45 Fonctions de plusieurs variables : dérivés partielles, di érentielle

C1.
E F Lc(E, F )
E F,
∀u∈ Lc(E, F ),∥u∥= sup
x∈E\{0}
∥u(x)∥
∥x∥= sup
∥x∥=1 ∥u(x)∥
∥·∥EE∥·∥F
F. ∥·∥
E F.
Lc(E, F )E F
E F E F
Lc(E, F ) = L(E, F ).
fOE
F a ∈ O u∈ Lc(E, F )
V0E
∀h∈ V, f (a+h) = f(a) + u(h) + o(∥h∥)
f a,
u∈ Lc(E, F )ε B (0, η) 0
η > 0
lim
h→0ε(h) = 0
∀h∈B(0, η), f (a+h) = f(a) + u(h) + ∥h∥ε(h)

lim
h→0∥f(a+h)−f(a)−u(h)∥
∥h∥= 0
η > 0B(a, η)⊂ O.
E F
f a
γ∥f(a+h)−f(a)−u(h)∥
δ∥h∥≤∥f(a+h)−f(a)−u(h)∥′
∥h∥′≤β∥f(a+h)−f(a)−u(h)∥
α∥h∥
α, β, γ, δ
OE f :O → F. f
a∈ O,
f(x)−f(a) = u(x−a) + ∥x−a∥ε(x−a)→
x→a0
u
OE f :O → F. f
a∈ O,(45.1)
u, v Lc(E, F ).
lim
h→0∥(v−u) (h)∥
∥h∥= 0
x∈E\ {0},
0 = lim
t→0∥(v−u) (tx)∥
∥tx∥=∥(v−u) (x)∥ ∥x∥
(v−u) (x) = 0, u (x) = v(x).
x= 0, u =v.
df (a)
E F (45.1) f a.
E n ≥1
F=R, f :O → Ra∈ O, df (a)
E gf(a)∈E df (a) (x) =
⟨gf(a)|x⟩x∈E. gf(a)f a.
E=R,

ORf:O → F. f
a∈ O τa:x7→ 1
x−a(f(x)−f(a)) O \ {a}
a. f′(a)
ORf:O → F. f
a∈ O df (a) (h) = f′(a)h
h∈Rf′(a) = df (a) (1)
f a, x ∈ O \ {a}
f(x) = f(a) + (x−a)τa(x)
=f(a) + (x−a)f′(a)+(x−a) (τa(x)−f′(a))
x=a τa(a) = f′(a).lim
x→aτa(x) = f′(a),
f(x) = f(a) + (x−a)f′(a) + o(x−a)
f a
df (a) : h∈R7→ f′(a)h
f a, x ∈ V \ {a},V
aOf(x) = f(a) + df (a) (x−a) + o(x−a)
1
x−a(f(x)−f(a)) = 1 x−adf (a) (x−a) + o(1)
=df (a) (1) + o(1) →
x→adf (a) (1)
f a df (a) (1) .
OE f :O → F. f
a∈ O f a u ∈ L(E, F )
f(a+h) = f(a) + u(h) + o(∥h∥).
E, F
u(h) = f(a+h)−f(a) + o(∥h∥)f a,
lim
h→0u(h) = 0, u 0, E.
f a df (a) = u.
OE f :O → F. f
O,O.
f:O → FO,
df :O → Lc(E, F )
x7→ df (x)
f.
df Lc(E, F )
fC1O.

f:O → FC1df (x)=0
x∈ O.
f:E→FC1E
df (x) = f x ∈E. x, h E,
f(x+h) = f(x) + f(h) = f(x) + f(h) + ∥h∥ε(h)
ε= 0 f df (x) = f.
E, F
C1E.
n≥2, E1,··· , EnE=
n
k=1
Ek
∀x= (x1,··· , xn)∈E, ∥x∥= max
1≤k≤n∥xk∥
n f E F C1E
x, h E
df (x) (h) =
n
k=1
f(x1,··· , xk−1, hk, xk+1,···, xn)
x, h E, n f,
f(x+h) = f(x1+h1,··· , xn+hn)
=f(x1,··· , xn) +
n
k=1
f(x1,···, xk−1, hk, xk+1,··· , xn) + R(x, h)
R(x, h)f(y1,··· , yn)
j̸=k(yj, yk) = (hj, hk).
f n M ≥0
∀y∈E, ∥f(y)∥ ≤ M
n
k=1 ∥yk∥
j̸=k(yj, yk) = (hj, hk),
∥f(y)∥ ≤ M∥hk∥∥hk∥
n
i=1
i/∈{j,k}
∥yk∥ ≤ M∥h∥2
n
i=1
i/∈{j,k}
∥yk∥
αx∥h∥ ≤ 1,
∥R(x, h)∥ ≤ αx∥h∥2
f(x+h) = f(x1,··· , xn) +
n
k=1
f(x1,··· , xk−1, hk, xk+1,··· , xn) + ∥h∥o(h)
f x
n f h 7→
f(x1,··· , xk−1, hk, xk+1,··· , xn), df (x)

det : Mn(R)→R
nC1Mn(R)
d(det) (X) (H) =
n
k=1
det (X1,··· , Xk−1, Hk, Xk+1,··· , Xn)
X∈ Mn(R)Xkk
(E, ⟨· | ·⟩)f:x7→ ∥x∥=
⟨x|x⟩E\ {0}x∈E\ {0}h∈E
df (x) (h) = ⟨x|h⟩
∥x∥
x, h E,
∥x+h∥2=∥x∥2+ 2 ⟨x|h⟩+∥h∥2
∥x+h∥2− ∥x∥2= (∥x+h∥−∥x∥) (∥x+h∥+∥x∥)
= 2 ⟨x|h⟩+∥h∥2
x̸= 0,∥h∥<∥x∥x+h̸= 0
∥x+h∥−∥x∥=2⟨x|h⟩
∥x+h∥+∥x∥+∥h∥2 ∥x+h∥+∥x∥
=⟨x|h⟩
∥x∥
2∥x∥
∥x+h∥+∥x∥+∥h∥2 ∥x+h∥+∥x∥
∥x+h∥ − ∥x∥ − ⟨x|h⟩
∥x∥=⟨x|h⟩ ∥x∥2∥x∥ ∥x+h∥+∥x∥−1+∥h∥2 ∥x+h∥+∥x∥
=⟨x|h⟩
∥x∥∥x∥−∥x+h∥ ∥x+h∥+∥x∥+∥h∥2 ∥x+h∥+∥x∥
∥x+h∥−∥x∥ − ⟨x|h⟩
∥x∥≤|⟨x|h⟩| ∥x∥|∥x∥−∥x+h∥| ∥x+h∥+∥x∥+∥h∥2 ∥x+h∥+∥x∥
≤ ∥h∥|∥x∥−∥x+h∥|
∥x+h∥+∥x∥+∥h∥2 ∥x+h∥+∥x∥
≤ ∥h∥|∥x∥−∥x+h∥| +∥h∥
∥x+h∥+∥x∥