Green ìÜD@à

Šœþ

“bç³ÓÜuÜä, ÓܳbçuEä”

1. ‡k:

ç3bçí%ðµsAГbçu'ñ

qí” ¥w2íŸÄ4uÄÑBbFçí

bçuì2, ìÜ, „p, ¥ú¨íbç

³œ, ÿˇòh, à¤íbçà‹5.

H¢Ð2B´öí6, Ê¥¹dıBb

øJòhíi*SDÓÜíhõVn

Green ìÜ

!…ìܪzu|½bí

!‹5ø, 7(uø&íR,Ĥ

Bb½ú(u´óéNí!‹:

“(D½ (double in-

tegral) 5É[ÑS?”

w!u'ìí—Green ìÜ:

•O¥ £ (C5(ª“

ÑCFˇ– R 5øO½

(Þ)

¥_ìÜ|o|ÛuâLÅAB`>íbç

ÓÜçð George Green (1793- 1841)

k 1828'û˝Úç (electricity) D~ç

(magnetism) FêÛí, çÍ(Vòg

(Gauss) $A6êÛ¥!‹, Green í!‹

éBbg)Ý/5EU, 6yñ‚IF

k:

bçunj×AÍíxk

çbçJø−ø<ÓÜég (5) ¯±

.>ƒ‚À

Green ìÜubç&2|½bíì

Ü5ø, 7Êú&Dyò&˛ÈíR—

Stokes ìÜDàìÜ (Divergence The-

orem) †ZA7@àbçí!

2. !…ìÜ:

DíÉ[¥uí3bl

iù, õÒ,¥£uâDqÓ−cú

|½bíõ., N¬¥_½b!‹—

!…ìÜ (fundamental theorem of cal-

culus), BbpëDõÒ,uÑ

øñsÞ, ˛¤uóªLí (inverse)

!…ìÜ: f : [a, b] → R Ñø

©/ƒb/ F ′ = f †

∫ b

aF ′(x)dx =

∫ b

af(x)dx = F (b)− F (a)

(1)

¥ìܵsBbbl4M

25

26 bçfÈ 21»4‚ ¬86'12~

∫ baf(x)dx cÛ°)ƒb f 5Ÿƒb (prim-

itive, C antiderivative) F ¹ªÇÕ6z

p7?-59õ:

“ø_ƒb5íMkv

ƒb5iäMíÏ”

²Æuzj˙ (1) z–ÈíDTà

kw“É&” (zero dimension) iä5,í

“” (É&íuvõ5M) ©û–V,

¥É&íiäus_«õ a D b

f(x)=F ′(x)

a b

!…ìÜ5ÓÜ<2:

t (1)BbªJªÒà-: àÇFý,

cqø;òí<äwiÞ$k Auì

.‰, ®Ê<q¼D¼§ÑF (x), ÇÕ

âk<7Ýêr¥£Ä¤<7û¶°v6

®Ñp (CÑ|), Bb;½íu¤Ñp0Ñ

Öý?(cq®íòá@$k1)Bb¦w2

ø¨ [x, x+∆x] Võ, ÊÀPvÈqÑpƒ

¥¨<äí®¾.â$k•<äj²¼|í

®¾D¼ªí®¾5Ï

(F (x+∆x)−F (x))× A (ò × ?Þ)(2)

ĤÀP<Åq®íÑp0Ñ

f(x) = lim∆x→0

(F (x+∆x)−F (x))A

∆x ·A(®¾

ñ

)

= lim∆x→0

F (x+∆x)− F (x)

∆x= F ′(x) (3)

Juc_–È[a, b]†r¶Ñp<q5®¾Ñ∫ b

af(x)dx =

∫ b

aF ′(x)dx

]∫ b

af(x)dx =

∫ b

aF ′(x)dx = F (b)− F (a)

âJ,5&ªø!…ìÜ5Ó

Ü…”ÿuè0 (conservation law)

3. (5ÓÜ<2:ø”õ§ø‰“í‰Tà7•ø˛ø

(G, 7°wFTíŠ (work), ÿAÍû

_F‚í(

NÞ,L<²¾ F = (u, v), 7

w•O(~²¾ (cos τ, sin τ), ¶²¾

(cos ν, sin ν) 5¾MÑ

Fτ = F · (cos τ, sin τ)=u cos τ+v sin τ

Fν = F · (cos ν, sin ν)=u cos ν+v sin ν

(4)

ÄÑ τ, ν Ñì

cos τ = − sin ν, sin τ = sin ν (5)

Green ìÜD@à 27

7ÀP~²¾N ÀP¶²¾Ñ

(cos τ, sin τ) =(dx

ds,dy

ds

),

(cos ν, sin ν) =(dy

ds,−dx

ds

)(6)

à‹ø F eщ†(

∮CFτds =

∮Cudx+ vdy (7)

(‰× PG=Š)

FH[í<2ÿuŠ (work) wŸuJø F

eÑÚ¼ò (current density) C¼ñ§

†(

∮CFνds =

∮Cudy − vdx (8)

(Ú¼ × PG=Ú¦¾)

FH[í<2ÿuÚ¦¾ (flux) C¼ñ¼¦

¬(C5¦¾,BbÊ(Þ´‡ú¥s_

¾Ty¿p5«n

4. Green ìÜ

GreenìÜ!…,u(DÞ5

É[, õÒ,ÿu!…ìÜ5R

Green ìÜ: ICÑÞ,ø¨Ëí

¥£(7wFˇ–Ñ R, cqƒb

P (x, y), Q(x, y) Ñ©//øŸRûb6©

/†A∮CPdx+Qdy =

∫ ∫R

(∂Q

∂x−∂P

∂y)dA

=∫ ∫

R

∣∣∣∣∣∣∂∂x

∂∂y

P Q

∣∣∣∣∣∣ dA (9)

bçíi:

âk( C 퉓ª×ªüøO7

k1³péí¡b, Ĥ̶òQ°(

, Í7Bbª‚àV¡ (approxima-

tion) í–1VTÜ, ¥£ubç«wu&

(analysis) í3bxÍ

Åj$ =⇒ Öi$ =⇒ RI. RÑøÅj$

R5iä C = C1 + C2 + C3 + C4

C1 : a0 ≤ x ≤ a1, y = b0

C2 : x = a1, b0 ≤ y ≤ b1

C3 : a0 ≤ x ≤ a1, y = b1

C4 : x = a0, b0 ≤ y ≤ b1

Ĥ‚à!…ìܪ)∮CF · dr=

∮CP (x, y)dx+Q(x, y)dy

28 bçfÈ 21»4‚ ¬86'12~

=∫ a1

a0

P (x, b0)dx+∫ b1

b0Q(a, y)dy

+∫ a0

a1

P (x, b1)dx+∫ b0

b1Q(a0, y)dy

=∫ b1

b0[Q(a, y)−Q(a0, y)]dy

+∫ a1

a0

[P (x, b0)− P (x, b1)]dx

=∫ b1

b0

∫ a1

a0

∂Q

∂xdx dy −

∫ a1

a0

∫ b1

b0

∂P

∂ydx dy

=∫ a1

a0

∫ b1

b0(∂Q

∂x− ∂P

∂y)dx dy

II.cq R ª[Ñ

R=(x, y)|a≤x≤b, g1(x)≤y≤g2(x),C=∂R

BbMTÜ P (x, y), Q(x, y)∮CP (x, y)dx

=∮

C1

P (x, y)dx+∮C2

P (x, y)dx

+∮

C3

P (x, y)dx+∮C4

P (x, y)dx

=∫ b

aP (x, g1(x))dx+

∫ b

aP (x, g2(x))dx

=−∫ b

a

∫ g2(x)

g1(x)

∂P

∂y(x, y)dy dx

°Üª)∮CQ(x, y)dy =

∫ b

a

∫ g2(x)

g1(x)

∂Q

∂xdy dx

s6¯9∮CPdx+Qdy =

∫∫R(∂Q

∂x−∂P

∂y)dA

=∫∫

R

∣∣∣∣∣∣∂∂x

∂∂y

P Q

∣∣∣∣∣∣ dA¥ÿu Green ìÜ, TX7BbÉk

(DÞ5É[, O¥_V¡j¶

_Ì„ÿu(C.âuÅ( (recti-

fiable curve), 7/´bàƒø_Y¹ (uni-

formly convergent) í–1úkyøOí–

Green ìÜEÍuúí, O˛%¬

í¸ˇ, è6Eª¡5SjÞ

íz

Wæ 1: ‚à Green ìÜl(∮C(2xy − x2)dx+ (x+ y2)dy

w2( C uâÓ( y = x2 Dò(

y = x Fˇ–5iä

j: bòQl4(‘.bø C “

Ñ¡b, Í7‚à Green ìܦ

P (x, y) = 2xy − x2, Q(x, y) = x+ y2

†∂Q

∂x− ∂P

∂y= 1− 2x

Ĥ

Ÿ(=∫∫

R(1− 2x)dA

=∫ 1

0

∫ x

x2(1− 2x)dy dx = 1

30

Green ìÜD@à 29

Green ìÜzpø¥£(C5(

DCFˇ–5Þ (½) 5É

[, ĤÊÔy8$5-, (5S<2

Ñ“Þ” : Wà¦

∂Q

∂x− ∂P

∂y= 1

†â Green ìÜø∮CPdx+Qdy =

∫∫R1 dA = |R|

¦/ª¦P , Q à-:

P = 0, Q = x

P = −y, Q = 0

P = −12y, Q = 1

2x

Í:–Râ¨Ëí¥£(C F

ˇA, wÞÑ:

|R|= 12

∮C−y dx+ x dy

=−∮

Cy dx =

∮Cx dy (10)

Wæ2: t°Æ x2

a2 +y2

b2= 1 5Þ

j: øÆ[Ñ¡b

x= a cos θ, y = b sin θ,

0 ≤ θ ≤ 2π

‚àÍ5!‹)

A=∮

Cx dy

=∫ 2π

0(a cos θ)(b cos θ dθ)

=1

2ab∫ 2π

0(1 + cos 2θ) dθ = πab

ÑOy¿p«n GreenìÜ,Bbyl

4ø_Wæ1*w2)ƒÑ>

Wæ3: ˛ø C ÑL<¥£NË(,

t°(

∮C

−ydx+ xdy

x2 + y2(0, 0) ∈ C

j: ílcq (0, 0) 1.¨ÖÊ C 5

q¶, †

P (x, y) =−y

x2 + y2Q =

x

x2 + y2

=⇒ ∂Q

∂x− ∂P

∂y= 0

Ĥâ Green ìÜø

∮C

−ydx+xdy

x2 + y2=∫∫

R0 dA=0

wŸ, J (0, 0) Ê C 5q¶, ¤v (0, 0) Ñ

øJæõ (singularity), s¥õíj¶ÿ

u “fÇ…”,Tø_šÑρ,J (0, 0)ÑÆ

-íÆ Bρ 5?–

R′ = R−Bρ, ∂R′ = C + ∂Bρ

âkR′1.¨Ö (0, 0), Ĥ‚à‡ÞíR

; ∮∂R′

−ydx+ xdy

x2 + y2= 0

30 bçfÈ 21»4‚ ¬86'12~

O ∂R′ = C + ∂Bρ, ]∮C

−ydx+xdy

x2+y2=∮

∂Bρ

−ydx+xdy

x2+y2

=∫ 2π

0

(−ρ sin θ)(−ρ sin θdθ+(ρ cos θ)(ρ cos θdθ)(ρ cos θ)2+(ρ sin θ)2

=∫ 2π

01 dθ = 2π

Å1: ¤TMÑ2π[ý(C ÷7Jæ

õ (0, 0) ø˛, 7M$k0†u³

÷ƒ (0, 0), ¥Fú@íZuµ‰ƒ

bÜí÷b (winding number), Ê

¼ñ‰ç†u=¼ (circulation)

Å2: „pí¬˙2BbêÛ•O( C 5

($k•Oƶ ∂Bρ 5(, ¥

³Þíbç…”ÿu°OÜ (homo-

topy theory), Ĥ GreenìܪR

ƒÀ©¦– (simply connected re-

gion), 7¥£uµ‰ƒbû˝íø½

b3æ, °v6zpµÆí(5(

ª“ÑWÀí(5( (WàÆ

¶í(), ¥ÿubçíXÿ—–ø

µÆí½æ“ÑWÀí½æ

Å3: J C1, C2 ÑLù‘.ó>í¨NË

¥£(7/·÷¬Ÿõ(0, 0), †

∮C1

−ydx+ xdy

x2 + y2=∮

C2

−ydx+ xdy

x2 + y2

¥Î7°OÜ5Õ, òQí<2ÿu

v(úk$‰ (deformation) u

ø.‰¾ÇÕBbªN¬”è™ (po-

lar coordinate) Võ; I

θ = tan−1 y

x

†\ƒb (integrand) AÑ

dθ =−ydx+ xdy

x2 + y2

Ĥ¥_(õÒ,ÿuÊ¿¾•O

(C(Lv‡j²) i5‰“¾, ç

ÍJu÷7ø˛†w‰“¾Ñ 2π, J

ußv‡j²÷7ø˛†w‰“¾Ñ

−2π, ¥_–1ÿu‡ÞFzí÷b

(winding number)

÷b= 1 ÷b= 0 ÷b= −1

Green ìÜD@à 31

ÓÜíi:

Green ìÜ6ªN¬ÓÜíiVw

…, IRÑNÞ,íNË( C Fˇ5À©

¦–, q Cª[Ñx = x(t), y = y(t) 5

¡b, 7²¾

F (x, y) = (P (x, y), Q(x, y)) (11)

[ý¼ñí§,Bb;l4¼ñ%¬iäC

5¦¾ (flux), "Î(ø( C Ñ

Jßü¨7õw2ø¨, ílu x ¾¦¬

si 5¦¾ (é(¶5Þ) Ñ

Pi(x, y) si cosαi w2 αi ÑÕ¶²¾ ν

D x W5Hi, yø®¶Mr¶‹–V1‚

à Riemann ¸ø x W¶5¾Ñ∮CP (x, y) cos(ν, x) ds (12)

°ÜyW5¾Ñ∮CQ(x, y) cos(ν, y) ds (13)

Ĥr¶5¦¾Ñ∮C[P (x, y) cos(ν, x)+Q(x, y) cos(ν, y)]ds

=∮

C[P (x, y)

dy

ds−Q(x, y)

dx

ds] ds

=∮

CP (x, y) dy −Q(x, y) dx (14)

=∮

CF · νds, ν =

(dy

ds,−dx

ds

)

ÇøjÞBbõüä$, âkâ¥ä$˝V

òi,í¼§Ñ P (x, y),ĤÀPvÈq

P (x, y) y í®¼p, 7°øvȆ_

P (x + x, y) y í®¼|, FJ•x Wj

²5ÀP´¼¾Ñ

[P (x+∆x, y)− P (x, y)]∆y

∆x∆y

I ∆x → 0, )”Ì ∂P/∂x °Ü• y W

5ÀP´¼¾Ñ ∂Q/∂y ĤÀP´¼¾ Ñ

∂P∂x+ ∂Q

∂y7 ¦¬c_– R 5r¶¦¾Ñ∫∫

R

(∂P

∂x+

∂Q

∂y

)dxdy (15)

ÄÑ®u.ª9ò (cqí), °øvÈí®

¾.â*iäC ¼| (”¾è0), ]

∮CP dy −Qdx

=∫∫

R

(∂P

∂x+

∂Q

∂y

)dxdy (16)

¥yŸzp Green ìÜ5ÓÜ<2Ñ “è0

”

P (x, y)→ ∆y → P (x+∆x, y)

x ∆x x+∆x

.................................

.................................

32 bçfÈ 21»4‚ ¬86'12~

5. ²¾$5 Green ìÜ

â (9) D (16) s, BbAÍ7Íù

ª²¾í–1:

F (x, y) = (P (x, y), Q(x, y)) = Pi+Qj

(²¾Ò)

τ =(dx

ds,dy

ds

)(17)

(ÀP~²¾)

ν =(dy

ds,−dx

ds

)(ÀPÕ¶²¾)

ÇÕ²¾Ò F íì (curl) Ñ

curlF =∇× F =

∣∣∣∣∣∣∣∣∣

i j k∂∂x

∂∂y

∂∂z

P Q 0

∣∣∣∣∣∣∣∣∣=

(∂Q

∂x− ∂P

∂y

)k (18)

à (divergence) †Ñ

divF = ∇ · F = ∂P

∂x+

∂Q

∂y(19)

FJ Green ìÜ ((9), (16)) ªZŸÑ∮CF · dr=

∮CF · τds=

∫∫R∇× F · kdA

(20)∮CF · ν ds =

∫∫R∇ · F dA (21)

M[ý~²¾D¶²¾$í Green ì

Ü, 7wÓ܆Mu“ Š” (work) D“ ¦

¾” (flux), (20), (21) sªRByò&

b˛È,ÿu¦˚í Stokes ìÜDàìÜ

(Divergence Theorem)

ÑS curlF,∇F \˚ÑìDà

á? Bb5?Ôyí– R uJ (x0, y0)

ÑÆ-, šu r íÆ, †â©/4ø

∇×F (x0, y0)·k = limr→0

1

πr2

∮CF · τds

∇F (x0, y0) = limr→0

1

πr2

∮CF · νds (22)

FJìuÀPÞq5|×=¼ (circula-

tion), 7à†uÀPÞq5¦¾ (¼¾

5‰“), ¥_$|×íßT.§è™í

à, à (18), (19) s, 7/N¬ (22)

(¹‚à Green ìÜ) Bbª„pì¸à

uDè™ÌÉ

Å1: à‹ F = ∇φ = (φx, φy), ²¾ (vec-

tor) F ÑÓ¾ (scalar) φ5G (gra-

dient) † (21) ªZŸÑ

∮C

∂φ

∂νds =

∮C∇φ · ν ds

=∫∫

R∇∇φ dA =

∫∫R∆φ dA (23)

ƒbφ˚Ñ P?ƒb (potential func-

tion) 7

∆ = ∇ · ∇ = ∂2

∂x2+

∂2

∂y2(24)

†u|±í Laplace 4ä, ¥uR

j˙2|½bí4ä

Å2: J¦F = u∇v, †â²¾5l4)

∇·F =∇·(u∇v)=u(∇·∇v)+∇v·∇u

Ĥ (21) AÑ Green dø$

∮Cu∂v

∂νds

=∫∫

R[u∆v +∇u · ∇v] dA (25)

Green ìÜD@à 33

J u, v ²,(sóÁ†ª) Green

dù$

∮C

∣∣∣∣∣∣∣∣∣u v

∂u∂ν

∂v∂ν

∣∣∣∣∣∣∣∣∣ds =

∫∫R

∣∣∣∣∣∣u v

∆u∆v

∣∣∣∣∣∣ dA(26)

¥<$ÊRj˙2rÆOÝ/½

bíiH

Å3: GreenìÜí|ßTÜG¶u‚à

$ (differencial forms), BbWÀz

pà-: (∮CP dx+Qdy

5\ƒb (integrand) Ñø¼$

(first order differential form)

L = P (x, y)dx+Q(x, y)dy (27)

L5r (total differential) Ñ

dL= dPdx+ dQdy

= (Pxdx+ Pydy)dx

+(Qxdx+Qydy)dy

= (Qx − Py)dxdy (28)

Ĥ Green ìܪ½hŸÑ$

(differential form)∮CP dx+Qdy

=∫∫

R

(∂Q

∂x− ∂P

∂y

)dxdy

⇔∮

CL =

∫∫RdL (29)

¥_t°v6zpÊ–R5£wiä5s65É[, 7õÒ,ÿ

u!…ìÜ5R, çÍwXÿ

†u — ¶ (integration by

part)

6. (í!…ìÜ

ÊdügBbùª7P?ƒb (poten-

tial function)F = ∇φ5–1, õÒ,ª%

âr (total differential) VÜjF · dr = Pdx+Qdy

dφ = φxdx+ φydy

⇒ F · dr = ∇φ · dr = dφ (30)

Ĥ

P = φx Q = φy, ⇒ Py = Qx

²Ñj˙íxk†u “£¯ (exact) ”

(í!…ìÜ: cq φ Ñø

ªƒb,/wG ∇φ Ñ©/, r Ñ©Q

a,b sõ5L<(, †

∫ b

a∇φ · dr = φ(b)− φ(a) (31)

¥ìÜíÓÜ<2 (Úçíi): •O

ÚÒ,sõÈLø(ÚÒFTíŠÑsõ

íÚPÏ, â Green ìܪøJ F = ∇φ,

C1, C2 Ñ©Q a,b 5L<s‘(†∮CF · dr =

∫C1

F · dr −∫

C2

F · dr

=∫∫

R∇× (∇φ) · k dxdy = 0

Ĥ ∫C1

F · dr =∫

C2

F · dr

34 bçfÈ 21»4‚ ¬86'12~

¹(D˜ÌÉ (path-independent),

²Æuz, øøÓñ (C”õ) âP0 a G

BP0 b ‰úÓñ (”õ) FTíŠ, c¸Ó

ñ (”õ) –õP0£@õP0É, 7Dw

«Fcí«˜ÌÉ, ¤vBb˚²

¾Ò F Ñ\èí (conservative), 7φ†˚

ÑF5P?ƒb, ¥ìÜ6µsBb

φ(x)− φ(a) =∫ x

a∇φ · dr =

∫ x

aF · dr(32)

¹P?ƒbªâ\è‰í(7)

?ìÜ:

r(t) = (x(t), y(t)) ªeÑP0²¾ƒ

b, †§D‹§Mu:

v(t) =dr

dt=

(dx

dt,dy

dt

)

a(t) =d2r

dt2=

(d2x

dt2,d2y

dt2

)(33)

qÓñí”¾Ñ m, wF§Õ‰í¯‰Ñ F

†ââìø

F = ma = md2r

dt2(34)

†ú F 7k* r(t1) B r(t2) FTíŠÑ

W =∫ r(t2)

r(t1)F · dr

=∫ t2

t1F (r(t)) · dr(t)

dtdt

=∫ t2

t1mdv

dt· v(t)dt

=1

2m∫ t2

t1

d

dt

(|v(t)|2

)dt

=1

2m |v(t2)|2 − 1

2m |v(t1)|2 (35)

w2 12m|v(t)|2 Ñ? (kinetic energy),

Ĥ (35) ÿu“?ìÜ”, wÓÜ<2:

¯‰úÓñFTíŠk ?íZ‰¾

à‹ F Ñø\è‰Ò, F = −∇φ, †â(

í!…ìÜø:

W = −φ(t2) + φ(t1) (36)

† (32) ZŸÑ

φ(t1) +1

2m|v(t1)|2

= φ(t2) +1

2m|v(t2)|2 (37)

w2 φ ÑP? (potential energy), Ĥ

(37) ÿu?¾è0ì (conservation of

energy)

à5ÓÜ<2:

7§í–15(, Bbg)¥u'

ßívœV«„à (divergence) íÓÜ

Green ìÜD@à 35

<2, BbªJ¥ó;d: cq£Êø¯

øWXJp*q, |n$AíÇ$Ñ Ω0, 7

(%âµ.CwFÄÖU) Ω0 ‰“, wP

0²¾Ñ (x(t), y(t)) = (ξ, η), §†Ñ

v(t) =(

dxdt, dy

dt

)= (u, v), %¬ t vÈ5

( Ω0 o²Ñ Ωt, BbEí½æu Ω0

D Ωt, s65Þ‰“ÑS? (õÒ,â

‰ç…™ÿuø虉²) Bbõw2ø

ü) (dV0 → dVt) s6É[Ñ

dV0 = dxdy =∂(x, y)

∂(ξ, η)dξdη

= J(t)dξdη = J(t)dVt (38)

w2 J(t) ÿu Jacobian

J(t) =

∣∣∣∣∣∂(x, y)∂(ξ, η)

∣∣∣∣∣ =∣∣∣∣∣∣

∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

∣∣∣∣∣∣=dV0

dVt(39)

Ĥ Ω0 → Ωt 5‰“8$, $ku«n

J(t) 퉓, Bbcq J, J−1 = 0, ¹³

¢“í8$ (ªq 0 < J < ∞): âr

(total differential) ø

d

dt

(∂x

∂ξ

)=

∂

∂ξ

(dx

dt

)

=∂u

∂ξ=

∂u

∂x

∂x

∂ξ+

∂u

∂y

∂y

∂ξ

°Üª)

d

dt

(∂x

∂η

)=

∂u

∂x

∂x

∂η+

∂u

∂y

∂y

∂η

d

dt

(∂y

∂ξ

)=

∂v

∂x

∂x

∂ξ+

∂v

∂y

∂y

∂ξ

d

dt

(∂y

∂η

)=

∂v

∂x

∂x

∂η+

∂v

∂y

∂y

∂η

J(t) ú t 1‚àWq54”ª)

dJ

dt=

∣∣∣∣∣∣ddt

∂x∂ξ

ddt

∂x∂η

∂y∂ξ

∂y∂η

∣∣∣∣∣∣+∣∣∣∣∣∣

∂x∂ξ

∂x∂η

ddt

∂y∂ξ

ddt

∂y∂η

∣∣∣∣∣∣=

∣∣∣∣∣∣∂u∂x

∂x∂ξ+ ∂u

∂y∂y∂ξ

∂u∂x

∂x∂η+ ∂u

∂y∂y∂η

∂y∂ξ

∂y∂η

∣∣∣∣∣∣+

∣∣∣∣∣∣∂x∂ξ

∂x∂η

∂v∂x

∂x∂ξ+ ∂v

∂y∂y∂ξ

∂v∂x

∂x∂η+ ∂v

∂y∂y∂η

∣∣∣∣∣∣=

∣∣∣∣∣∣∂u∂x

∂x∂ξ

∂u∂x

∂x∂η

∂y∂ξ

∂y∂η

∣∣∣∣∣∣+∣∣∣∣∣∣

∂x∂ξ

∂x∂η

∂v∂y

∂y∂ξ

∂v∂y

∂y∂η

∣∣∣∣∣∣=

∂u

∂x

∣∣∣∣∣∣∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

∣∣∣∣∣∣+∂v

∂y

∣∣∣∣∣∣∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

∣∣∣∣∣∣=

(∂u

∂x+

∂v

∂y

)J(t) = ∇ · vJ(t)

FJ J(t) Å—ø¼j˙;

EulerÇt:

dJ

dt=(∇ · v)J ⇐⇒ d lnJ

dt=∇ · v (40)

¥_t˚Ñ Euler mÇt (Euler ex-

pansion formula), âj˙ø

J(t) = et(∇·v)J(0) (41)

36 bçfÈ 21»4‚ ¬86'12~

â¤éͪ)

∇ · v > 0⇔ J(t) > J(0)⇔ Þ‰×

∇ · v = 0⇔ J(t) = J(0)⇔ Þ.‰

∇ · v < 0⇔ J(t) < J(0)⇔ Þ‰ü

ĤBb˚ø¼ñÑ.ª9ò (incom-

pressible) wöõ<2 :

¼ñ%¬LS퉲w$ÕÖÍ

Z‰7, OwÞ (Cñ) ºá

\M.‰

ì5ÓÜ<2:

âWæ3í%ðø(

∮CF · dr =

∮Cudx+ vdy

=∫∫

R(vx − uy)dA

Ѳ¾Ò F = (u, v) =÷¥£( C 5=

¼, ‚à$

vx−uy = curlF · k = ∇× F · k

=

∣∣∣∣∣∣∣∣∣

i j k∂∂x

∂∂y

∂∂z

u v 0

∣∣∣∣∣∣∣∣∣· k

Bbªø‡ø[Ñ

∮CF · dr =

∫∫R(vx − uy)dA

cqRÑ¨Ö P0(x0, y0) 5–, †

1

|R|∮

CF · dr

[ýÀ“Þ5=¼, Ĥ

curlF · k =∇× F · k= lim

R→P0

1

|R|∮

CF · dr

= limR→P0

1

|R|∫∫

R(∇× F · k)dA

FJ²¾Ò F 5ì ∇ × F (Ê (x0, y0)

5ì) ÿuÀPÞßÞ|×=¼í¾/

wj²òk R

7. Green ìÜʵ‰ƒb5

@à :

@àø:

Cauchy ìÜ: R ѵbÞ,5À©

¦– (simply connected domain), f Ñ

ì2Ê R,5ÀM (single value) ©/ª

/j&íƒb, †∮Cf(z)dz = 0 (42)

w2 C ѨÖÊ R qL<¨Ë5 Jor-

dan (

j: cqf = u + iv, z = x+ iy †µ

‰ (õÒ,ÿu() Ñ∮Cf(z)dz =

∮C(u+iv)d(x+iy)

=∮

Cudx−vdy+i

∮Cvdx+udy

Green ìÜD@à 37

O f ª/

f ′(z) =∂u

∂x+ i

∂v

∂y

=∂v

∂x− i

∂u

∂y

ÄÑ u, v 5RûbÑ©/, Ĥª‚à

Green ìÜMTàƒõ¶D™¶)∮C(udx−vdy) =

∫∫R

(−∂v

∂x−∂u

∂y

)dx dy

∮C(vdx+udy) =

∫∫R

(∂u

∂x− ∂v

∂y

)dx dy

â Cauchy-Riemann j˙ (f Ñj&ƒb)

∂u

∂x=

∂v

∂y,

∂u

∂y= −∂v

∂x

ø,Þs_ÑÉ, ]ª!∮Cf(z)dz = 0

µ‰íÓÜ<2ªâdúgín

7), " Cauchy ìÜ5„p:∮Cf(z)dz

=∮

C(u− iv)d(x+ iy)

=∮

Cudx+ vdy + i

∮C−vdx+ udy

=Š(work) + i¦¾(flux) (43)

çͪJ‚àGreenìÜø5“Ñù½

(ªœ (9),(16))∮Cf(z) dz

=∫∫

R

(∂v

∂x− ∂u

∂y

)dA

+i∫∫

R

(∂u

∂x+

∂v

∂y

)dA

=∫∫

R(curlF+i∇ · F )dA (F=(u, v)) (44)

f 5µ‰o“Ѳ¾Ò F = (u, v)5ì

Dàíù½, ²Æuz, N¬ú²¾

Ò F = (u, v) íìDàíû˝ª6Œ

Bbúƒb f C f íw…, õÒ, Cauchy

- Riemann j˙ÿuàDìí ¯:ux = (−v)y

uy = −(−v)x

⇐⇒∇ · F =ux+vy=0

curlF =vx−uy=0(45)

Ê¡H&2”Šð±í ^k'K

¶ (compensated compactness method)

5XÿZuà‹²¾Ò5àDì'ß

í4”, †Ÿ²¾ÒªJ)ƒyßí'K

4 (ÿuF‚í àìùÜ (div-curl

lemma))

@àù:

(40)ªeѵb$íGreenìÜ:

Bblõõ虉²: (x, y) −→ (z, z)

z = x+ iy

z = x− iy=⇒

x =

12(z + z)

y = 12i(z − z)

(46)

ârC© (chain rule) ª)

∂∂x= ∂

∂z+ ∂

∂z

∂∂y= i

(∂∂z

− ∂∂z

)

=⇒∂ ≡ ∂

∂z= 1

2

(∂∂x

− i ∂∂y

)∂ ≡ ∂

∂z= 1

2

(∂∂x+ i ∂

∂y

) (47)

Ĥ Laplace 4äª[Ñ

∆ = ∂xx + ∂yy = 4∂∂

38 bçfÈ 21»4‚ ¬86'12~

ÛÊ5?ªƒb (differentiable func-

tion), f = u− iv , †

2i∂f

∂z= 2i

1

2

(∂

∂x+ i

∂

∂y

)(u− iv)

= (vx − uy) + i(ux + vy)

= curlF+i∇ · F (F =(u, v)) (48)

ÇøjÞ‚à (differential form)

5tª)

d[fdz] =∂f

∂zdz ∧ dz +

∂f

∂zdz ∧ dz

=∂f

∂zdz ∧ dz

=∂f

∂z(dx− idy) ∧ (dx+ idy)

= 2i∂f

∂zdxdy (49)

µb$íGreen ìÜ: f = f(z, z)

Ñ©//wRûb?Ñ©/†∮Cf(z)dz = 2i

∫∫R∂f

∂zdx dy

=∫∫

R∂f

∂zdz ∧ dz (50)∮

Cf(z)dz = 2i

∫∫R∂f

∂zdx dy

=∫∫

R∂f

∂zdz ∧ dz (51)

w2 C = ∂R, R Å— Green ìÜ5b°

à‹ f Ñj&ƒb (analytic func-

tion) †

2i∂f

∂z= (−vx − uy) + i(ux − vy) = 0

M¦õ¶D™¶$kÉ, ¥ÿuO±í

Cauchy - Riemann j˙:

ux = vy uy = −vx (52)

yøŸ)ƒ Cauchy ìÜ∮Cf(z)dz = 2i

∫∫R∂f

∂zdx dy

= 2i∫∫

R∂fdx dy=0

ĤBbªJz: Green ìÜÿu Cauchy

ìÜíR#

@àú:

Cauchy t: cq f ∈ C1(R),†

f(ξ) =1

2πi

∫∂R

f(z)

z − ξdz

+1

2πi

∫∫R∂f/∂z

z − ξdz ∧ dz (53)

¥_téÍuÊøO×綵‰ƒb

Ü5 Cauchy t5R

„p: Bb"‡Þ5Wæ3l4(

5j¶5?–

Rρ = R− Bρ(ξ)

∂Rρ = ∂R− ∂Bρ = C − ∂Bρ

1/¦ƒb

F (z)=f(z)

z − ξ

=⇒ ∂F

∂z=∂f

∂z

1

z − ξ

Green ìÜD@à 39

†â Green ìÜ (50) ø

∮∂Rρ

F (z)dz =∫∫

Rρ

∂F

∂zdz ∧ dz

Ĥ ∮C

f(z)

z − ξdz −

∮∂Bρ

f(z)

z − ξdz

=∫∫

Rρ

∂f

∂z

1

z − ξdz ∧ dz (54)

I ρ → 0 /â©/45Ô”ª)

∮∂Bρ

f(z)

z − ξdz → 2πif(ξ) (55)

ÇøjÞâ”è™ø

dz ∧ dz = 2idx dy = 2irdr dθ1

|z − ξ|dz ∧ dz = 2idr dθ (56)

FJ (55) AÑ Cauchy t (Iρ →0)

f(ξ) =1

2πi

∮C

f(z)

z − ξdz

+1

2πi

∫∫R∂f/∂z

z − ξdz ∧ dz

Å1: Caucly tu_©4/i1í!

‹,¦/Bb”ñqÿwì,¥ÿuw@

”$,Í7¥_“ÿu”ºk1978'H

â Kerzman D Stein Êû˝Öµ‰ƒ

b (Several Complex Variables) F

Hú, ¥_!‹, Bb˚Ñ Kerzman-

Stein t, E5è6ª¡©óÉ

’e

Å2: à‹fÑj&ƒb (analytic func-

tion), ∂f∂z= 0, †vtÿƒf$í

Caucly t, …µsBb, úRq

Løõ ξ ∈ Ω, ƒbMf(ξ), ªâ[Ñ

iä C = ∂R5˜,7¥_[Û

ìÜ (representation theorem) 6T

X7 Cauchy-Riemann j˙ ∂f∂z= 0

jít

@àû:

à‹5?j˙ (∂½æ)

∂f

∂z= h (57)

Cauchyt (53)µsBb¬idø

ªeÑ5Ÿj (homogeneous solution), 7

dùªeÑÝ5Ÿj (nonhomogeneous

solution), Ĥ Cauchy t (53) °

vT‡∂½æ (58) 5jÑ

f(ξ) =1

2πi

∫∂R

f(z)

z − ξdz

+1

2πi

∫∫R

h

z − ξdz ∧ dz (58)

∂-½æ: cq f ∈ C∞(R), †Ý5Ÿ

Cauchy-Riemann j˙

∂u

∂z= f(z, z)

5jª[Ñ

u(ξ) = v(ξ) +1

2πi

∫ ∫Rf(z, z)

ξ − zdz ∧ dz

(59)

v(ξ) ÑL<íj&ƒb (analytic func-

tion)

j: ¥½æuj²ªâµ‰ƒb5Ü

7V, OBbªœ¿¡àRj˙íxÍ,

40 bçfÈ 21»4‚ ¬86'12~

â Cauchy j, òh7k, BbÛb

TÜíu∂

∂z

1

z − z0= ?

éͪø

∂

∂z

1

z − z0

= 0 ∀z = z0

ĤBbªJcq

∂

∂z

1

z−z0

= cδ(z−z0)δ(z−z0)

= cδ(x−x0)δ(y−y0)

c Ñø&°5/b, .ÜøO4ª z0 = 0 †

1

z= lim

ε→0

z

zz + ε2, ε > 0

òQ)

∂

∂z(

z

zz + ε2) =

1

|z|2 + ε2− |z|2(|z|2 + ε2)2

=ε2

(|z|2 + ε2)2

Ĥ

c=∫∫

R2

∂

∂z

(1z

)dz ∧ dz

= limε→0

∫∫R2

∂

∂z

( z

zz + ε2

)dz ∧ dz

= limε→0

∫∫R2

ε2

(|z|2 + ε2)2dz ∧ dz

= limε→02i∫∫

R2

ε2

(|z|2 + ε2)2dxdy

= limε→02πi

∫ ∞

0

2rdr

(r2 + 1)2= 2πi

²Æuz

∂

∂z

1

z − z0= 2πiδ(z − z0)δ(z − z0)

FJ 12πi

1z−z0

]ø!…j(fundamental so-

lution) ] ∂ ½æ5jÑ

u(ξ) = v(ξ) +1

2πi

∫ ∫Rf(z, z)

ξ − zdz ∧ dz

= (5Ÿj) + (Ý5Ÿj)

@àü:

ʵ‰ƒbíêmvÍ2, uD¼ñ

‰çÝ/ò~5É[, ÔMuù& ì, .

ª9ò, Ìì, .xò4í¼ñ (steady

two-dimensional flow of an incompress-

ible, irrotational, inviscid fluid) w2|O

±íu Bernoulli ì, ¥ìzp7ìœ

ìWíŸÜ, uN˛Ídø_|½bíì:

Bernoulliì:

P +1

2ρ|w|2 = constant

(9‰+ ?=/b) (60)

cq Ω Ñ xy -NÞ,íø_À©¦–

(simply connected domain) 7(

C=z=z(s)=x(s)+iy(s), 0≤s ≤ L

†H[ Ω 2ø‘ŨNËí¥£–

(, Ĥ( C ,Løõ5ÀP~²¾DÀ

PÕ¶²¾Ñ

τ = z′(s) =dx

ds+ i

dy

ds≡ eiθ(s)

ν =dy

ds− i

dx

ds= −ieiθ(s) (61)

ÇÕcq¼ñí§Ñ

w = u+ iv (62)

Green ìÜD@à 41

†â (43) ªø∮Cwdz

=∮

Cudx+ vdy + i

∮C−vdx+ udy

=∮

Cw · τds+i

∮Cw · νds

==¼(circulation)

+i Ø"(expansion) (63)

õ¶[ý•( C 5=¼, ™¶†[ýÊ

(qÞ5‰“0, âkBbF5?íuÔ

yí¼ñ ( ì, .ª9ò, Ìì, .xò

4) ¥_cqµsBbÊ( C qñøT

àƒ¼ñí‰u¼ñ…™í9‰ (¶²¾j

²).â$k¦¬C5¾¦¾ (momentun

flux), Ĥ∫C[Pν + ρw(w · ν)]ds = 0 (64)

ρ [ý¼ñ5ò, PÑw9‰, ‚àÉ[

dz = eiθds, iν = eiθ,

iw · ν = 12(weiθ + we−iθ)

ªJø (65) ZŸÑ∮C(P +

1

2ρ|w|2)dz+1

2

∮Cρw2dz=0 (65)

|(Bbcq¼ñuÌGí (homogene-

ous) Ĥρ =/b, ÄÑ w uj&ƒb, F

J w2 6uj&ƒb, ]â Cauchy ìܪ

ø ∫Cw2dz = 0 ⇒

∫Cw2dz = 0

FJ (66) “WÑ

∮C(P +

1

2ρ|w|2)dz = 0 (66)

â Morera ìܪ!\ƒbP +

12ρ|w|2Ñj&ƒb, Oâ Green ìÜ (51)

, ªøñøª?íõMj&ƒb (real-

valued analytic function) u/bƒb, Ä

¤ª!

P +1

2ρ|w|2 = /b

¡5’e

1. bç5qñj¶£<2 (I, II, III); «!

À| (1970)

2. The Cauchy Transform, Potential The-

ory and Conformal Mapping; Steven R.

Bell; CRC PRESS (1992).

3. Introduction to Calculus and Analy-

sis(I, II); R. Courant and F. John;

Springer-Verlag (1989)

4. Complex Variables; G. Polya and G.

Latta; John Wiley & Sons, Inc. (1974)

5. Vector and Tensor Analysis; 2nd ed., E.

C. Young; Marcel Dekker, Inc. (1993)

—…dT6L`ÅAŠ×çbçû˝F—