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P7→ Z1
0
P(t)2dt
E=Rn[X]
f:C∗→Cf(z) = 1/z
det: Mn(R)→R
InM
M
det M∈ Mn(R)
E F Rϕ:E×E→F
ϕdϕ
p∈N
fp: (x, y)∈R2\ {(0,0)} 7→ (x+y)psin 1
px2+y2
fp
(0,0)
(0,0)
f:E→F f(λx) = λf(x)λ∈R
x∈E f
E=C∞(Rn,R)E∗E
D=d∈E∗| ∀(f, g)∈E2, d(fg) = f(0)d(g) + g(0)d(f)
DE∗
D {0}
d∈ D h d(h)
f∈E
∀x∈Rn, f(x) = f(0) +
n
X
i=1
xiZ1
0
∂f
∂xi
(tx) dt
x7→ R1
0
∂f
∂xi(tx) dt E
d∈ D (a1, . . . , an)∈Rn
∀f∈E, d(f) =
n
X
i=1
ai
∂f
∂xi
(0)
D

f(P) = R1
0P(t)2dt f(P+H) = f(P)+2R1 0P(t)H(t) dt+R1 0H(t)2dt
`(H)=2R1
0P(t)H(t) dt ` E
Ek.k=k.k∞
R1
0H(t)2dt
≤ kHk2
∞= o(kHk)
f(P+H) = f(P) + `(H) + o(kHk)
f P df(P): H7→ 2R1
0P(t)H(t) dt
a∈C∗h∈C
f(a+h)−f(a) = −h
a(a+h)=`(h) + α(h)
`:h7→ − h
a2
α(h) = −h
a(a+h)+h a2=h2 a2(a+h)= O(h2) = o(h)
f a
`:h7→ − h
a2
det C∞
det M=X
σ∈Sn
ε(σ)
n
Y
i=1
aσ(i),i
det(I+H) = 1 + ϕ(H) + o(kHk)ϕ= dI(det)
H=λEi,j
det(In+λEi,j ) = 1 + λδi,j = 1 + λϕ(Ei,j ) + o(λ)
ϕ(Ei,j ) = δi,j
ϕ= tr
M
det(M+H) = det Mdet(I+M−1H) = det M+ det Mtr(M−1H) + o(H)
d(det)(M): H7→ det Mtr(M−1H)
M
d(det)(M): H7→ det Mtr(M−1H) = tr(tCom M.H)
M7→ d(det)(M)M7→ tr(tCom M×.)
GLn(R)Mn(R)
ϕ(a+h, b+k) = ϕ(a, b)+ϕ(h, b)+ϕ(a, k)+ϕ(h, k) = ϕ(a, b)+ψ(h, k)+o (k(h, k)k)
ψ: (h, k)7→ ϕ(h, b) + ϕ(a, k)ϕ(h, k) = o (k(h, k)k)
|ϕ(h, k)| ≤ Mkhkkkk
ϕ(a, b) dϕ(a, b) = ψ
x=rcos θ y =rsin θ
fp(x, y) = (cos θ+ sin θ)prpsin 1
r
p≥1|fp(x, y)| ≤ 2prp−−−−−−−→
(x,y)→(0,0) 0f
(0,0)
p= 0 f0(x, y) = sin 1
√x2+y2+∞
p≥1
p= 2
f2(x, y)=(x+y)2sin 1
px2+y2= O(k(x, y)k2)
(0,0)
f2(0,0)

p > 2
fp(x, y)=(x+y)p−2f2(x, y)
fp
p= 1
h→0+
1
h(f1(h, 0) −f1(0,0)) = sin 1 h
f(0,0) (1,0)
f(0) = f(0.0) = 0.f(0) = 0
`= df(0)
f(λx) = f(0) + `(λx) + o(λkxk)
f(λx) = λf(x)
λ`(x) + o(λkxk) = λf(x)
λ λ →0+f(x) = `(x)
f
dh:f7→ dhf(0) h∈Rn
h λ f ∈E
d(fh) = λd(f)
D
d(fh) = f(0)d(h) + λd(f)
f
d(h)=0
x∈Rnϕ:t∈[0 ; 1] 7→ f(tx)C1
ϕ(1) = ϕ(0) + Z1
0
ϕ0(t) dt
f(x) = f(0) + Z1
0
n
X
i=1
xi
∂f
∂xi
(tx) dt
KRn
x(x, t)7→ ∂f
∂xi(tx)
K×[0 ; 1]
fi:x7→ R1
0
∂f
∂xi(tx) dtC∞
pi:x7→ xi
d
d(f) =
n
X
i=1
d(pifi) =
n
X
i=1
d(pi)fi(0)
d(f(0)) = 0 pi(0) = 0
ai=d(pi)
fi(0) = Z1
0
∂f
∂xi
(0) dt=∂f
∂xi
(0)
∀f∈E, d(f) =
n
X
i=1
ai
∂f
∂xi
(0)
h∈RndhRnD
f:x7→ (h|x)
dh= 0 =⇒h= 0
dim D=n