215: applications di érentiables sur un ouvert de Rn

Rn
C1Ck
Ω⊂RnRma∈Ω
t f(a+th) 0
h ∂hf i
∂if
h f(a+t0h)t0
y2/x f(0, y) = 0
x6=y
a df(a)f(a+
h) = f(a) + df(a)(h) + o(h)
Ω
Rf
R2
df(a)(h) = ∂hf=Phi∂if(x)
xy/(x2+y2)
∂jfiDf(x)(h) = Jf(x)h
0
f g ⇒f+g d(f+g)(a) = df(a) + dg(a)
F g =F(f1, . . . , fn)fia
dg(a) = F(df1(a), f2,...fn) + . . .
f a g f(a)⇒g◦f d(g◦f)(a)(h) = dg(f(a))(df(a)(h))
f a f df(a)→f−1a d(f−1)(a) =
(df)−1(f(a))

f=g+hi f
R2∂1g=∂2h ∂2g=−∂1h f
C1Ck
f C1n
f(n−1) f(n)Ck
f(n)
n
C∞
[x, y] Ω Df(x)6k
||f(x)−f(y)|| 6k||x−y||
C1
C1
C1
fnΩa
g fnf g
C1C+
FR[a, b]U
f(b)−f(a) = Df(c)(b−a)
fRpc||f(b)−f(a)|| 6
||Df(c)||.||b−a||
R
eit

f a ⇒df(a)=0
f(x, y) = x3+y30
x2(x−1)2
f a ⇒d2f(a)a
a⇒d2f(a)
x2−y3x2+y4
f C Rnf
f
f
x
df(x)=0
Rn
Rnf f
f C2ΩRn
a x0∈ΩF(x) := x−df(x)−1(f(x))
xn+1 =F(xn)a

U E a U f C1
df(a)f
a f(a)
Ck
f(x, y)=(x2−y2,2xy)
(y−z)∂xf+(z−x)∂yf+(x−y)∂zf= 0
ϕ(x+y+z, x2+y2+z2)ϕ C1
u2<3v
f(x, y) = u g(x, y)v dϕ 2 (a, b)
U1
(x, y)
f U
f f C1
Ck
f(x, y) = (x+1, xy)=(s, p)U={x>y}
V=s2>4pf
U f(U)
f C1CkRnF
f
CkRn
U×V G C1Ckf(a, b)=0
f(x, y)=0 x∈U0, y ∈V0, f(x, y)=0⇔(x∈U0, y =ϕ(x)
ϕ Ck
x2+y2−1 = 0 y= 0
ϕ0(x) = −x/y =−xϕ(x)

f
df(M)
M0Γf= 0
df(M0)(M−M0)
H(x, y) = 0 H
H(x, y) = f2
xFxy −fxfy(fx2+fy2) + f2 yfxy
Q(x, y) = d2f(M0)(x, y)2=rx2+ 2sxy +ty2
ϕ rs −t2>0f
ϕ rs −t2<0
Q(x−x0, y −y0)=0