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)