Operatori differenziali: - Consorzio Elettra 2000

Principali relazioni differenziali:
1) ∇( Φ + ψ) = ∇Φ + ∇ψ
2) ∇ ⋅(
A + B)
= ∇ ⋅ A + ∇ ⋅B
3) ∇ × ( A + B)
= ∇× A + ∇×
B
4) ∇
5) ∇
2
( Φ + ψ) = ∇ Φ + ∇
2
2
2
( A + B)
= ∇ A + ∇
2
6) ∇ ⋅ ( ∇Φ)
= ∇ Φ
7) ∇(
Φ ψ) = Φ∇ψ
+ ψ∇Φ
2
2
ψ
B
8) ∇ ⋅(
ΦA)
= Φ∇ ⋅ A + ∇Φ ⋅ A
9) ∇ × ( ΦA ) = Φ∇× A + ∇Φ × A
10) ∇ ⋅(
A × B)
= B ⋅∇× A − A ⋅∇×
B
11) ∇×∇Φ = 0
12) ∇ ⋅∇× A = 0
2
13) ∇ × ( ∇ × A)
= ∇(
∇ ⋅ A)
− ∇ A
2
14) ∇ ⋅(
Φ∇ψ)
= ∇Φ ⋅∇ψ
+ Φ∇ ψ
15) ∇ ( A ⋅B)
= A×
( ∇× B)
+ B×
( ∇× A)
+ ( A ⋅∇)
B + ( B ⋅∇)
A
16) ∇ × ( A × B)
= ( ∇ ⋅B)
A − ( ∇ ⋅ A)
B + ( B ⋅∇)
A − ( A ⋅∇)
B
17) ∇
2
18) ∇
2
( Φψ)
= Φ∇ ψ + ∇Φ ⋅∇ψ
+ ψ ∇
2
2
2
( ΦA ) = Φ∇ A + A∇
Φ + 2( ∇Φ ⋅ ∇)
A
19) ∇∇ ⋅( ΦA ) = ( ∇Φ)
∇ ⋅ A + Φ∇∇⋅ A + ∇Φ ×∇× A + ( A ⋅∇)
∇Φ + ( ∇Φ ⋅∇)
A
20) ∇ × ∇ × ( ΦA
) = ∇Φ × ∇ × A − A∇
2
2
Φ
Φ + ( A ⋅∇)
∇Φ + Φ∇ × ∇ × A + ∇Φ ∇ ⋅ A − ( ∇Φ ⋅∇)
A
Le relazioni 11, 12 e 13 sono particolarmente importanti.