poly de cours

R Rp
RpR
C1
RpRq
C2
R3

R Rp
[a, b]Rp
f:I→E E f
t0∈I−→ V∈E
f(t0+h) = f(t0) + h−→ V+o
−→
V
f(t0+h)−f(t0) + h−→
V
=o −→
V
−→
V f0(t0) = −→ V
f:I→Rnf= (f1, ..., fn)f t fi
f0(t) = (f0
1(t), ..., f0
n(t))
f E −→ V1, ..., −→ Vnf
f1−→ V1+···+fn−→ Vnf fkf0=
n
P
k=1
f0
k−→
Vk
E=R3t7→ (t2,1/t)R∗t7→ (2t, −1/t2)
E=Ct7→ t2+isin t t 7→ 2t+icos t
E=M2(C)t7→ tsin t
t−1t7→ tcos t 1 0 
Ck
Ck
f:I→Rp
fC1f I
f0:I→Rp
k1fCkf
C1f0Ck−1
fC∞Ckk > 0
f= (f1, ..., fp)Ckfi
f0(t0)f00(t0)
t0
R R

f, g :R→Rpλ:R→Rα∈Ru∈ L(Rp,Rq)B
Rp
αf +g(αf +g)0=λf0+g0
λf λ0f+λf0
t7→ f(λ(t)) t7→ λ0(t)f0(λ(t))
t7→ u(f(t)) t7→ u(f0(t))
t7→ B(f(t), g(t)) t7→ B(f0(t), g(t)) +
B(f(t), g0(t))
B
2 3
k k t7→ kf(t)k2
t7→ 2<f(t)|f0(t)>
Ck
fCn+1 I E I E
a, b ∈I
f(b) = f(a) +
n
X
k=1
(b−a)k
k!f(k)(a) + Zb
a
(b−t)n
n!f(n+1)(t)dt
RpR
RpRERE
f1: (x, y)∈R27→ x2y−cos(y)xf2:M∈ Mn(R)7→ det(M)f3:M∈R27→ ΩM2
1
RpR
f a a
a+h h
f(a+h) = f(a) + ϕ(h) + o(khk)
ϕ a E
h
f a
f(a+t−→ d) = f(a) + Kt +o(t)
t
Rp
Rp

C1
C1
fRpR
f i a = (a1, ..., an)
t7→ f(a1, ..., ai−1, t, ai+1, ..., an)
ai∂if(a)
fC1
∂if E R
f i
E=Rp
∂if(a)∂f
∂xi
(a)p= 3 x y z
∂f
∂x (a)∂f ∂y (a)∂f ∂z (a)
∂f
∂x (3,2,1) ∂f ∂x (x, y, z)
x∂f
∂x (z, x, y)
f(z, x, y)
(x, y, z)7→ f(z, x, y)
(∂1f)(z, x, y)6=∂
∂x (f(z, x, y))
f(x2)0f0(x2)
C1
fC1Ωa= (a1, ..., an)∈Ω
h= (h1, ..., hn)
f(a+h) = f(a) +
n
X
k=1
hi∂kf(a) + o(khk)
−→
d= (d1, ..., dn)
fa+t−→ d=f(a) + n
X
k=1
di∂kf(a)!t+o(t)
C1C0
a a Ω Ω

fC1Ωa∈Ωdf(a)h7→
p
P
k=1
∂kf(a)hkdf(a)
f a
f(a+h) = f(a) + df(a).h +o(khk)
df(a).h (df(a))(h)
fa+t−→ d=f(a) + df(a).−→ dt+o(t)
ΩRnf, g ∈ C1(Ω,R)λ∈R
Φ = λf +gC1Ωi∈[[1, p]] ∂iΦ =
λ∂if+∂ig
∀a∈Ω, d(λf +g)(a) = λdf(a) + dg(a).
Ψ = fg C1Ωi∈[[1, p]] ∂iΨ=(∂if)g+
f∂ig
∀a∈Ω, d(fg)(a) = df(a)g(a) + f(a)dg(a).
f(a+h) = f(a) + df(a).h +o(khk)
(h1, ...hn)7→
p
P
k=1
∂kf(a)hk
df(a).h =
∂1f(a)
∂pf(a)
·
h1
hp
f∈ C1(Ω,R)a∈Ωf a ∇f(a) = 
∂1f(a)
∂pf(a)
f(a+h) = f(a)+ <∇f(a)|h> +o(h)
fa+t−→ d=f(a)+ <∇f(a)|−→ d > t +o(t)
f∈ C1(Ω,R)∇fΩRp
f g C1ΩRpRλ∈R
λf +gC1∇(λf +g) = λ∇f+∇g
fg C1∇(fg) = g∇f+f∇g
f(a)6= 0 f a 1/f C1
∇(1/f) = −1
f2∇f
−→
E=−∇V M M +d−→ ` dV =−−→ E .d−→ `