Vector definitions, identities, and theorems

Identities
1. (A × B) · C = A · (B × C)
2. A × (B × C) = B (A · C) − C (A · B)
3. ∇ (fg) = f∇g + g∇f
4. ∇ (a/b) = (1/b)∇a − a/b 2 ∇b
5. ∇ · (A · B) = (B · ∇)A + (A · ∇)B + B × (∇ × A) + A × (∇ × B)
6. ∇ · (fA) = (∇f) · A + f (∇ · A)
7. ∇ · (A · B) = B · (∇ × A) − A · (∇ × B)
8. ∇ · ∇ × A = ∇ 2 A
9. ∇ × (∇f) = 0
10. ∇ · (∇ × A) = 0
11. ∇ × (fA) = (∇f) × A + f (∇ × A)
12. ∇ × (A × B) = (B · ∇)A − (A · ∇)B + (∇ · B)A − (∇ · A) B
13. ∇ × (∇ × A) = ∇ (∇ · A) − ∇ 2 A
14.
References
∂Bx
(A · ∇)B = Ax
∂x
∂By
+ Ax
+
Ax
∂x
∂Bz
∂x
∂Bx
+ Ay + Az
∂y
∂By
+ Ay + Az
∂y
∂Bz
+ Ay + Ax
∂y
∂Bx
∂z
ˆx
∂By
ˆy
∂z
ˆz
[1] P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves, Includinger Electric Circuits, 3rd
ed, W. H. Feeman and Company, New York, 1988
2
∂Bz
∂z