Some relationships from vector analysis
∇=µˆı∂
∂x + ˆ∂
∂y +ˆ
k∂
∂z ¶: nabla operator
∇ · φ= grad φ:φscalar field →vector field
=³ˆı∂
∂x φ+ ˆ∂
∂y φ+ˆ
k∂
∂z φ´
∇ · ~
A= div ~
A:~
Avector field →scalar field
=³∂
∂x Ax+∂
∂y Ay+∂
∂z Az´
∇ × ~
A= curl ~
A:~
Avector field →vector field
=¯
¯
¯
¯
¯
¯
¯
ˆıˆˆ
k
∂
∂x
∂
∂y
∂
∂z
AxAyAz
¯
¯
¯
¯
¯
¯
¯
=³∂
∂y Az−∂
∂z Ay´ˆı+³∂
∂z Ax−∂
∂x Az´ˆ+³∂
∂x Ay−∂
∂y Ax´ˆ
k
∇2=∇·∇=µ∂2
∂x2+∂2
∂y2+∂2
∂z2¶: laplacian operator
applied on electric potential U:
∇2U= div grad U= 0 “Laplace equation”
∇2U=³∂2
∂x2U+∂2
∂y2U+∂2
∂z2U´= 0
Next: ∇ × (∇ × ~
A) = ∇(∇ · ~
A)− ∇2~
A
where: ∇ · ~
A=∂
∂x Ax+∂
∂y Ay+∂
∂z Az≡S
so that ∇(S) = ˆı∂
∂x S+ ˆ∂
∂y S+ˆ
k∂
∂z S
and
∇2~
A=µ∂2
∂x2Ax+∂2
∂y2Ax+∂2
∂z2Ax¶ˆı
+µ∂2
∂x2Ay+∂2
∂y2Ay+∂2
∂z2Ay¶ˆ
+µ∂2
∂x2Az+∂2
∂y2Az+∂2
∂z2Az¶ˆ
k