Esercizi svolti di analisi reale e complessa - Dipartimento di ...

δ = 1
100
∂f
∂x = ∇f = 2f(x) x
∂f
∂x
= 2xy
z−x 2
∂f
∂x =
∂f
= 2xi (x
∂xi
2 j) = 2xi f(x) ;
j=i
1 2x2 0 . . . 0
−x2 sin (x1x2) −x1 sin (x1x2) 0 . . . 0
aβ =
β
| (−1)
PN(x; x0) =
2 |−1 | β
2 |!
(|β|−1)! ( β
2 )!
N
β∈N n |β|=1
aβ(x) β
βi ∀i = 1..n
0
G(x, y, t) ≡ t3 −2xy +y
z = z(x, y) (1, 1)
G(x, y, z(x, y)) ≡ 0 z(1, 1) = 1
(x, y)
z
0 = ∂G(x,y,z(x,y))
(−2y + 3z(x, y)
∂G
∂G
∂z
∂x = ( |(1,1) ∂x (x, y, z(x, y)) + ∂t (x, y, z(x, y)) ∂x (x, y)) |(1,1) =
2 ∂z
∂x (x, y)) |(1,1) = −2 + 3 ∂z
∂x (1, 1)
2 (1, 1) = 3 .
∂z
∂x
P2(x, y; 1, 1) = 1 + 2
1
4
3 (x − 1) + 3 (y − 1) − 9 (x − 1)2 − 1
9 (y − 1)2 + 2
9 (x − 1)(y − 1).
G(x, y, t) = t3 − 2xy + t