Formulario Matemático de Electromagnetismo
r = r r̂
x = r sen θ cos ϕ
y = r sen θ sen ϕ
z = r cos θ
r = rr̂ + zk̂
x = r cos ϕ
y = r sen ϕ
z=z
3. Rotores
Cartesianas
î
∂
∇×A =
∂x
Ax
Cilíndricas
r̂
1 ∂
∇×A =
r ∂r
Ar
ˆj
∂
∂y
Ay
rϕˆ
∂
∂ϕ
rA ϕ
k̂
 ∂A z ∂A y
∂
= 
−
∂z
∂z
 ∂y
Az
  ∂A z ∂A x
 î + 
−
∂z
  ∂x
ˆ  ∂A y ∂A x
−
 j + 
∂y
  ∂x
k̂
∂ (rA ϕ ) 
 ∂ (rA ϕ ) ∂A r  
∂
1  ∂A
 ∂A ∂A r 
r̂ + r z −
k̂ 
=  z −
−
ϕˆ + 
∂z r î  ∂ϕ
∂z 
∂z 
∂ϕ  
 ∂r
 ∂r
Az
Esféricas
r̂ rθˆ
r senθϕˆ
∂
∂
∂
1
∇× A = 2
∂ϕ
r senθ ∂r ∂θ
Ar rAθ r senθAϕ
=

1  ∂(r senθAϕ ) ∂(rAθ )   ∂(r senθAϕ ) ∂Ar  ˆ  ∂(rAθ ) ∂Ar 
ˆ




−
−
−
θ
+
−
θ
ϕ
r̂
r
sen




∂θ
∂ϕ  
∂r
∂ϕ   ∂r
∂θ 
r 2 senθ î 


 k̂

donde:
en cartesianas
A = Axiˆ + Ay ˆj + Az kˆ
en cilíndricas
A = Ar rˆ + Aϕ ϕˆ + Az kˆ
en esféricas
A = Ar rˆ + Aϕ ϕˆ + Aθθˆ
6. Identidades Vectoriales
∇ × (∇φ ) = 0
∇ ⋅ (∇ × A ) = 0
∇ × (∇ × A ) = ∇(∇ ⋅ A )− ∇ 2 A
∇(φψ ) = φ∇ψ + ψ∇φ
∇ ⋅ (φA ) = φ∇ ⋅ A + A ⋅ ∇φ
∇ × (f (r )r ) = 0
r
1
(con r = r )
∇  = − 3
r
r
∇ ⋅ (A × B) = B ⋅ (∇ × A )− A ⋅ (∇ × B)
∇⋅ r = 3
∇× r = 0
∇(A ⋅ r ) = A
∇ × (φA ) = ∇φ × A + φ(∇ × A )
( )
∇ r n = nr n − 2 r
1
∇ 2   = δ(r )
r
1
∇2  = 0
(para
r ≠ 0)
r
∇ × (A × B) = (B ⋅ ∇ )A − (A ⋅ ∇ )B + A∇ ⋅ B − B∇ ⋅ A
∇ × (φA ) = φ∇ × A − A × ∇φ