∗

∗

† ‡

†

‡

(apu)n ∗ )

(apu)n

u2x (apu)n u2x

vx = 0 f = anx = bnx

∗)

(apuv)n

vx

x

y

λx − y;

λ1 λ2 . . . λn ,

(λax − ay )n = f · {λn + a1 λn−1 + a2 λn−2 + . . . an } = 0. λκ λκ ux − uy = 0. λκ (λκ ax − ay )n = 0. λκ (ax uy − ux ay )n = (auv)n = 0, (auv)n

u

v

v

f λnκ (−1)κ

! " n f y κ = f aκ . κ

aκ

λκ (−1)κ aκ =

f

#

λ1 λ2 . . . λκ .

n α1,x α2,x . . . αn,x .

f = α1,x α2,x . . . αn,x ; (aακ u)n = 0.

v

f

λκ λκ ux − uy = 0 ακ λκ ακ,x − ακ,y = 0, λκ

λκ =

(−1)κ aκ =

ακ,y . ακ,x

# α1,y α2,y ακ,y ... . α1,x α2,x ακ,x

λκ sν = λν1 + λν2 . . . λνn aκ sn = (−1)n an1 + (−1)n−1 an−2 a2 . . . . 1 aκ ! " κ n f aκ = (−1) fy κ κ sn f

fn

fy ν

n $ f sn = np % n α1,y αn2,y αnκ,y n + n ... n =f · αn1,x α2,x ακ,x

f p = nn−1 fyn − nn−1 f fyn−2 fy2 · · · f2 p = pny .

p

n

y

y

x n−κ f n sκ = npκ . y px

(apu)n = 0. f x

f

fy (af u)n fy

(af u)n

f u

fy

u2x (apu)n f

(apu)n

u2x # = un" um x sn aκ

(apu)n

u2x

(af u)n , (apu)n (af u)n = Aunx + Bf, B f, (abu), (af u), bx . (apu)n

u2x

u2x

#

# = un" um x

ux m = 1.

(#xy)n = 0 # #

ux

ux ϑ = un−1 ϑ

un" = ux un−1 . ϑ u

x &y

y

0 = (#x& uv)n = (u" vx − v" ux )n ! " κ n = (−1) (u" vx )n−κ (v" ux )κ , κ κ=0 κ=n #

#vxn

u &v

! " n = −ux v" (−1) (u" vx )n−κ (v" ux )κ−1 , κ κ=1 κ=n #

κ

m=2

v" (#xy)n−1 # #

u2x ϑ = un−2 ϑ

u2x

un" = u2x un−2 ϑ u

v nun−1 v" = 2ux vx un−2 + (n − 2)u2x un−3 vϑ . " ϑ ϑ x &y

u

v # ϑ = un−2 ϑ

x &y

ux

un" = u2x un−2 . ϑ u

v nun−1 v" = vx un−1 + (n − 1)ux vϑ un−2 " ϑ ϑ x &y

u

0 = vx (ϑxy)n−1 . ϑ

ux

u2x

#

m = 3, 4, 5, . . . # um x um−1 (#xy)n−m+1 = 0. " #

um x ϑ = un−m ϑ

n−m un" = um . x uϑ

u

x &y

! " ! " n − m m n−m−1 n n−1 (#xy) = ux uϑ (ϑxy), u ! 1 " ! 1" n − m m n−m−2 n n−2 ux uϑ (ϑxy)2 , u" (#xy)2 = 2 2 ! " ! " 3 n−3 n − m m n−m−3 3 u (#xy) = ux uϑ (ϑxy)3 , 1 " 3 · · · · · · ·!· · · · · " ! " n−m m n m n−m = ux uϑ (ϑxy)n−m , u" (#xy) n − m n−m ! " n um−1 (#xy)n−m+1 = 0, n−m+1 " # um−1 x ϑ = un−m+1 ϑ

un" = um−1 un−m+1 . x ϑ x &y !

u

" ! " n − m + 1 m−1 n−m n n−1 (#xy) = ux uϑ (ϑxy), u 1 ! " ! 1" n − m + 1 m−1 n−m−1 n n−2 ux uϑ (ϑxy)2 , u" (#xy)2 = 2 2 · · · · · · · ·! ···· " ! " n − m + 1 m−1 n m n−m = ux uϑ (ϑxy)n−m , u" (#xy) n − m n − m ! " ! " n − m + 1 m−1 n n−m+1 (#xy) = u (ϑxy)n−m+1 , um−1 n−m+1 x n−m+1 " (ϑxy)n−m+1 = 0. ϑ

ux

#

um−2 (#xy)n−m+2 = 0, " um−2 (#xy)n−m+1 (#yz) = 0, " um−2 (#xy)n−m+1 (#zx) = 0, "

um x

uz (#xy) + ux (#yz) + uy (#zx) = u" (xyz) #

u2x (#xy)n = 0, (#xy)n−1 (#yz) = 0, (#xy)n−1 (#zx) = 0. sn

a

(apu)n .

κ=n # κ=0

an−κ sκ = 0,

xn + a1 xn−1 + a2 xn−2 + · · · an = (x − λ1 )(x − λ2 ) · · · (x − λn ) aκ sn

n

nsn =

κ=n # κ=1

κaκ

∂sn ∂aκ

g0 g1 g2 · · · g0 = −1

∂sn . ∂aκ

κ=n # κ=0

an−κ gκ = 0.

∂sn = ngµ . ∂an−µ ∂sn = −n ∂an µ = 0 µ = 1, 2, 3, . . . µ

κ>0 ∂sn−κ = (n − κ)gµ−κ ∂an−µ

sµ =

κ=µ # κ=1

κ=n # κ=0

an−µ sn−κ

κaκ gµ−κ .

aκ sn−κ

κ≤µ κ=µ # κ=0

aκ

∂sn−κ + sµ = 0, ∂an−µ

∂sn−κ ∂an−µ

sµ

κ=µ κ=µ # # ∂sn + (n − κ)gµ−κ aκ + κaκ gµ−κ = 0 ∂sn−µ κ=1 κ=1 κ=µ # ∂sn +n gµ−κ aκ = 0. ∂sn−µ κ=1

n

κ=µ # κ=0

gµ−κ aκ = 0,

∂sn = ngµ . ∂an−µ

dsκ

W =

κ=n # λ=n # κ=1 λ=1

daκ

κaκ gn−κ−λ+1 daλ

λ κ sn−λ+1 =

κ=n # κ=1

dsn−κ+1 =

λ=n # λ=1

κaκ gn−κ−λ+1 , λ=n

# ∂sn−κ+1 daκ = (n − κ + 1) gn−κ−λ+1 daλ , ∂aκ λ=1

W κ=n # κ=1

κ=n # κ=1

κ=n #! κ=1

κ aκ dsn−κ+1 , n−κ+1

κ aκ dsn−κ+1 − sn−κ+1 daκ n−κ+1

κ=n−1 # ! κ=1

sn−κ+1 daκ

κ+1 aκ+1 dsn−κ − sn−κ daκ+1 n−κ

"

"

= 0,

= 0.

aκ

sκ

f

! " n κ n−κ f aκ = (−1) , a a κ y x κ

f n sn−κ = npn−κ pκ y x.

y

z

dy = z

' n ( κ−1 f daκ = (−1)κ κ , κay az an−κ x n κ f dsn−κ = n(n − κ)pn−κ−1 p p z y x. aκ , sn−κ , daκ , dsn−κ

κ=n−1 # ! κ=1

κ=n # κ=0

an−κ sκ = 0,

κ+1 aκ+1 dsn−κ − sn−κ daκ+1 n−κ

"

=0

! " n κ n−κ n−κ κ ay ax py px = 0, κ κ=0 ! " κ=n−1 # )κ + 1 n κ+1 (−1) an−κ−1 (n − κ)pn−κ−1 pz pκ = aκ+1 y x y x κ + 1 n − κ κ=0 ! " * n n−κ κ κ+1 κ n−κ−1 −py px (−1) (κ + 1)ay az ax κ+1 ! " κ=n−1 # n = (ay pz − py az ) (−1)κ+1 (κ + 1) aκ+1 an−1−κ pn−1−κ pκ y x y x κ + 1 κ=0 ! " κ=n−1 # n−1 = −n(ay pz − py az ) (−1)κ (ay px )κ (ax py )n−1−κ κ κ=0 κ=n #

(−1)κ

0

(ay px − py ax )n = 0,

(ay pz − py az )(ay px − py ax )n−1 = 0. y

z

(az px − pz ax )(ay px − py ax )n−1 = 0. (apu)n u2x (af u)n pny fy f f =0

x fy

f

(apu)

n

f

fy n (af u)n x x

(af u)n

u2x (af u)n = Au2x + Bf

(af u)n (A + Cf )u2x + (B − Cu2x )f. (af u)n f f, (abu), (af u), bx (af u)n

(af u)n

(af u)n (af u)n = Au2x + Bf. B B = Cu2x + D (af u)n (af u)n = (A + Cf )u2x + Du2x , D f, (abu), (af u), bx .

(af u)n

(af u)n (af u)n = Au2x + Bf

+ , 1 (abu) ((af u)bx )n−1 − ((bf u)ax )n−1 2 κ=n−2 # 1 n−2−κ κ = (abu)((af u)bx − (bf u)ax ) {(af u)bx } {(bf u)ax } 2 κ=0

(af u)n = (abu)(af u)n−1 bn−1 = x

=

κ=n−2 # 1 n−2−κ κ (abu) {f (abu) − ux (abf )} ((af u)bx ) ((bf u)ax ) . 2 κ=0

κ=n−2 # 1 (abu)2 ((af u)bx )n−2−κ ((bf u)ax )κ 2 κ=0

κ=n−2 # 1 n−2−κ κ (abu)(abf ) ((af u)bx ) ((bf u)ax ) 2 κ=0

ux L = −2Aux.

K=

#

(abc)2 (af u)κ (bf u)λ (cf u)µ an−2−κ bn−2−λ cn−2−µ x x x κ, λ, µ κ + λ + µ = n − 2. ux (abc) = ax (bcu) + bx (cau) + cx (abu), ux K a, b, c f

Kux = 3(abc)(abu)

# (af u)κ (bf u)λ (cf u)µ an−2−κ bn−2−λ cn−2−µ . x x x

µ=0

3L

µ=ν+1

Kux − 3L # bn−2−λ cn−2−ν , = 3(abc)(abu)(cf u) (af u)κ (bf u)λ (cf u)ν an−2−κ x x x κ, λ, ν

κ+λ+ν =n−3 a, b, c

= (abc) = 0.

-

Kux − 3L ((abu)(cf u) + (bcu)(af u) + (cau)(bf u)) # κ · (af u) (bf u)λ (cf u)ν an−2−κ bn−2−λ cn−2−ν x x x L=

1 Kux , 3

1 A=− K 6

=−

1# (abc)2 (af u)κ (bf u)λ (cf u)µ an−2−κ bn−2−λ cn−2−µ . x x x 6

B=

κ=n−2 # 1 n−2−κ κ (abu)2 ((af u)bx ) ((bf u)ax ) 2 κ=0

B = Cu2x + D D f, (abu), (af u), bx

xn + y n = − y n−1 ) = n(x n−1

(y + (x − y))n n−1 n (y + (x − y))

+ (x − (x − y))n , n−1 − n (x − (x − y))

" ( n ' n−κ y n(x −y )= + (−1)κ xn−κ (x − y)κ−1 , κ κ=2 κ=n−2 κ=n # # !n" ' ( y n−κ + (−1)κ xn−κ (x − y)κ−2 , xn−2−κ y κ = n κ κ=0 κ=2 κ=n # !n − 1" ' ( y n−κ + (−1)κ xn−κ (x − y)κ−1 , 2n(xn−1 − y n−1 ) = n κ−1 κ=2 ! " κ=n # n ( ' = κ y n−κ + (−1)κ xn−κ (x − y)κ−1 , κ κ=2 κ=n # !n" ( ' 0= (κ − 2) y n−κ + (−1)κ xn−κ (x − y)κ−3 . κ κ=3 n−1

κ=n #!

n−1

bx (af u),

x

x−y

(abu)

2nB

= (abu)

2

κ=n #! κ=2

2

y

ax (bf u)

f (abu) − ux (abf )

(abu)2 (abf )

" ( n ' (ax (bf u))n−κ + (−1)κ (bx (af u))n−κ κ

0 = (abu)2 ux (abf )

κ=3

a

κ−2

(f (abu) − ux (abf )) , " ( ' n (κ − 2) (ax (bf u))n−κ + (−1)κ (bx (af u))n−κ κ

κ=n #!

κ−3

(f (abu) − ux (abf ))

.

b

! " n κ−2 nB = (abu) (−1) , (ax (bf u))n−κ + (−1)κ (bx (af u))n−κ (f (abu) − ux (abf )) κ κ=2 ! " κ=n # n 0 = (abu)2 ux (abf ) (−1)κ (κ − 2)(bx (af u))n−κ κ κ=3 2

κ=n #

κ

(f (abu) − ux (abf ))κ−3 .

nB

0

! " . n n−κ f (abu)κ−2 = (abu) (−1) (bx (af u)) κ κ=2 ! " λ=κ−2 / # κ−2 + (−1)λ (f (abu))κ−2−λ (ux (abf ))λ , λ λ=1 ! " ! " κ=n # λ=κ−2 # n κ−2 n−κ = (abu)2 ux (abf ) (−1)κ+λ (κ − 2) (bx (af u)) κ λ − 1 κ=3 2

κ=n #

κ

λ=1

= (abu)2

κ=2

nB

(f (abu))κ−2−λ (ux (abf ))λ−1 " ! "! κ−2 n n−κ κ−2−λ (−1)κ+λ (f (abu)) λ (bx (af u)) λ κ

κ=n # λ=κ−2 # λ=1

λ

(ux (abf )) ,  κ=n ! " # n  n−κ κ−2   (−1)κ (f (abu)) (bx (af u))   κ   κ=2 ! "! " κ=n # λ=κ−2 # = (abu)2 κ−2 2 2 κ+λ n  −ux(abf ) (−1) (λ − 1)   κ λ   κ=2 λ=2   n−κ κ−2−λ λ−2 (bx (af u)) (f (abu)) (ux (abf )) . B = Cu2x + D,

nC = −(abu)2 (abf )2

nD =

κ=n # λ=κ−4 #

(−1)κ+λ

κ=4

κ=n #

λ=0

(−1)κ

κ=2

! "! " n κ−2 (λ + 1) κ λ+2 n−κ

(bx (af u))

κ−4−λ

(f (abu))

λ

(ux (abf )) ,

! " n n−κ κ−2 κ f (f (abu)) . (bx (af u)) κ

(af u)n = (A + Cf )u2x + Df,  n # 2 2  − u (abc) bn−2−λ cn−2−µ (af u)κ (bf u)λ (cf u)µ an−2−κ  x x x x  6   ! "! " κ=n λ=κ−4  # #  n κ−2   (−1)κ+λ (λ + 1)  −f u2x (abu)2 (abf )2 κ λ+2 κ=4 λ=0 n(af u)n = n−κ κ−4−λ λ   (f (abu)) (ux (abf )) (bx (af u))   ! " κ=n  #  n  n−κ κ−2 κ   (−1)κ f (f (abu)) . (bx (af u))  +f κ κ=2

q=

qyn

=

nfyn

! " n κ−2 −f (−1) fyn−κ fyκ ; f κ κ=2 κ=n #

κ

 # n  − (abc)2 bn−2−λ cn−2−µ (af u)κ (bf u)λ (cf u)µ an−2−κ  x x x  6  ! "! "  κ=n λ=κ−4 # # n κ−2 (aqu)n = u2x −f (abu)2 (abf )2 (−1)κ+λ (λ + 1)  κ λ+2   κ=0 λ=0   n−κ κ−4−λ λ (f (abu)) (ux (abf )) . (bx (af u)) u2x pny

qyn

n − 1 n−2 f fy fy 2 , nn−1 fyn − nn−1 ! " 2 n nfyn − f fyn−2 fy2 , 2 f2 pny − nn−2 qyn pny = nn−2 qyn + f 2 ryn . f (apu)n = nn−2 (aqu)n + f 2 (aru)n ; (aru)n

u2x (ay rx − ry ax )n = 0,

(ay rx − ry ax )n−1 (aru) = 0.

(n ' ' (n 0 = (abu)rx − (rbu)ax = (abr)ux + bx (aru) , ' (n−1 ' (n−1 0 = (abu)rx − (rbu)ax (abr)ux = (abr)ux + bx (aru) (abr)ux , n−1

0 = bnx (aru)n + n (bx (aru)) ux (abr) ! κ=n # n" n−κ κ−2 + u2x (abr)2 (ux (abr)) , (bx (aru)) κ κ=2

0 = n (bx (aru))

n−1

ux (abr) κ=n # !n − 1" n−κ κ−2 +u2x (abr)2 n (bx (aru)) (ux (abr)) , κ − 1 κ=2 κ=n # !n" n−κ κ−2 n 2 2 f (aru) = ux (abr) (ux (abr)) . (κ − 1) (bx (aru)) κ κ=2 (aqu)n

(aru)n

 n#  − (af u)κ (bf u)λ (cf u)µ an−2−κ bn−2−λ cn−2−µ  x x x  6   ! "! " κ=n λ=κ−4  # #  n κ−2   (−1)κ+λ (λ + 1)  −f (abu)2 (abf )2 κ λ+2 κ=4 λ=0 (apu)n = u2x   (bx (af u))n−κ (f (abu))κ−4−λ (ux (abf ) λ   ! " κ=n  #  n  2   (κ − 1) (bx (aru))n−κ (ux (abr))κ−2 .  +f (abr) κ κ=2 C C K

C

ξ K

x f

#

fy