Fonctions de plusieurs variables - Classe B/L

URnR
3,4,...n
A(xA, yA)B(xB, yB)R2A B d(A, B)
d(A, B) = (xB−xA)2+ (yB−yA)2
R2
n A(x1, x2, . . . , xn)B(y1, y2, . . . , yn)
A B
d(A, B) =
n
p=1
(xp−yp)2
r > 0ARnA r
B(A, r)
B(A, r) = {M∈Rn/ d(M, A)< r}

UR2M∈U
M U
∀M∈U, ∃r > 0/ B(M, r)⊂U
R
]−1,3[×]0,+∞[
R2
R2\ {(0,0)}
{0} × R
UR2f:U→RA= (a, b)U
f1A
f1:t7→ f(t, b)
f2A
f2:t7→ f(a, t)
k k f
URnf:U→RA= (a1, a2, . . . , an)U
k fkA
fk:t7→ f(a1, . . . , ak−1, t, ak+1, . . . , an)
f U RA= (a, b)R2
f A
lim
(x,y)→(a,b)f(x, y) = f(a, b)
f U ⊂R2f U

1
f U R2A(a, b)U
•f1x(a, b)
t7→ f(t, b)alim
x→a
f(x, b)−f(a, b)
x−a
∂f
∂x(a, b)
•f1y(a, b)
t7→ f(a, t)blim
y→b
f(a, y)−f(a, b)
y−b
∂f
∂y (a, b)
f n U RnA(a1, . . . , an)∈U
f: (x1, x2, . . . , xn)7−→ f(x1, x2, . . . , xn)
k f1A= (a1, . . . , an)akf
1k A
∂f
∂xk
(a1, . . . , an)
C1
∂f
∂x
∂f
∂y U f C1
UfU
C1
C1
fR2∀(x, y)∈R2, f(x, y) = ye−x2−y
(x, y)7→ −x2−y x y
y f
R2
∀(x, y)∈R2,∂f
∂x(x, y) = −2xye−x2−y,∂f ∂y (x, y) = (1 −y)e−x2−y

1
fC1U(a, b)U
(a, b)ε(x, y)
f(x, y) = f(a, b)+(x−a)∂f
∂x(a, b)+(y−b)∂f ∂y (a, b) + (x−a)2+ (y−b)2ε(x, y)
lim
(x,y)→(a,b)ε(x, y) = 0
H(h, k)O(0,0) (a+h, b+k)
U
f(a+h, b +k) = f(a, b) + h∂f
∂x(a, b) + k∂f ∂y (a, b) + o
H→0d(O, H)
1f(a, b)
1g
g(a+h) = g(a) + hg0(a) + o
h→0(h)
fC1U A = (a, b)U
(h, k)7−→ h∂f
∂x(a, b) + k∂f ∂y (a, b)
f A(a, b)dAf(h, k)
f A
(h, k)7→ αh +βk
α β
1
A= (a, b)H= (h, k)O= (0,0)
f(a+h, b +k) = f(a, b) + dA(h, k) + o(d(O, H))

2
f1∂f
∂x
∂f
∂y 1
2f
∂2f
∂x2=∂ ∂x
∂f
∂x
∂2f
∂x∂y =∂ ∂x
∂f
∂y
∂2f
∂y∂x =∂ ∂y
∂f
∂x
∂2f
∂y2=∂ ∂y
∂f
∂y
f: (x, y)7→ ye−x2−y(x, y)R2
∂f
∂x(x, y) = −2xye−x2−y,∂f ∂y (x, y) = (1 −y)e−x2−y
∂f
∂x
∂f
∂y 1x
y(x, y)R2
∂2f
∂x2(x, y) = −2y(1 −2x2)e−x2−y∂2f ∂y∂x(x, y) = −2x(1 −y)e−x2−y
∂2f
∂x∂y (x, y) = −2x(1 −y)e−x2−y∂2f ∂y2(x, y)=(−2 + y)e−x2−y
fC1U∂f
∂x
∂f
∂y C1U
fC2U
fC2U(a, b)U
∂2f
∂x∂y (a, b) = ∂2f ∂y∂x(a, b)
fC2
f xixj
f xjxi