇F =

∂F/∂x1

∂F/∂xi

∂F/∂xn

P1

P2

P3

P4

P5

r tt♦♥r② ♣♦♥ts ♣♦♥ts P1 ♥ P4 r ♦ ♠①♠ P2 s ♥♥①♦♥ ♣♦♥t P5 s ♦ ♠♥♠♠ ♥ P3 s t ♦ ♠♥♠♠

♥♦t♦♥ ♦ r♥t s r② s ♥ ♠♥♠③t♦♥ s♥ ♠♥♠♠ s rtr③ ② t t tt t r♥t ♦ F s ③r♦ F s C1 ♦r t ③r♦st ♦ tr♥t s♦ t st ♦ stt♦♥r② ♣♦♥ts s♦ ♦♥t♥s ♦tr ♣♦♥ts s s♦♥ ♥r ♦r t ♥rt s rst ♠♥♠♠ ♥ ♦ rtr t♥ ♦t t stt♦♥ ♥ ♥ ♦rs stt♦♥r② ♣♦♥t ♥ s♦ ♦ ♦r ♦♠①♠♠ t ♥ s♦ ♥ ♥①♦♥ ♣♦♥t ♦ rtr rtr③ stt♦♥r② ♣♦♥tt s s♦ ♥ssr② t♦ t s♥ ♦ t s♦♥ ♦rr rt

❲ ♥♦ tt t♦r♠s tt tt t♦♥ t r♥ts

(1) ‖x‖2 = xtx

(2) ∇(btx) = ∇(xtb) = b

(3) ∇(xtAx) = (A + At)x = 2Ax A s s②♠♠tr

s st ♥ s② ② ♦ ♥②♥ t sqr ♥♦r♠ ♦ t♦r ♥ ♠tr① ♣r♦ts s t tr s s♦ tr ♦ ♣r♦ ①♣♥ xtAx ♥ ♦♠♣t tsrts rt t♦ t rs xi

A B C D

r tt♦♥r② ♣♦♥ts ♦ ♠♥♠♠ ♣♦st ♥t ss♥ ♦♠①♠♠ ♥t ♥t ss♥ ttr ♣♦st t ♥♦♥♥t ss♥ s ♣♦♥t ss♥ t ♣♦st ♥ ♥t ♥s

xtAx =∑

k1

∑

k2

ak1,k2xk1

xk2

∂xtAx∂xi

= ∂∂xi

(

ai,ix2i + xi

∑

k2 6=i

ai,k2xk2

+ xi

∑

k1 6=i

ak1,ixk1

)

= 2ai,ixi +∑

k2 6=i

ai,k2xk2

+∑

k1 6=i

ak1,ixk1

=∑

k2

ai,k2xk2

+∑

k1

ak1,ixk1

=∑

j

(ai,j + aj,i)xj

♥ ♥ tt ts ①♣rss♦♥ ♦rrs♣♦♥s t♦ t t r♦ ♦ (A + At)x

ss♥

r♥t ♥♦r♠t♦♥ t♦ ♦rr s r♣rs♥t ② t ♠tr① ♦ s♦♥ ♦rrrts rt t♦ ♦♣s ♦ rs (xi, xj) t ss♥ ♦ F ♥♥♦t ② ∇2F

∇2F =

∂2/∂x1,x1. . . ∂2/∂x1,xn

∂2/∂xn,x1. . . ∂2/∂xn,xn

s ♦r ♥rt ♥t♦♥s t s ♣♦ss t♦ ♦♠♣t ②♦r srs ①♣♥s♦♥ ♦r ♠trt ♥t♦♥ r♦♥ ♣♦♥t x0 s s ♦♦ s♦♥ ♦rr ♣♣r♦①♠t♦♥♦ F (x0 + p) s ♥t♦♥ ♦ t s♣♠♥t t♦r p r♦♠ x0

F (x0 + p) ≃ F (x) + pt (∇F (x0)) + 1/2pt(∇2F (x0)

)p

s ♣r♦s② ♠♥t♦♥ t ss♥ s ♠♦r ♥♦r♠t♦♥ ♦t stt♦♥r②♣♦♥t r s♦s s♦♠ ♦♥rt♦♥s tt ♥ ♥♦♥tr ♥ t s ♦ ♥t♦♥ ♦ t♦ rs ♦r♥ t♦ t s♥ ♦ t ♥s ♦ t ss♥ t♦♥rt♦♥ s ♦ ♠♥♠♠ ♦ ♠①♠♠ ttr ♦r s♣♦♥t r♦r t♦ ♠♥♠③ ♥t♦♥ F ♥♥ t ③r♦st ♦ t r♥t s ♥♥r ♥♦t s♥t ♥ s♦ ♥s t♦ t s♥ ♦ t ♥s ♦ t ss♥♥ t② r ♣♦st ts ♠② ♦ ♠♥♠♠

rt ♣t♠③t♦♥

rt ♦r♠s

qrt ♦r♠ s ♣♦②♥♦♠ ♥t♦♥ s tt t r ♦ ts tr♠s s ♥♦t rrt♥ t s ②s ♣♦ss t♦ rt qrt ♦r♠ s ♦♦s

F (x) = 1/2xtGx + xtc + α

r G s n × n s②♠♠tr ♠tr① c s t♦r ♦ Rn ♥ α sr ♥♥ t

stt♦♥r② ♣♦♥ts ♦ qrt ♦r♠ s qt s② ② st②♥ ts r♥t

∇F (x) = 0⇔ Gx + c = 0

s ♠♥♠③♥ qrt ♦r♠ ♦r s♦♥ s②♠♠tr ♥r s②st♠ r t♦ ♣r♦♠stt r q♥t

st sqrs

❲ s♣♣♦s tt ♥t t♦ s♦ ♥r s②st♠ ♦ qt♦♥s s tt t ♥♠r m♦ qt♦♥s s rtr t♥ t ♥♠r n ♦ rs

a1,1x1 + . . . + a1,nxn = b1

am,1x1 + . . . + am,nxn = bm

♦r Ax = b ♥ t ♥r s t s②st♠ s ♥♦ s♦t♦♥ t s t♥ ♣♦ss t♦ tr②t♦ ♥ t st s♦t♦♥ t t♦r x tt ♠♥♠③s t s♠ ♦ t sqrrss

F (x) =m∑

i=1

(n∑

j=1

ai,jxj − bi

)2

= ‖Ax− b‖2

♥t♦♥ F s qrt ♦r♠ tt ♥ rtt♥ s ♦♦s ♦r♠

F (x) = ‖Ax− b‖2 = (Ax− b)t(Ax− b) = xtAtAx− 2xtAtb + btb

t t♦ tr♠s xtAtb ♥ btAx r q s♥ t② r srs ♦r 1 × 1 ♠trstt r s②♠♠tr t♦r x tt ♠♥♠③s F s s tt ∇F = 0 r♦r♦r♠s ♥

∇F (x) = 2AtAx− 2Atb = 0

❲ rtr t ss ♦r♠ ♦r t st sqrs s♦ ♥r rrss♦♥ t♦r x tt ♠♥♠③s ‖Ax− b‖2 stss AtAx = Atb

st sqrs t r rs ♦ r♦♠

♦r s♦♠ ♦rt♠s t ♠② s t♦ r♠♦ s♦♠ rs ♦ r♦♠ ♦ t s②st♠② ①♥ s♦♠ ♣r♠trs ♦ t ♦t ♥t♦♥ ♦r♠② ts ♠♥s tt t t♦rx ♦ ♣r♠trs s ♣rtt♦♥ ♥t♦ t♦ st♦rs x = [xf |xl] rst nf ♦♠♣♦♥♥tsxf = [x1 . . . xf ] ♦rrs♣♦♥ t♦ t fr ♣r♠trs ♥ t r♠♥♥ n−nf ♦♠♣♦♥♥tsxl = [xnf+1 . . . xn] ♦rrs♣♦♥ t♦ t l♦ ♣r♠trs s t ♥t♦♥ F s♦②♣♥s ♦♥ t t♦r xf ♥ rrts

F (xf ) = ‖Ax− b‖2 =

∥∥∥∥∥∥∥∥

[Af | Al]

xf

xl

− b

∥∥∥∥∥∥∥∥

2

Prtt♦♥♥ t t♦r x ♥t♦ t♦ st♦rs [xf , xl] ♥tr② ♣rtt♦♥s t ♠tr①A ♥t♦ t♦ s♠trs Af , Al ♥ t ♣r♦t Ax t ♦♥ts ♦ A tt t♣r♠trs ♥ xf r ♥ Af ♥ t♦s tt t ♣r♠trs ♥ xl r ♥ Al t s t♥♣♦ss t♦ s♦t t tr♠s tt ♣♥ ♦ xf

F (xf ) = ‖Afxf + Alxl − b‖2

② ♥tr♦♥ b′ = Alxl − b ♥ ② s♥ t stsqrs ♦r♠ ♣r♦ ♥ t♣r♦s st♦♥ ♦♥ rtrs t qt♦♥ sts ② t ♠♥♠③r xf ♦ F

AtfAfxf = At

fb′ ♦r At

fAfxf = Atfb− At

fAlxl

♦♥♥r ♦♣t♠③t♦♥

❲ ♥♦ ♦s ♦♥ t ♠♦r t ♣r♦♠ ♦ ♠♥♠③♥ ♥♦♥♥r ♥t♦♥s ♥ tsst♦♥ ♣rs♥t t t♦♥ ♠t♦ tt ♥ ♦♠♣t stt♦♥r② ♣♦♥t ♦ ♥♦♥♥r ♥t♦♥ tr ♠t♦s ①st ♥ t ♥trst rr ♠② r rr♥ ♦♦s❬♦ ♥ ❲rt ❪ ♦r ♠♦r ts

❯♥rt ♥t♦♥s

❲ rst ♦♥sr t s ♦ ♥rt ♥t♦♥ ♦♦♥ ♦rt♠ trt②♦♠♣ts stt♦♥r② ♣♦♥t ♦ t ♥t♦♥ f

|f ′(x)| > ǫp← −f ′(x)/f ′′(x)x← x + p

♥

♦rt♠ ♦rs s ♦♦s t st♣ ♦♠♣t t s♦♥ ♦rr ♦ t♥t♦♥ f r♦♥ t rr♥t ♣♦♥t x s ♣♣r♦①♠t♦♥ s ♠♦ ♥t♦♥s♥ t s ♦♦ ♣♣r♦①♠t♦♥ ♦r ♦♦ ♠♦ ♦ t rt♦♥s ♦ f r♦♥ x♥ ♥t♦♥ ♦ s♣♠♥t p

f(x + p) ≃ fx(p) = f(x) + pf ′(x) + p2/2f ′′(x)

♥ t ♦rt♠ ♦♠♣ts t s♣♠♥t p tt ♠♥♠③s t ♠♦ ♥t♦♥s s

f ′x(p) = 0

f ′(x) + pf ′′(x) = 0p = −f ′(x)/f ′′(x)

♥② t rr♥t ♦t♦♥ s ♣t x ← x + p ♥ t ♦rt♠ trts t♣r♦s st♣s

trt ♥t♦♥s

♣r♦s ♥rt ♦rt♠ ♥ s② ♣t t♦ t ♠trt s s s♥ ♥ t ♣r♦s st♦♥ t s♦♥ ♦rr ②♦r ①♣♥s♦♥ ♦ ♠trt♥t♦♥ F rts

F (x + p) ≃ Fx(p) = F (x) + pt (∇F (x)) + 1/2pt(∇2F (x)

)p

t s r② qrt ♦r♠ ❲ t♥ ♥ t s♣♠♥t p p tt ♠♥♠③s tsqrt ♦r♠

∇Fx(p) = 0(∇F (x)) + (∇2F (x)) p = 0

p = − (∇2F (x))−1

(∇F (x))

t♦♥ ♦rt♠ ♦r ♠trt ♥t♦♥s ♥ t♥ ♥ ②

‖∇F (x)‖ > ǫs♦ (∇2F (x))p = −(∇F (x))x← x + p

♥

♥ ♦tr ♦rs t♦ ♠♥♠③ ♥♦♥♥r ♥t♦♥ t t♦♥ ♦rt♠ ♠♥♠③s srs ♦ qrt ♦r♠ ② s♦♥ srs ♦ ♥r s②st♠s ♦ s♦ t♦s ♥rs②st♠s ♦♥ ♠② s ♦♥ ♦ t ♠t♦s st ♥ t♦♥

♦♥str♥ ♣t♠③t♦♥ r♥ t♦

❲ ♥♦ ♦s ♦♥ t ♣r♦♠ ♦ ♦♥str♥ ♦♣t♠③t♦♥ ♦♥str♥ ♦♣t♠③t♦♥♣r♦♠ ♥ ♦r♠③ ②

♠♥♠③F (x) t

G1(x) = 0G2(x) = 0

Gm(x) = 0

r F ♥ G1, G2 . . . Gm r ♥t♦♥s r♦♠ Rn t♦ R ♥t♦♥s Gi r

♦♥str♥ts ♥ ♠♣♦rt♥t t♦r♠ ♥♦♥ s t ♦♥t♦♥ stts tt t♠♥♠③r ♦ F tt stss t Gi ♦♥str♥ts s s♦ stt♦♥r② ♣♦♥t ♦ t ♥t♦♥L ♥ ②

L(x, λ) = F (x) +m∑

i=1

λiGi(x)

♥t♦♥ L t r♥♥ ♦ t ♦♥str♥ ♦♣t♠③t♦♥ ♣r♦♠ ♣♥s♦ m t♦♥ λi ♣r♠trs ss♦t t t m ♦♥str♥ts Gi

stt♦♥r② ♣♦♥ts ♦ t r♥♥ L ♥ ♦♥ ② t t♦♥ ♠t♦♣rs♥t ♥ t ♣r♦s st♦♥

♠ tr ts t♦rs rs ♥ ♥ r

♦r♣②

r s♦♥ ♥ ❲s t♦r② ♦ s♣♥s ♥ tr ♣♣t♦♥s♠ Prss ❨♦r

s♦② ♦♦s② ♥ P rör t s♦rs ♦r ♥strtr sr♠ss ♦r♥ ♦♥ ♥t ♦♠♣t♥

① r♥ ♣♦②r s♣s t sttr trs ❱s ♦♠♣tr

① ♥t ♥s ♥ ♠s ♠♦r♣♥ ♦♠♣tr r♣s ♦r♠

♥ rss ♥ ❩ P♦♣♦ s♣ ♦ ♠♥ ♦② s♣s r♦♥strt♦♥♥ ♣r♠tr③t♦♥ r♦♠ r♥ s♥s r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

P ③ ♥ ♦ts♠♥ ♥t ♥s ♥ ♦♠♣rss♦♥ ♦ ♠ss ♥ ♦s♦♥ ♦tr ♥ ♥ t♦rs ♥s ♥ trs♦t♦♥ ♦r♦♠tr ♦♥ t♠ts ♥ ❱s③t♦♥ ♣s ♣r♥r r♥r

P ③ ♦♥t♥r rs é② ♥ sr♥ ♥s♦tr♦♣ ♣♦②♦♥ r♠s♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦P

P ③ ❯ ♦ts♠♥ ♥ tt♥ ♥t ♥s ♥ r♠s♥ ♦srs sr r♣♦rt P t♦r ♦ ①♥

♥♦ P r♥s♥ ♦r r♥ ♦rs ♥ s P♣ ♦♠♣t♦♥ ♥ ♥♠t♦♥ ♦ ♣♦♣ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

r ♥ r s♦♠tr t①tr ♠♣♣♥ ♦r r♦r♠ srs ♦♠♣trr♣s ♦r♠

P ③rs ♥ s♣rt♦s ♥ s♥ ♣♥r ♦♣♠♥ts t♦ ♣r♦r♠ t①tr♠♣♣♥ ♦♥ rtrr② r srs ♦♠♣trs r♣s

♠ ♥ ♥ r♥r♥ ♣♦♣t♠③ t①tr ♠♣s ♦♠♣trr♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

② ♥ tr♥s♥t r②♥tr ♦♦r♥ts ♥ Pr♦♥s ♦ t t r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣sss♦t♦♥

♥♥s ❱é③♥ ♥ éss Ps sr tt♥♥ ♦r ♥♦♥st♦rtt①tr ♠♣♣♥ P ♦♠♣tr r♣s Pr♦♥s ♦ P

r ♥ ♦♥ ♦♣rt t①tr ♠♣♣♥s ♦♠♣tr r♣s ♥♣♣t♦♥s

r♠♥♥ rt♥ r♥r♥ ♥ ❩♦r♥ t♥♣st t♥ ♦ ♠trs♦t♦♥ srs r♥st♦♥s ♦♥ r♣s Pr♦♥s♦ P

r♦③ ♣♣rsr♥ ♣r♠tr③t♦♥ ♦ ♥s srs ♥ Pr♦♥s ♦ tt ♥tr♥t♦♥ ♦♥r♥ ♥ ♥tr r♦♣ ♦♥ ♦♠♣tr r♣s ❱s③t♦♥♥ ♦♠♣tr ❱s♦♥ ❲ ♣s

❱ ♥③ ♥ ❱ttr ♠♦r♣ ♠♦ ♦r t s②♥tss ♦ s ♥ Pr♦♥s♦ P ♣s Prss

❱ ♥③ ss♦ P♦♦ ♥ ❱ttr ♥♠t♥ s ♥ ♠s ♥ ♦♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

❱ ♥③ r♠ ❱ttr ♥ P ①♥♥ s ♥ ♠s♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

♥♥ ♠t♦♥ ♦ r♥ srs P ♦♠♣tr r♣s Pr♦♥s ♦ P

♦rrt♥ s♠r ♥ ♥ Pr♠tr③t♦♥ ♦ tr♥ ♠ss ♦rqrtr ♦♠♥s ♥ Pr♦♥s ♦ t ♦♥ r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

♦ts ♦♠♠s ♥ ♦t ♥t ♥r s②st♠ s♦rs ♦r ♠s ♣r♦ss♥ ♥ t♠ts ♦ rs ❳ ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr♥ ♣s ♣r♥r r♥ r Pr♦♥s ♦ t t ♥tr♥t♦♥ ♦♥r♥

P ♦rs ♥ r P♥r ♦♥♦r♠ ♠♣♣♥s ♦ ♣s t srs ♥ ♥ P♦tr t♦rs ❱s③t♦♥ ♥ t♠ts ♣s ♣r♥r r

r♥♥r ♥ ♦tt ♠t♠t t♦r② ♦ ♥t ♠♥t ♠t♦s♦♠ ♦ ①ts ♥ ♣♣ t♠ts ♣r♥r s♦♥ t♦♥

♠♣♥ ♥ P Pr♠tr③♥ ♠ss t rtrr② t♦♣♦♦② ♥Pr♦♥s ♦ ♠ ♥ t♠♥s♦♥ t ♥ Pr♦ss♥ ♣s

rr ♥ rt s tss ♦r rt♠ ♣r♦r s♦ t①tr♥ r♥st♦♥s ♦♥ r♣s

rr ♥ rt P♥t♥ t r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

rr ♦r♦ r♥ ♥ rt t♥r ♠trt ♦♠tr②♠s ♥ Pr♦♥s ♦ t t r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥P ♣s r♦r♣s ss♦t♦♥

♦qt r ♥ t②♣ tr♥s♦r♠t♦♥ ♥②tq é♥érs♥t r♣rés♥tt♦♥♦♥♦r♠ t é♥é ♠♦②♥ ♦♥t♦♥s r♠♦♥qs t♥ s ♥s té♠tqs

♥ ♣tr r♣ ♦r② ♠r ♥ ♦♥ ♦♥r♥rs ♥ t♠ts ♠r♥ t♠t ♦t②

♣r ❨♦ ♥t ②s r t s♣ ♦ r♠ ♥ ❲ts ♣♣♥♥ ♥ tt♠t ♥s ♦♠ ♠r♥ t♠t ♦t② Pr♦♥

❯ r♥③ t ♥ ♠♣ ①♦♠s ♥ rt♦♥ ♣r♦♠s ♥ sr ♣r♠tr③t♦♥ ♦♠♣tr ♦♠tr s♥

♦♥t♥r P ③ ♥ sr♥ ❱rt♦♥ s♣ ♣♣r♦①♠t♦♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

♦♥s ♥ t♣♥s♦♥ r ♣♥ ♦rt♠ ♦♠♣tt♦♥ ♦♠tr②♦r② ♥ ♣♣t♦♥s

♦r♥t rts Pr♥♣ ♦♥♦r♠ ♣♣♥ ♥ ♥♠ rs ♥trs♥ ❨♦r

❨ ❱rèr ♥ ♥ r♣ ♥r♥t ♥ rtr♦♥ ♦r ♣♥rt② ♥ r♣trtr ♦r② ♦♠ ♦ ♦♥t♠♣♦rr② t♠ts ♣s ♠r♥ t♠t ♦t②

P ♥r st ♥ ♥ ♥ ♣t sr ♣r♠tr③t♦♥ ♠t♦ ♥Pr♦♥s ♦ t t ♥tr♥t♦♥ s♥ ♦♥t ♣s

sr♥ ②r ♥ P ③ ♥tr♥s ♣r♠tr③t♦♥s ♦ sr ♠ss♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

♦ r♠♦ r♥t ♦♠tr② ♦ rs ♥ srs Pr♥t

♦♥ P r♠r r♥ ❱ Ps ♥ rt r♥t♥ ♠s s♥ ♣♥ ♥t♦rs ♥ ♣♦rt ❯❯ ❯♥rst② ♦ ♥♦s ♥

♦♥ P r♠r r♥ ❱ Ps ♥ rt ♣tr srqr♥t♦♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s♦ P

♦s ♦t♦♥ ♦ t ♣r♦♠ ♦ Pt r♥st♦♥s ♦ t ♠r♥ t♠t ♦t②

♠♣ rt♥ ♦s ♥ ❲ tt③ rr ♦♠♣tt♦♥ ♦ Pr♠♦♥ ♠♥s ♥ r♣♦rt ❯♥rst② ♦ ❲s♥t♦♥ ②

♦s ♠♣ ♦♣♣ ♦♥sr② ♥ ❲ tt③ trs♦t♦♥ ♥②ss ♦ rtrr② ♠ss ♥ Pr♦♥s ♦ P ♣s Prss

st♥ ❱ r③s② ♥ ♦ts♠♥ ①tr ♠♣♣♥ t r ♦♥str♥ts♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

rs♦♥ ♥ rP ♣t♠② tt♥ sr ♥t♦ s srt ♦♠♣tt♦♥ ♦♠tr②

rs♦♥ ♥ ❲tts② r② ♦♣t♠ ♦♠♦t♦♣② ♥ ♦♠♦♦② ♥rt♦rs♥ Pr♦♥s ♦ t t ②♠♣♦s♠ ♦♥ srt ♦rt♠s ♣s

r♥ rs ♦r rt tsst♦♥s ♦♠♣tr ♦♠tr s♥

rs♦♥ ♥ ♦♦♦ t♣r♦ ♥t♦♥s ♦r sr s♥ ♦♠♣tr ♦♠tr s♥

r r ♦♥♥tt② ♦ r♣s ③♦s♦ t♠t ♦r♥

r ♣r♦♣rt② ♦ ♥t♦rs ♦ ♥♦♥♥t s②♠♠tr ♠trs ♥ ts ♣♣t♦♥ t♦ r♣ t♦r② ③♦s♦ t♠t ♦r♥

♠ ♦r♥r ♥ ❱ ♥ ♦♥♦r♠ t①tr ♠♣♣♥ ♥ Pr♦♥s ♦r♦r♣s ♣s

♦tr Pr♠tr③t♦♥ ♥ s♠♦♦t ♣♣r♦①♠t♦♥ ♦ sr tr♥t♦♥s♦♠♣tr ♦♠tr s♥

♦tr Pr♠tr t♥s ♥ sttr t ♣♣r♦①♠t♦♥ ♥tr♥t♦♥♦r♥ ♦ ♣ ♦♥

♦tr sss ♣r♠tr③t♦♥ ♥ s♣♥ sr ♣♣r♦①♠t♦♥ ♥ ♣♦ ♥ rt♥ t♦rs t♠ts ♦ rs ❳ ♣s ♦♥♦♥ ♣r♥r

♦tr ♦♥① ♦♠♥t♦♥ ♠♣s ♥ s② ♥rs♦♥ ♥ s♦♥ t♦rs ♦rt♠s ♦r ♣♣r♦①♠t♦♥ ❱ ♣s

♦tr ♥ ♦♦r♥ts ♦♠♣tr ♦♠tr s♥

♦tr ♥t♦♦♥ ♣s ♥r ♠♣♣♥s ♦r tr♥t♦♥s t♠ts♦ ♦♠♣tt♦♥

♦tr ♥ ♦r♠♥♥ Pr♠tr③t♦♥ ♦ tr♥t♦♥s ♥ ♥♦r♥③♣♦♥ts ♥ s ♥ ♦tr t♦rs t♦rs ♦♥ trs♦t♦♥♥ ♦♠tr ♦♥ t♠ts ♥ ❱s③t♦♥ ♣s ♣r♥rr♥ r

♦tr ♥ ♦r♠♥♥ r ♣r♠tr③t♦♥ tt♦r ♥ sr② ♥ ♦s♦♥ ♦tr ♥ ♥ t♦rs ♥s ♥ trs♦t♦♥♦r ♦♠tr ♦♥ t♠ts ♥ ❱s③t♦♥ ♣s ♣r♥rr♥ r

♦tr ♥ ♠rs sss ♣r♠tr③t♦♥ ♥ sr r♦♥strt♦♥♦♠♣tr ♦♠tr s♥

♦tr ♦r♠♥♥ ♥ ♠rs Pr♠tr③t♦♥ ♦ ♠♥♦ tr♥t♦♥s ♥ ♠r ♥ tör t♦rs ♣♣r♦①♠t♦♥ ♦r②❳ strt ♥ ss ♥②ss ♥♥♦t♦♥s ♥ ♣♣ t♠ts ♣s ❱♥rt ❯♥rst② Prss s

♦tr ós ♥ ♠rs ♥ ♦♦r♥ts ♥ ♦♠♣tr ♦♠tr s♥

♦tr ♦r♠♥♥ ♥ ós ♥r ♦♥strt♦♥ ♦ r②♥tr ♦♦r♥ts ♦r ♦♥① ♣♦②♦♥s ♥s ♥ ♦♠♣tt♦♥ t♠ts

♦② ♥ ♠ ♥r ♥ s ♦♠♣tr r♣s Pr♥♣s♥ Prt s♦♥❲s② ♦st♦♥ s♦♥ t♦♥

❱ r♥③ ①♠♠ ♥♦r♠ ♦♣t♠③t♦♥ ♦ qss♦♠tr ♠♣♣♥s ♠r♥r r t ♣♣t♦♥s

r♥ ❲♠♦tt ♥ P rt rr str♥ ♦♥ ♣♦②♦♥srs ♥ Pr♦♥s ♦ t ②♠♣♦s♠ ♦♥ ♥trt r♣s ♣s Prss

ÿ sqst♦♥s ♥rs r s♣rs rs ♦♠♠♥tt♦♥s ♦ttsæ ♥tr♠ ♦tt♥♥ss ♥t♦rs

♦rtr ♦ts♠♥ ♥ rst♦♥ srt ♦♥♦r♠s ♦♥ ♠ss ♥ ♣♣t♦♥s t♦ ♠s ♣r♠tr③t♦♥ ♦♠♣tr ♦♠tr s♥

♦ts♠♥ ♥ r♣ ♣rtt♦♥♥ s♣tr ♥②ss ♥ t ♠s ♣r♦ss♥ ♥Pr♦♥s ♦ ♣ ♦♥ ♥tr♥t♦♥ ♣s ♦♠♣tr ♦t②

♦ts♠♥ ❳ ♥ r ♥♠♥ts ♦ s♣r ♣r♠tr③t♦♥ ♦r ♠ss r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦P

r♥r ❱rt♦♥ s♥ ♥ r♥ ♦ s♣♥ srs ♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

r♥r ♥ ♦r♠♥♥ ♥tr♣♦t♥ ♥ ♣♣r♦①♠t♥ sttr t trr t♥s♦r ♣r♦t s♣♥s ♥ été t ♥ ♠r t♦rs r tt♥ ♥ trs♦t♦♥ t♦s ♥♥♦t♦♥s ♥ ♣♣t♠ts ♣s ❱♥rt ❯♥rst② Prss s

r♠♠ ♥ s Pr♠tr③♥ N ♦ t♦r ♥ t♠ts ♦rs ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr ♥ ♣s ♣r♥rr♥ r Pr♦♥s ♦ t t ♥tr♥t♦♥ ♦♥r♥

r♠♠ ♥ s ♦♥ srs ♦ rtrr② t♦♣♦♦② s♥ ♠♥♦s♥ Pr♦♥s ♦ P ♣s Prss

❳ ♥ ❨ ♦♠♣t♥ ♦♥♦r♠ strtrs ♦ srs ♦♠♠♥t♦♥s ♥♥♦r♠t♦♥ ♥ ②st♠s

❳ ♥ ❨ ♦ ♦♥♦r♠ sr ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ trst r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

❳ ♦rtr ♥ ♦♣♣ ♦♠tr② ♠s r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

❳ ❨ ❲♥ ♥ ❨ ♦♠♣t♥ ♦♥♦r♠ ♥r♥ts Pr♦ ♠trs♦♠♠♥t♦♥s ♥ ♥♦r♠t♦♥ ♥ ②st♠s

❳ ❨ ❲♥ ♥ ❨ ♦♠tr ♦♠♣rss♦♥ s♥ ♠♥♥ sr strtr ♦♠♠♥t♦♥s ♥ ♥♦r♠t♦♥ ♥ ②st♠s

❳ ❨ ❲♥ ♥ P ♦♠♣s♦♥ ♥ ❨ ♥s ③r♦ sr♦♥♦r♠ ♠♣♣♥ ♥ ts ♣♣t♦♥ t♦ r♥ sr ♠♣♣♥ r♥st♦♥♦♥ ♠♥

❳ ❨ ❲♥ ♥ ❨ trs♦t♦♥ ♦♠♣tt♦♥ ♦ ♦♥♦r♠ strtrs♦ srs ♦r♥ ♦ ②st♠s ②r♥ts ♥ ♥♦r♠ts

s♦ ♥ ♥s♦tr♦♣ ♠s ♣r♠tr③t♦♥ s♠ ♥♥r♥ t ♦♠♣trs

s♦ ❲ ♥s ♥ P rör trs♦t♦♥ s♥ ♣r♦ss♥ ♦r ♠ss♥ Pr♦♥s ♦ P ♣s Prss

s♦ ❱♠↔ ❲ ♥s ♥ P rör ♦r♠ ♠ss ♥ Pr♦♥s♦ P ♣s Prss

s♦ ♦♦s② P rör ♥ ❲ ♥s ②r ♠ss trs♦t♦♥ s♥ rr ♥ rrr r♥♠♥t ♥ Pr♦♥s ♦ t t ♥♥②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s Prss

r ♥♥♥t ♥♥♥♠ ♥s ♣r♦ ♥ ♦♥♦r♠sr ♣r♠tr③t♦♥ ♦r t①tr ♠♣♣♥ r♥st♦♥s ♦♥ ❱s③t♦♥♥ ♦♠♣tr r♣s

❨ ❳ ♥ ♥ t♦♥ s♣r s♣♥s ♦r ♥s ③r♦ s♣ ♠♦♥ ♥Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣ ♦♥ ♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

r♥ srt ①tr♦r s P tss ♦r♥ ♥sttt ♦ ♥♦♦②Ps♥

②♦s ♥ r ❱♦r♦♥♦s ♥tr♣♦t♦♥ t r ♦♥t♥t② ♥Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s Prss

♦♣♣ Pr♦rss ♠ss ♥ Pr♦♥s ♦ P ♣s Prss

♦♣♣ ♥ Pr♥ ♣ ♦♠♣rss♦♥ s♥ s♣r ♦♠tr② ♠s ♥ ♦s♦♥ ♦tr ♥ ♥ t♦rs ♥s ♥ trs♦t♦♥ ♦r♦♠tr ♦♥ t♠ts ♥ ❱s③t♦♥ ♣s ♣r♥r r♥r

♦r♠♥♥ tt♥ r ♦r♠ srs ♥ r♦ r♥r ♥ ♠♥♥t♦rs Pr♥♣s ♦ ♠ ♥②ss ♥ ②♥tss ♦♠ ♦ ♥tr♥t♦♥ rs ♥ ♥♥r♥ ♥ ♦♠♣tr ♥ ♣tr ♣s r ♠ Psrs ♦st♦♥

♦r♠♥♥ ♦r② ♥ ♣♣t♦♥s ♦ Pr♠tr③♥ r♥t♦♥s P tss♣rt♠♥t ♦ ♦♠♣tr ♥ ❯♥rst② ♦ r♥♥ ♦

♦r♠♥♥ ♥ ♦tr ♥ ♦♦r♥ts ♦r rtrr② ♣♥r ♣♦②♦♥s r♥st♦♥s ♦♥ r♣s

♦r♠♥♥ ♥ r♥r P ♥ ♥t ♦ ♣r♠tr③t♦♥ ♠t♦ ♥ P r♥t P ♦♥♥èr ♥ ♠r t♦rs r ♥ r s♥♥t♦ ♥♥♦t♦♥s ♥ ♣♣ t♠ts ♣s ❱♥rt❯♥rst② Prss s

♦r♠♥♥ ♥ r♥r rtr r♠s♥ ♥ r♦ r♥r ♠♥♥ ♥ P t♦rs Pr♦♥s ♦ ❱s♦♥ ♦♥ ♥ ❱s③t♦♥ ♣s rrü♥ r♠♥② ♥①

♦r♠♥♥ ♥ ♠rs r♥t♥ ♣♦♥t ♦s t s♣r t♦♣♦♦② ♥ ② ③r ♥ ♠r t♦rs r ♥ r s♥♥t♦ ♦r♥ t♦s ♥ ♣♣ t♠ts ♣s s♦r♦Prss r♥t♦♦

♦r♠♥♥ ♥ r♥ qrtr r♥r♥ ♣r♠t ♥ Pr♦♥s ♦ t ❲♦rs♦♣ ♦♥ r♣s rr r♦r♣s ②♠♣♦s♠ Pr♦♥s ♣s r♦r♣s ss♦t♦♥

♦r♠♥♥ r♥r ♥ ♠♣♥ rr ♣r♠tr③t♦♥ ♦ tr♥tsrs ♥ r♦ ♠♥♥ ♥ P t♦rs Pr♦♥s ♦ ❱s♦♥♦♥ ♥ ❱s③t♦♥ ♣s r♥♥ r♠♥② ♥①

♦r♠♥♥ ❯ s ♥ r♥r ♠s♥ tr♥t srs t ♦♣t♠♣r♠tr③t♦♥s ♦♠♣tr s♥

r P ♦rs t♣♥s♦♥ ❲ ♠♥rs ♠ ♣r♥ ♦tt♥r s♦♥♦r♠② t ♠♣♣♥ t ♠♥ r♠ ♥ ♠ ♦♠♣t♥ ♥ ♦♠♣trssst ♥tr♥t♦♥ ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr ♥ ♣s ♣r♥r r♥r

rs ♥ ♦sr♦ ♣t ♥r♣♣♥ ♦r ♥trt t①tr ♣♥t♥♥ Pr♦♥s ♦ t ②♠♣♦s♠ ♦♥ ♥trt r♣s ♣s Prss

s♥r ♥ P ♥str♦♠ tr♠♥ ♠ss ♥ Pr♦♥s ♦ ❱s③t♦♥ ♣s ♦♠♣tr ♦t②

s♥r ♠♦ ♥ ♦ts♠♥ ♦♥♥tt② s♣s ♥ Pr♦♥s ♦ ❱s③t♦♥ ♣s ♦♠♣tr ♦t②

♥ ❨ ❲♥ ❨ ♥ ❳ ♣t♠ ♦ ♦♥♦r♠ sr ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ ❱s③t♦♥ ♣s ♦♠♣tr♦t②

♥ ♦ ♥ ❳ ♦♠♣t♥ sr ②♣r♦ strtr ♥ r ♣r♦tstrtr ♥ Pr♦♥s ♦ t ②♠♣♦s♠ ♦♥ ♦ ♥ P②s ♦♥P ♣s Prss

r ♥ ❲rr♥ ♥ ♦♦r♥ts ♦r ♦s tr♥r ♠ss r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

r ❲rr♥ ♥ sr♥ ♦♠tr ♦♥strt♦♥ ♦ ♦♦r♥ts♦r ♦♥① ♣♦②r s♥ ♣♦r s ♥ Pr♦♥s ♦ t r r♦r♣s②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

P ♣ ♥ ❲rr♥ ♥r ♦♠tr ♦♥strt♦♥ ♦ ♦♦r♥ts ♥ ♦♥① s♠♣ ♣♦②t♦♣ ♦♠♣tr ♦♠tr s♥

s ❱ r♦② ♥ r rts s♦♣ ♠s s♠♥tt♦♥♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

♥ ♦♥ r t s♣ ♦ r♠ ♠r♥ t♠t ♦♥t②

❩ r♥ ♥ ♦ts♠♥ ♣tr ♦♠♣rss♦♥ ♦ ♠s ♦♠tr② ♥ Pr♦♥s ♦P ♣s Prss

❩ r♥ ♦ts♠♥ ♥ ♦rtr r♦♥r② ♥r ♣r♠tr③t♦♥ ♦ ♠ss ♥ t ♣rs♥ ♦ ♦♥str♥ts ♥ Pr♦♥s ♦ t ♥tr♥t♦♥♦♥r♥ ♦♥ ♣ ♦♥ ♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

r② ♣r♥♦r♥ ♥ P rör srt ♦♥♦r♠ ♠♣♣♥s r♣ttr♥s r♥st♦♥s ♦♥ r♣s

♦♦s② t ♥ P rör ♦② s♠♦♦t ♣r♠tr③t♦♥s t♦ st♦rt♦♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦P

❲ ♥♥r ♦rs ♥ r♥t ♦♠tr② ♣r♥r r♥ r

♥sr ös♥ r rsrt r ts♥ t♠tr❱r♥♥

P ♦t ❱♦rst③ ❯ s ♥ P sr♥r♣♣♥ ♣♣r♦t♦ r♠s♥ ♣♦②♦♥ srs ♦♠♣tr r♣s ♦r♠ Pr♦♥s ♦ r♦r♣s

♦♠♥↔ ♥ tt♥♥ ♦ rtrr② srs ② ♣♣r♦①♠t♦♥ t♦♣ str♣s ♥ ❯ ♥ ♥ ❲♦③♥② t♦rs r♦♠ ♦♠tr ♠♦♥t♦ s♣ ♠♦♥ ♦♠ ♦ ♥tr♥t♦♥ rt♦♥ ♦r ♥♦r♠t♦♥ Pr♦ss♥♣s r ♠ Psrs ♦st♦♥

❨ ♦r♥ ♥ s♣tr r♣ r♥ ♥ ♦♠♣t♥ ♥ ♦♠♥t♦rs ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr ♥ ♣s ♣r♥r r♥ r Pr♦♥s ♦ t t ♥♥ ♥tr♥t♦♥ ♦♥r♥

ós ♥ ❱ár② Pr♠tr③♥ ♦♠♣① tr♥r ♠ss ♥ ② ③r ♥ ♠r t♦rs r ♥ r s♥ ♥t♦ ♦r♥ t♦s ♥ t♠ts ♣s s♦r♦ Prss r♥t♦♦

❱ r♦② ♥ r r♦ss♣r♠tr③t♦♥ ♥ ♦♠♣t r♠s♥ ♦ ♠♦s r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

❱ r♦② ♥ r ♠♣t s ♠s ♦♠♣t♦♥ ♥ Pr♦♥s ♦ tr r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

❱ r♦② r ♥ ♦ts♠♥ t♠r ♦♥strt♥ ♦♥str♥ t①tr ♠♣s r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦P

r②s③ r♥t ♦♠tr② ♦r ❨♦r

❯ s ♦r♠♥♥ ♥ r♥r ❯s♥ ♠♦st s♦♠tr ♣r♠tr③t♦♥s ♦rr♠s♥ ♣♦②♦♥ srs ♥ rt♥ ♥ ❲ ❲♥ t♦rs Pr♦♥s ♦♦♠tr ♦♥ ♥ Pr♦ss♥ ♣s ♦♥ ♦♥ ♥ ♦♠♣tr ♦t②

♠rt ②trä ③♠ r r t♠t ♥ r♥ ♥♥♥ ♦♠ ♥♥ r s r♥

♥r ② ♥ P ♣r r②♥tr ♦♦r♥ts ♥ Pr♦♥s ♦ t t r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

♥r ② ♥ P ♥ ♦♦r♥ts ♦r rtrr② s♣r♣♦②♦♥s ♥ ♣♦②r ♥ R

3 ♥ P ♥♥ ② ♥ ♠r t♦rsr ♥ r s♥ ♥♦♥ ♦r♥ t♦s ♥ ♣♣ t♠ts♣s s♦r♦ Prss r♥t♦♦

③rs P♦♦ ❱tr ♥ ❱rr♦st ♦♠♣t♥ ♥♦♥ ♣♦②♦♥ s♠ ♦ ♥ ♦r♥t tr♥t sr ♥ Pr♦♥s ♦ t t ♥♥②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s Prss

♦rt♦♥ ♥ ♦♣♣ s♣ ss♦♥ srs ♥ Pr♦♥s ♦P ♣s Prss

❲ ❲ ♥s P rör ♦sr ♥ ♦♥ P trs♦t♦♥ ♣t ♣r♠tr③t♦♥ ♦ srs ♥ Pr♦♥s ♦ P ♣s Prss

❲ ♦♥ ❲ ♥s ♥ P rör trs♦t♦♥ ♠s ♠♦r♣♥♥ Pr♦♥s ♦ P ♣s Prss

❨ ♦♦s♥ ♥♦s ♥ ♣r♠tr r ♥tr♣♦t♦♥ ♦♠♣tr s♥

❨ ♠ ♥ s ♣r♠tr③t♦♥ t rt ♦♥r② ♦♠♣trs r♣s

♥② Pr♥ ♥st♥ ♥ ♦♣♣ t♠ r ♦r rtrr② srs♥ Pr♦♥s ♦ t ②♠♣♦s♠ ♦♥ ♥trt r♣s ♣s Prss

é② ♦♥str♥ t①tr ♠♣♣♥ ♦r ♣♦②♦♥ ♠ss ♥ Pr♦♥s ♦ P ♣s Prss

é② ♦♠♥ ①tr♣♦t♦♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

é② ♦r♠t♦♥ ♥②ss ♦♥ ♠♥♦ ♥ r♣♦rt tt♣♦rr♣t♦♥s

é② ♣tr♠ ♥♥t♦♥s ♦rs ♥ ♦rt♠ tt ♥rst♥s♦♠tr② ♥ Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣ ♦♥♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

é② ♥ t ♦♥st♦rt t①tr ♠♣♣♥ ♦r sr tr♥t♠ss ♥ Pr♦♥s ♦ P ♣s Prss

é② Ptt♥ ② ♥ ♦t st sqrs ♦♥♦r♠ ♠♣s ♦r t♦♠t t①tr ts ♥rt♦♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

❲ ② ♥ é② t♦♠t ♥ ♥trt ♠s t♦ s♣♥ ♦♥rs♦♥♥ Pr♦♥s ♦ t t r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

s♥ trr r ❨ ②♥ ♥ rt Pr♦♥t♦♥rs ♦r ♥♥t ♥r s②st♠s rs♥ ♥ sr ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ t t♥tr♥t♦♥ s♥ ♦♥t ♣s

t r♦s ♠♣ ♥ P rör ♥ ♠ ♣r♦ss♥ ♣♣r♦ t♦sr ♠t♥ ♥ Pr♦♥s ♦ t r r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr②Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

♦sss♦ ♦♣♣ r ♥ ❲rr♥ ♠♦♦t ♦♠tr② ♠s ♥ Pr♦♥s ♦ t rst r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

♦ás③ ♥ rr ♥ t ♥ s♣ ♦ ♦♥ ❱rèr ♠tr① ♥♥s ♥sttt ♦rr

♥ ♥ ♣t♠ t①tr ♠♣♣♥ ♥ Pr♦♥s ♦ r♦r♣s ♣s

❲ ♥ P rt Pr♠tr③t♦♥ ♦ r♥♦♠② ♠sr ♣♦♥ts ♦r st sqrstt♥ ♦ s♣♥ rs ♥ srs ♦♠♣tr s♥

♦t ❨ ♥ ❱rr♦st ♥trt t①tr ♠♣♣♥ ♥ Pr♦♥s ♦P ♣s Prss

s ♥ s♣t ♥tr♣♦t♦♥s ♦r t♠♣rtr strt♦♥s ♠t♦♦r ♥♦♥♦♥ ♣♦②♦♥s ♥tr♥t♦♥ ♦r♥ ♦ ♦s ♥ trtrs

s ♥ s♣t r ♦♥strt♦♥ ♦ s♠♦♦t ♥tr♣♦♥ts ♦♥ ♣♦②♦♥ ♦♠♥s t♠t ♦r♥

s ♥ ♥ s♣t ♠♦♦t t♦ ♠♥s♦♥ ♥tr♣♦♥ts r♣ ♦r ♣♦②♦♥s ♦r♥ ♦ r♣s ♦♦s

rs♥r ♥tr ♥ ♣t② ♦♥ ♥ r♥r♥ ♦r rst ♥♠t♦♥ ♥ Pér♦ ♥ s♠r t♦rs ♥r♥ ♥qs ♣s ♣r♥r ❲♥ ❨♦r Pr♦♥s ♦ t t r♦r♣s ❲♦rs♦♣ ♦♥ ♥r♥

rt♥② ♥s ♥ ♦ tt♥♥ ♦ tr♥t srs♥♦r♣♦rt♥ rts ♥ ssts ♦♠♣tr s♥

❲ s sr② ♦ t ♦♠tr rsts ♥ t ss t♦r② ♦ ♠♥♠srs t♥ ♦ t r③♥ t♠t ♦t②

rt♦r ♦ t t ♦rs trr sr♣t♦ s♠ ♥♥t♠ ♠♥t♦♠♠♦t sr

sr r♦♥② ♥ ♦♦ ♥ ♦t♦♦r s♣rs s②♠♠tr ♥♥t t♦r③t♦♥ ♠t♦ r♥st♦♥s ♦♥ t♠t ♦tr

②r rr ♥ sr♥ ♥r③ r②♥tr ♦♦r♥ts ♦♥rrr ♣♦②♦♥s ♦r♥ ♦ r♣s ♦♦s

❱ ♥♦ ♦tt♦♥ ♣♦②♦♥ ♦♥t♥♠♥t ♥ ♠♥♠♠ ♥♦sr ♥ Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s Prss

t♥ ♥ ③ ♥ ♣♣rrt t♦②s r♦♠ ♠ss s♥ str♣s ♣♣r♦①♠t ♥♦♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s♦ P

♦r♥ ♠♥♥♥ ♦♠tr② ♥♥rs Ptrs ❲s② s♦♥ t♦♥

♠ ❱s ♦♠♣① ♥②ss ①♦r ❯♥rst② Prss ①♦r ❨♦r

❳ r♥ ♥ rt r ♠♦rs ♥t♦♥s ♦r ①trt♥ t t♦♣♦♦strtr ♦ sr ♠s r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

♦ ♥ ❲rt ♠r ♣t♠③t♦♥ ♣r♥r rs ♥ ♣rt♦♥ssr ♣r♥r ❨♦r s♦♥ t♦♥

P ♥ ❲♥r ♠♥ ♣♥r r♣s t ① rt① ♦t♦♥s ♥ r♣r♥ ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr ♥ ♣s ♣r♥rr♥ r Pr♦♥s ♦ t t ♥tr♥t♦♥ ②♠♣♦s♠

Pr ♥ P r ♦♥str♥tsts②♥ ♣♥r ♦♣♠♥t ♦ ♦♠♣① srs ♦♠♣tr s♥

Prs♥ ♦rt♥ ♠♣t srs ♥ Pr♦♥s ♦ P ♣s Prss

P♥ rst♥ss♦♥ ♥ ❩♦r♥ ♥trt ♠♦♥ ♦ t♦♣♦♦② ♦♠♣①♦♠tr t r♥st♦♥s ♦♥ r♣s Pr♦♥s♦ P

❯ P♥ ♥ P♦tr ♦♠♣t♥ srt ♠♥♠ srs ♥ tr ♦♥ts①♣r♠♥t t♠ts

P♣♦♥ ♥ ♦rs♦ ♠ss t①tr ♠♣♣♥ ♦ ss♦♥ srs ② ♠♦♣t♥ ♥ t①tr ♥♥ ♥ Pr♦♥s ♦ P ♣s Prss

Pt ttstq ①♣ér♠♥t t é♦rq s qs ♦♠s ① s♦rs ♦érs tr❱rs Prs

P♦r♠s ♥ ♥ ♦② ♠♣s r♥st♦♥s♦♥ r♣s Pr♦♥s ♦ P

Pr♥ ♥ ♦♣♣ ♣r ♣r♠tr③t♦♥ ♥ r♠s♥ r♥st♦♥s♦♥ r♣s Pr♦♥s ♦ P

Pr♥ ♥st♥ ♥ ♦♣♣ ♣♣ t①trs ♥ Pr♦♥s ♦ P ♣s Prss

Pr♥ ❲ ♥s ♥ P rör ♦♥sst♥t ♠s ♣r♠tr③t♦♥s ♥ Pr♦♥s ♦ P ♣s Prss

Pt♦♠② ♦r♣② ♦r r♥st ② t♥s♦♥

Pr♥♦♠♦ ♦♥ ♥ ♠r ♠ss t①tr tss ♥ Pr♦♥s ♦ t♦♥ r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

ó rsrt r ts♥ t♠tr❱r♥♥

ó ♣r♦♠ ♦ st r ♥ t ♣r♦♠ ♦ Pt t♠ts❩tsrt

② ♥ é② rr st sqrs ♦♥♦r♠ ♠♣s ♥ Pr♦♥s ♦ tt P ♦♥r♥ ♦♥ ♦♠♣tr r♣s ♥ ♣♣t♦♥s P ♣s ♦♠♣tr ♦t②

② ❲ ❱t ♦t♥t♥ é② ♥ ♦rs r♣t tt♣♦rrs♦trr♣t

② ❲ é② r ♥ P ③ Pr♦ ♦ ♣r♠tr③t♦♥ r♥st♦♥s ♦♥ r♣s

tr ❲♦tr ♥ P♥ ♣tr♠ s♣tr s ❵♣ ♦srs ♥ s♦s ♦♠♣tr s♥

♠♥♥ r♥♥ ür ♥ ♠♥ ♦r r ♥t♦♥♥ ♥r rä♥r♥ ♦♠♣①♥ röÿ P tss ❯♥rstät ött♥♥

♦♦♦ ♥ Pr ♥trt s♥ ♦ s♠♦♦t ♥s N ♦ts s♥ ♠t♣r♦ ♥t♦♥s ♥ ♣♣t♦♥s ♥tr♥t♦♥ ♦r♥ ♦ ♣ ♦♥

♦♥ ♥ ♥ ♦♥r♥ ♦ r ♣♥s t♦ t ♠♥♥ ♠♣♣♥♦r♥ ♦ r♥t ♦♠tr②

♦s ♥ ♦♥♥r ♠♥s♦♥t② rt♦♥ ② ♦② ♥r ♠♥ ♥

❨♥ ♦ts♠♥ ♥ r Prt s♣r ♠♥ ♦♠♥♦ tr♥ ♠ss ♥ Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥♣ ♦♥ ♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

P ❱ ♥r ♥②r ♦rtr ♥ ♦♣♣ ①tr ♠♣♣♥ ♣r♦rss♠ss ♥ Pr♦♥s ♦ P ♣s Prss

P ❱ ♥r ♦rtr ♥②r ♥ ♦♣♣ ♥s♣③ ♣r♠tr③t♦♥♥ Pr♦♥s ♦ t t r♦r♣s ❲♦rs♦♣ ♦♥ ♥r♥ ❲ ♣s r♦r♣s ss♦t♦♥

P ❱ ♥r ❩ ❲♦♦ ♦rtr ♥②r ♥ ♦♣♣ trt ♦♠tr②♠s ♥ Pr♦♥s ♦ t rst r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥P ♣s r♦r♣s ss♦t♦♥

r ♥ ❲rr♥ ♥ ♥tr ♦♥strt♦♥ ♦r ♦♦r♥ts ♦r♦s rs ♦♠♣tr ♦♠tr s♥ ♦ ♣♣r

r♥r srt♠ Pr♥ ♥ ♦♣♣ ♥trsr ♠♣♣♥ r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

rt③ rr ❲♦s♦♥ ♥ ♣♣t♦♥s ♦ ♦♠♣tr r♣s♥ ♠ ♣r♦ss♥ t♦ ♥ ♠♦♥ ♦ t ♥t♦♥ rttr ♦ s♦rt① ♦♠♣tr r♣s ♥ ♣♣t♦♥s

rt③ ♥ ❲♦s♦♥ ♥♠r s♦t♦♥ t♦ t ♥r③♠♣♠rs ♣r♦♠ tt♥♥ ♥♦♥♦♥① ♣♦②r srs r♥st♦♥s♦♥ Pttr♥ ♥②ss ♥ ♥ ♥t♥

♠r ♦r♠t♦♥ ♦ ♦♥r② ♠s s♠♥tt♦♥ ♥ Pr♦♥s ♦ t ♦♥ ♥tr♥t♦♥ ②♠♣♦s♠ ♦♥ t Pr♦ss♥ ❱s③t♦♥ ♥ r♥s♠ss♦♥P❱ ♣s ♦♠♣tr ♦t②

♣r♦ ♥ P♦②r♦♥ r③t♦♥ ♦r s♣ tr♥s♦r♠t♦♥ ❱s♦♠♣tr

r ♣♥♥♥ tr s♠s ♦r r♥ ♣r♠tr③t♦♥ st♦rt♦♥ ♦ tr♥tsrs ♥ Pr♦♥s ♦ ♣ ♦♥ ♥tr♥t♦♥ ♣s ♦♠♣tr ♦t②

r ♦♥♦♣t♠ ♣r♠tr③t♦♥ ♥ sr ♦♥tr♦ ♥ ② ③r♥ ♠r t♦rs r ♥ r s♥ ♥t♦ ♦r♥t♦s ♥ ♣♣ t♠ts ♣s s♦r♦ Prss r♥t♦♦

r ♥♥♥ ♠ss r♣ ♦s

r ♥ trr r ♣r♠tr③t♦♥ ♦r ♠s♥ ② tr♥t♦♥tt♥♥ ♥ Pr♦♥s ♦ t t ♥tr♥t♦♥ s♥ ♦♥t ♣s

r ♥ trr Pr♠tr③t♦♥ ♦ t srs ♦r ♠s♥ s♥♥s tt♥♥ ♥♥r♥ t ♦♠♣trs

r ♥ trr ♠♦♦t♥ ♥ ♦r② r t♦ ♠♥♠③ ♥r st♦rt♦♥ ♥t①tr ♠♣♣♥ r♥st♦♥s ♦♥ r♣s

r ♥ rt ♠str ♥♦♥s♣♦s ♦st♦rt♦♥ t①tr s♠ ②♦t♥ Pr♦♥s ♦ ❱s③t♦♥ ♣s ♦♠♣tr ♦t②

r ♦ts♠♥ ♥ ②♥ ♦st s♣r ♣r♠tr③t♦♥ ♦ tr♥r♠ss ♥ ♦♥r ②♥ r ♥ ♠r t♦rs Pr♦♥s ♦t t sr♦r t♦♥ ♦♥r♥ ♦♥ ♦♠tr ♦♥ ♥ ♦♠♣trr♣s ♣s sr

r é② ♦♥ts② ♥ ♦♦♠②♦ st ♥ r♦st♥ s tt♥♥ r♥st♦♥s ♦♥ r♣s

r Pr♥ ♥ ♦s s ♣r♠tr③t♦♥ ♠t♦s ♥ tr ♣♣t♦♥s ♦♥t♦♥s ♥ r♥s ♥ ♦♠♣tr r♣s ♥ ❱s♦♥

♥ ♥tr♦t♦♥ t♦ t ♦♥t ♠t♦ t♦t t ♦♥③♥ ♣♥♥ r♣♦rt ♦♦ ♦ ♦♠♣tr ♥ r♥ ♦♥ ❯♥rst② tt♣s♠⑦q♣♣rs♣♥ss♦♥tr♥t♣

♠ ♥ ❨ ♣♣r♦①♠t tr♥s♦r♠t♦♥ ♦ ♥ rtrr② r sr♥t♦ ♣♥ s♥ ②♥♠ ♣r♦r♠♠♥ ♦♠♣tr s♥

s♦♥ t♦r ♥tt② ♦r t rt tsst♦♥ t♠t Pr♦♥s ♦t ♠r P♦s♦♣ ♦t②

s♦♥ r sr♣t♦♥ ♦ ♥tr ♥♦r ♥tr♣♦t♦♥ ♥ ❱ r♥tt t♦r♥tr♣♦t♥ trt t ♣s ❲② ❨♦r

♦r P ♥ ♥ ♥s rr ♣ttr♥ ♠♣♣♥ r♥st♦♥s♦♥ r♣s Pr♦♥s ♦ P

♦r♥ ♦♥r ♦♥t ♥ s♥s ♦♥st♦rt♦♥ ♣s ♠s ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ ❱s③t♦♥ ♣s ♦♠♣tr ♦t②

♦r♥ ♦♥r ❨ ♣♠♥ ① öss ♥ P ♣♥sr t♥ ♥ Pr♦♥s ♦ t ♦♥ r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr②Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

t♥r ♥ sr P♥r ♣r♠tr③t♦♥ ♦r ♦s ♠♥♦ ♥sg ♠sss♥ ♥② t②♣ ♦ ♣♦st ts ♦r♥ ♦ ♦♠♣t♥ ♥ ♥♦r♠t♦♥ ♥ ♥♥♥r♥

♠r ♥ s ♥t ♥s ♥ t ♦♥strt♦♥ ♦ ♣♦②♦♥ ♥t♠♥t ♥tr♣♦♥ts rs ♦ ♦♠♣tt♦♥ t♦s ♥ ♥♥r♥

❲ ♠♥r ♥ P♦♣♦ ♦r♠t♦♥ tr♥sr ♦r tr♥ ♠ss r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

❱ r③s② ♥ ♦ts♠♥ ①♣t sr r♠s♥ ♥ Pr♦♥s ♦ t rstr♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

r♥ ♦r♠♥♥ P ♥♦♥ ♥ ♦♥t♥ P♦②♣s r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦ P

♥ s♥ ♣r♦ss♥ ♣♣r♦ t♦ r sr s♥ ♥ Pr♦♥s ♦ P ♣s Prss

♥♥♠ ❱ ♥ ♥♦r ♦ ♦♠tr r♠♦r ♦r♥♦♥♥r ♠♥s♦♥t② rt♦♥ ♥

r ♥②r P ❱ ♥r ♦rtr ♥ ♦♣♣ ♥s♣③♣r♠tr③t♦♥ ♦r ♣s ♥r r♦♥strt♦♥ ♥ Pr♦♥s ♦ t ♦♥r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

❨ ♦♥ P ③ ♦♥t♥r ♥ sr♥ s♥♥ qr♥t♦♥st srt r♠♦♥ ♦r♠s ♥ Pr♦♥s ♦ t t r♦r♣s ②♠♣♦s♠ ♦♥♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

r ①tr s②♥tss ♦♥ srs ♥ Pr♦♥s ♦ P ♣s Prss

❲ tt ♦♥① r♣rs♥tt♦♥s ♦ r♣s Pr♦♥s ♦ t ♦♥♦♥ t♠t♦t②

❲ tt ♦ t♦ r r♣ Pr♦♥s ♦ t ♦♥♦♥ t♠t ♦t②

❱t ♥ é② ♣tr ♦♠tr② ♣r♦ss♥ t ♠♥♦ r♠♦♥s ♥r♣♦rt

❲s♣rss t♦♥ ♥t ♠♥t ss ♠ Prss ❨♦r

❲♥ ♠t ♥ ❨♥ r tt♥♥ s ♦♥ ♥r②♠♦ ♦♠♣tr s♥

❲rr♥ r②♥tr ♦♦r♥ts ♦r ♦♥① ♣♦②t♦♣s ♥s ♥ ♦♠♣tt♦♥t♠ts

❲rr♥ ♥ t ♥q♥ss ♦ r②♥tr ♦♦r♥ts ♥ ♦♠♥ ♥ rsss t♦rs ♦♣s ♥ r ♦♠tr② ♥ ♦♠tr ♦♥ ♦♠ ♦ ♦♥t♠♣♦rr② t♠ts ♣s ♠r♥ t♠t ♦t②

❲rr♥ r r♥ ♥ sr♥ r②♥tr ♦♦r♥ts ♦r♦♥① sts ♥s ♥ ♦♠♣tt♦♥ t♠ts ♦ ♣♣r

❨ ❲ ♥ ♦② ①tr s②♥tss ♦r rtrr② ♠♥♦ srs ♥ Pr♦♥s ♦ P ♣s Prss

❨♥ ♥ ❩♦r♥ s♠♣ ♠♥♦s ♦♥strt♦♥ ♦ srs ♦ rtrr②s♠♦♦t♥ss r♥st♦♥s ♦♥ r♣s Pr♦♥s ♦P

❨♥ rt③♠♥♥ r♠♥♥ ♥ ❩♦r♥ ①tr ♥ s♣ s②♥tss ♦♥srs ♥ ♦rtr ♥ ②s③♦s t♦rs ♥r♥ ♥qs ♣s ♣r♥r ❲♥ ❨♦r Pr♦♥s ♦ t t r♦r♣s❲♦rs♦♣ ♦♥ ♥r♥

❨♦s③ ② ♥ P st ♥ s♠♣ strt♠♥♠③♥♠s ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣♦♥ ♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

❲ ❨♦♥ t♠♥s♦♥ s♥ ♥ ♦t③ ♦♥s♦♥ ♥ t♦rs ♥②♦♣ ♦ ttst ♥s ♦♠ ♣s ❲② ❨♦r

❩②r öss ♥ P ♦♥① ♦♥r② ♥ s tt♥♥ ♥ rt r♦ r♥r ♠♥♥ P t♥ ♥ ❲str♠♥♥ t♦rs Pr♦♥s ♦ ❱s♦♥ ♦♥ ♥ ❱s③t♦♥ ♣s ü♥♥ r♠♥② ♦ ♥①

❩②r öss ♥ P srt t♥s♦r qsr♠♦♥ ♠♣s ♥ Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣ ♦♥ ♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

❩②r öss ♥ P tt♥ t ♦♥r② r ♦♠♣♦st ♣♣r♦t♦ sr ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ t r r♦r♣s ②♠♣♦s♠ ♦♥♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

❩②r öss ♥ P ❱rt♦♥s ♦♥ ♥ s tt♥♥ ♥ ♦s♦♥ ♦tr ♥ ♥ t♦rs ♥s ♥ trs♦t♦♥ ♦r♦♠tr ♦♥ t♠ts ♥ ❱s③t♦♥ ♣s ♣r♥r r♥r

❩②r öss ♥ P r♥r s♣r ♣r♠tr③t♦♥ ♥ Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣ ♦♥ ♥ ♣♣t♦♥s ♣s ♦♠♣tr ♦t②

❩♥ s♦ ♥ r trs sr ♣r♠tr③t♦♥ ♥t①tr ♠♣♣♥ r♥st♦♥s ♦♥ r♣s

❩♦ ②♥r ♦ ♥ ❨ ♠ s♦rts trtr♥ ♠s ♣r♠tr③t♦♥ s♥ s♣tr ♥②ss ♥ Pr♦♥s ♦ t ♦♥ r♦r♣s ②♠♣♦s♠ ♦♥ ♦♠tr② Pr♦ss♥ P ♣s r♦r♣s ss♦t♦♥

❩♦ ❳ ❲♥ ❨ ♦♥ sr♥ ♦ ♥ ❨ ♠ ①tr♦♥t♠ss t①tr♥ ♦ rtrr② srs r♦♠ ♠t♣ ♠s r♥st♦♥s ♦♥r♣s Pr♦♥s ♦ P

❩♠♥ ♠♠ ♥ r②t ①tr ♠♣♣♥ s♥ sr tt♥♥ ♠t♠♥s♦♥ s♥ r♥st♦♥s ♦♥ ❱s③t♦♥ ♥ ♦♠♣trr♣s