Vector definitions, identities, and theorems

1 Definitions
Vector definitions, identities, and theorems
Rectangular coordinates
1. ∇f = ∂f ∂f ∂f
∂x ˆx + ∂y ˆy + ∂z ˆz
2. ∇A = ∂Ax
∂x
3. ∇ × A =
∂Ay
+ ∂y
∂Az
∂y
+ ∂Az
∂z
− ∂Ay
∂z
4. ∇2f = ∂2f ∂x2 + ∂2f ∂y2 + ∂2f ∂z2
ˆx + ∂Ax
∂z
5. ∇ 2 A = ∇ 2 Axˆx + ∇ 2 Ayˆy + ∇ 2 Azˆz
Cylindrical coordinates
6. ∇f = ∂f 1 ∂f
∂ρ ˆρ + ρ ∂φ ˆ φ + ∂f
7. ∇ · A = 1
∂z ˆz
∂
ρ ∂ρ (ρAρ) + 1 ∂Aφ ∂Az
ρ ∂φ + ∂z
1 ∂Az ∂Aφ ∂Aρ
8. ∇ × A = ρ ∂φ − ∂z ˆρ + ∂z
9. ∇2f = 1
ρ ρ ∂f
∂ρ + 1
ρ2 ∂ 2 f
∂φ2 + ∂2f ∂z2 10. ∇ 2 A = ∇ (∇ · A) − ∇ × (∇ × A)
Spherical coordinates
11. ∇f = ∂f ∂f
∂r ˆr + r∂θ ˆθ + 1 ∂f
r sin θ ∂φ ˆφ
12. ∇ · A = 1
r 2
13. ∇ × A = 1
+ 1
r
∂(rAθ)
∂r
14. ∇ 2 f = 1
r 2
∂ 2
∂r r Ar
r sin θ
∂
∂r
∂
− Ar
∂θ
r
+ 1
r sin θ
Input by Tiao Lu
tlu@math.pku.edu.cn, 62759189(o)
Abstract
∂Az ∂Ay
− ∂x ˆy + ∂x
∂Ax
− ∂zy ˆz
∂Az
− ˆφ 1 ∂
∂ρ + ρ ∂ρ (ρAφ) − Aρ
∂φ ˆz
∂
∂θ (Aθ sin θ) + 1 ∂Aφ
r sin θ ∂φ
∂θ (Aφ sinθ) − ∂Aθ
∂ρ ˆr + 1
r
ˆφ
2 ∂f
∂r
+ 1
r 2 sin θ
∂
∂θ
sin θ ∂f
∂θ
1 ∂Ar
sin θ ∂φ
1 + r2 sin2 ∂
θ
2 f
∂φ2 1
∂(rAφ)
− ˆθ
∂r

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