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?E?????? ??Rn ?? ??????(U,?)??? ?????? ??? ????? ??U?? ??????? ?? ?? ?????? ?(p) = (x1(p),...,xn(p)). x
1:p?-→x1(p),
x n:p?-→xn(p), (U1,?1) = (U1,x1,...,xn), (U2,?2) = (U2,y1,...,yn),
21:?1(U1∩U2)-→?2(U1∩U2),
u?-→?21(u) =?2o?-11(u),????? ????p?U1∩U2? ???? (x1(p),...,xn(p)) =?1(p), (y1(p),...,yn(p)) =?2(p),
21(?1(p)) =?2(p),
??121(x1(p),...,xn(p)),...,?n21(x1(p),...,xn(p))?= (y1(p),...,yn(p)). y U y
J= det?∂xi∂yj?
21:?1(U1∩U2)-→?2(U1∩U2), u?-→?21(u) =?2o?-11(u),
12:?2(U1∩U2)-→?1(U1∩U2), u?-→?12(u) =?1o?-12(u),
??????? ?? ???? ?????12?? ?? ???????M??? ???? (Ui,?i)?? ????? ??M? ????? ??? ?O?? iUi? ?? ??????? ??? ???? ????i? M, ? R ?(t) =t3, t?R. S
R??????? ???
?? ??????? ??? ??? ???????(Ui,?i),(Vi,ψi), i= 1,2,???? ??? ?????? ??? S S
2={(x1,x2,x3)?R3:x21+x22+x23= 1},
x
21+x22+···+x2n+1= 1.
x
2+y2=z2,
d
1:y=x,
d
2:y=-x,
U i([X0,...,Xn]) =? X
0Xi,...,?XiXi,...,XnXi?
P n(C) ={[Z]?= 0?Cn+1}[Z]≂[λZ], C
I={i1,...,ik} ? {1,...,n}?? ??????
P
I= det(
(a a ik1···aikk) ??????? ??? ??? ???????PI?? ???? ??? ???? ???? ?? ?????? ???? ?????? ? ?? P U
I={π?Gn,k:PI(π)?= 0},
((((((1 0 0···0? ··· ··· ?
0··· ··· ···1? ··· ··· ?)
I:UI-→C(n-k)k.
g)?? ??????? ??? ?? ???????{(UI,?I)}??? ?? ????? ?? ?????? ??Gn,k?
M×N={(p,q) :p?M,q?N}.
?? ??????? ??? ??(U,?)??? ??? ????? ???M??(V,ψ)??? ??? ????? ???N, ?????(U×V,?×ψ)??? ??? ????? ???M×N? F
1(x1,...,xn) = 0,
F n-k(x1,...,xn) = 0. V=4? i=1?
Qi(z) =ci, z?C6?,
Q
1(z) =z1+z2-(z4+z5)2-z26,
Q
2(z) =z1z5+z2z4-z3z6-z24z5-z4z25,
Q
3(z) =z23-z1z25-z2z24+z24z25,
Q
4(z) =z1z2,
f(w,z) = 0,????? grad f|w0,z0≡?∂f∂w,∂f∂z? w
0,z0?= 0.
f(w0,z0) = 0, gradf|w0,z0?= 0. ??????? ??? ??∂f∂w?= 0, ????? ???f (w(z),z) = 0, w0=w(z0),∂w∂z≡0. f(w,z) =w2-Pn(z) = 0,?????
P(Z0,Z1,Z2) = 0,?????
???? ?g????? ??? ?????? ?g?????? ???? ?? ??????? ??????g≥0? ?? ?????? H
C-→C, z?-→w:w2=P3(z),
?? ????? ??????z=1t,???? w
2=P3?1t?
=?1t-e1??1t-e2??1t-e3? e θ= 0,?? ?w=r1/2?????? ??? ????θ= 2π,?? ?w=-r1/2? ?? ?? ?????? ?? ????? ?? ???????r1/2??? ??? ?? ?????? ?? ??????? ????z=e2??z=e3? ?? 1 ?????? ??σ1?????? ??????? ???w=?(z-e1)(z-e2)(z-e3)? ?? ??? ???? ?? ?????? ??????? ?? ?????e1?? ?????e2?? ??????? ?? ???? ?? ?? ?????? ??????? ????? ??????? ???? ?????? ???? ???? ?????? ???? ??? ??????e1,e2,e3,∞? ???? C
1??D2?? ???? ???D1??C2?
D e w
2= (z-e1)(z-e2)(z-e3)(z-e4),
y
2= (1 + (e1-e2)x)(1 + (e1-e3)x)(1 + (e1-e4)x).
w
2=Pn(z) =?P2g+1(z)??n= 2g+ 1
?P2g+2(z)??n= 2g+ 2
X={(w,z)?C2:F(w,z) = 0},
?g????? ??? ?????? ?g?????? ???? ?? ??????? ??????g≥0? ?????? ????? ?? X?
C-→C, z?-→w:F(w,z) = 0,
????? ??????? ??F(w,z) = 0??∂F∂w(w,z) = 0? ?? ???? ??? ??????? ????
F(w,z) =G(u,v,x,y) +iH(u,v,x,y),
F(w,z) = 0???G(u,v,x,y) = 0,
H(u,v,x,y) = 0.
u det det ?∂G∂u? 2 +?∂H∂u? 2 ????∂G∂u+i∂H∂u? ???2 ????∂F∂u? ???2 det? =????∂F∂v? ???2 det? ?= 0, ????∂F∂u? ???2 ?= 0??????∂F∂v? ???2 ?= 0, ???2 ?= 0. z???? ???????∂F∂w(w0,z0)?= 0. ∂F∂x=∂F∂w∂w∂x+∂F∂z= 0, ∂F∂y=-∂F∂w∂w∂y+i∂F∂z= 0. ?? ???????∂F∂w? ∂w∂x+i∂w∂y? = 0. ∂w∂x=-i∂w∂y, f(z) =f(a) +a1(z-a) +a2(z-a)2+··· f(z) =f(b) +b1(z-b) +b2(z-b)2+··· f(z) =f(a) +a1(b-a) +a2(b-a)2+··· +a1(z-b) + 2a2(b-a)(z-b) +···+a2(z-b)2+··· f(b) =f(a) +a1(b-a) +a2(b-a)2+··· b
1=a1+ 2a2(b-a) +···
b
2=a2+···
f(z) =f(b) +b1(z-a) +b2(z-a)2+··· ???rb≥ra- |b-a|>0? ?? ?? ?? ???? ????? ????? ???rb> ra- |b-a|>0? ?? ??? ?? ?????a?? ?????b? ??????? ?? ??????C? ????c1? C ∩D(a,ra)? ?? ???? c D
F(w,z) = 0?
????? ?? ?F(w(z),z) = 0???? ?? ??????D(b,rb)?????? ?? ?????b? ???? w ?? ???? ?????? ???? ??????? ???????C\{z1,...,zm} z f(x) =a0xm+a1xm-1+···+am=a0m k=1(x-αk), a0?= 0, g(x) =b0xn+b1xn-1+···+bn=b0n j=1(x-βj), b0?= 0, (((((((((((((((a
0a1... ... ... am0...0
0...0a0a1... ... ... am
b
0b1... ... bn0... ...0
0... ...0b0b1... ... bn)
f=Afm11fm22...fmrr, g=Bgn11gn22...gnss, f ???G=???g=n? ?? fG=fgg1=gff1=gF. ???f??g????? ??? ?? ??????? ??????? ???? ?? ???? ??????? Af m11fm22...fmrr.G=Bgn11gn22...gnss.F, ????? ???? ????j= 1,2,...,r?fmj ???F < m?
F(x) =A0xm-1+A1xm-2+···+Am-1,
G(x) =B0xn-1+B1xn-2+···+Bn-1,
(a0xm+···+am)(B0xn-1+···+Bn-1) = (b0xn+···+bn)(A0xm-1+···+Am-1). x m+n-1:a0B0=b0A0, x m+n-2:a0B1+a1B0=b0A1+b1A0, x
0:amBn-1=bnAm-1.
a
0B0-b0A0= 0,
a
1B0+a0B1-b1A0-b0A1= 0,
a mBn-1-bnAm-1= 0.
Δ≡det(
(((((((((((a
00...0-b00...0
a a m??????a0-bn??????-b0
0...0am0...0-bn)
)))))))))))= 0. fG-gF=???(f,g), =an0bm0m k=1n j=1(αk-βj), =an0m k=1g(αk), = (-1)mnbm0n j=1f(βj). fG-gF= 0 =???(f,g). ??f??g????? ??? ?? ??????? ??????? ????? ?? ???????F??G???? ??? ? fG-gF=???(f,g).
Disc(f) =a2m-20(-1)m(m-1)2?
i?=j(αi-αj) =a2n-20? ???(f,f?) = (-1)m(m-1)2a0Disc(f). f(x) =a0m k=1(x-αk), f?(x) =a0m k=1? j?=k(x-αj). f ?(αi) =a0? j?=i(αi-αj). ???(f,f?) =am-10m i=1f ?(αi) =a2m-10? j?=i(αi-αj). ??????? ???(-1)(αi-αj)2? ?? ?????? ?????? ?? ???? ????? ? ?m(m-1)2?????? ???(f,f?) = (-1)m(m-1)2a2m-10?
P(w,z) =p0(z)n?
i=1(w-wi(z)),
G(w,z) = (w-w1(z))...(w-wk(z)),
=wk-E1wk-1+E2wk-2- ···+ (-1)kEk, E 1=k? i=1w i(z), E2=k? i,j=1i
1,...,ij?{1,...,k} i 1<... i1...wij, L U??V???? ??? ?
UP+V Q|w=wj(z)=U.0 +V.0 = 0 =R(z).
w 2=n? k=1(z-zk), w 2+Pn(z)w+ 1 = 0,
w 4=z4-1.
w n+zn= 1, n≥2. ???? ??? ?? ????? ??X? g(X) =(n-1)(n-2)2. w 3+P2(z)w2+P4(z)w+P6(z) = 0,(z,w?C)
?????? ?? ???? ????? ?????(U,?)??M?? ????? ?????(V,ψ)??N? ?????? ??? ? ψofo?
-1:?(U)-→ψ(V), y y=ψofo?-1?ψ(V)?Rn, x??(U)?Rm. ?f(0) =g(0) =p??????? ????? ???? ?????? ?? ?????p? ?limt→0f(t)-g(t)t= 0 ???? ?? ?????p? ?? ????? ??????? ?? ???? ?????? ????? ?????(U1,?1)? ??? ?p?? ????? ??M? ???????α,β?K(=R??C)? ?αf??? ?????? ????U1? ?βg??? ?????? ????U2? v:C∞(p)-→R, f?-→v(f), ???v(αf+βg) =αv(f) +βv(g)? ????v(fg) =v(f)g(p) +f(p)v(g)? ?(x) =?1???A 0???B ?(x) = 0?? ?????? ??V? ?x?M??? i? i(x) = 1??x?M? ??????λv1+μv2? ?λ,μ?R?? ?????? ??? (λv1+μv2)(f) =λv1(f) +μv2(f), f? C∞(p) ?? ?????p? ?∂∂x1? p ,...,?∂∂xn? p ??p? ?? ???????∂∂xi? p f=?∂f∂xi? ??f? C∞(p)? ?? ???? ????? ?????? ?∂∂xi? p f=?∂f∂xi? ?∂∂x1? p ,...,?∂∂xn? p dimTpM= dimM. v=n? i=1α i?∂∂xi? p ?∂∂x1? p ,...,?∂∂xn? p? ??∂∂y1? p ,...,?∂∂yn? p? ???? ???? ????? ??TpM? ?∂∂xi? p =n? j=1∂y j∂xi(p)?∂∂yj? p ∂∂yi? p =n? j=1∂x j∂yi(p)?∂∂xj? p i=n? j=1∂x i∂yjβ j, i=n? j=1∂y i∂xjα j. t=t0,...,?dfndt? t=t0? t=t0=?dx i(f(t))dt? t=t0? ?????? v=n? i=1? dfidt? t=t0? ∂∂xi? f(t0). ?????v??? ?? ??????? ??????? ?M??f(t0)?? ?? ?????? ??? ?? ????? ?? g n i=1? dfidt? t=t0? ∂∂xi? f(t0)=n? i=1? dgidt? t=t0? ∂∂yi? ??f(t0)? T df=∂f∂xdx+∂f∂ydy. ?z=x+iyz=x-iy=??dz=dx+idy dz=dx-idy=??dx=12(dz+dz) dy=12i(dz-dz) df=12? ∂f∂x-i∂f∂y? dz+12? ∂f∂x+i∂f∂y? dz. ∂∂z=12? ∂∂x-i∂∂y? ,∂∂z=12? ∂∂x+i∂∂y?quotesdbs_dbs15.pdfusesText_21
Sous-variétés - Exo7 - Exercices de mathématiques