520 Appendix B
Thus the general form of the curl of the arbitrary vector is
∇×V =
1
h 2 h 3
(
∂
∂x 2
(V 3 h 3 )−
∂
∂x 3
(V 2 h 2 )
)
xˆ 1
+
1
h 1 h 3
(
∂
∂x 3
(V 1 h 1 )−
∂
∂x 1
(V 3 h 3 )
)
xˆ 2
+
1
h 1 h 2
(
∂
∂x 1
(V 2 h 2 )−
∂
∂x 2
(V 1 h 1 )
)
xˆ 3
=
1
h 1 h 2 h 3
∣ ∣ ∣ ∣ ∣ ∣
h 1 xˆ 1 h 2 ˆx 2 h 3 xˆ 3
∂/∂x 1 ∂/∂x 2 ∂/∂x 3
h 1 V 1 h 2 V 2 h 3 V 3
∣ ∣ ∣ ∣ ∣ ∣
. (B.14)
Generalized Laplacian of a scalar
The Laplacian of a scalar is
∇^2 ψ=∇·
(
xˆ 1
h 1
∂ψ
∂x 1
+
ˆx 2
h 2
∂ψ
∂x 2
+
xˆ 3
h 3
∂ψ
∂x 3
)
. (B.15)
Using Eq.(B.10) this becomes
∇^2 ψ=
1
h 1 h 2 h 3
(
∂
∂x 1
(
h 2 h 3
h 1
∂ψ
∂x 1
)
+
∂
∂x 2
(
h 3 h 1
h 2
∂ψ
∂x 2
)
+
∂
∂x 3
(
h 1 h 2
h 3
∂ψ
∂x 3
))
.
(B.16)
The Laplacian of a vector in general differs from the Laplacian of a scalar because there
are derivatives of unit vectors. However the general formula is overly complex and so the
Laplacian of a vector will only be calculated for cylindrical coordinates which are typically
of most interest.
Application to Cartesian coordinates
{x 1 ,x 2 ,x 3 }={x,y,z};{h 1 ,h 2 ,h 3 }={ 1 , 1 , 1 }
∇ψ = ˆx
∂ψ
∂x
+ˆy
∂ψ
∂y
+ˆz
∂ψ
∂z
(B.17)
∇·V =
∂Vx
∂x
+
∂Vy
∂y
+
∂Vz
∂z
(B.18)
∇×V =
(
∂Vz
∂y
−
∂Vy
∂z
)
xˆ+
(
∂Vx
∂z
−
∂Vz
∂x
)
yˆ
+
(
∂Vy
∂x
−
∂Vx
∂y
)
ˆz (B.19)
∇^2 ψ =
∂^2 ψ
∂x^2
+
∂^2 ψ
∂y^2
+
∂^2 ψ
∂z^2
(B.20)
∇^2 V =
∂^2 V
∂x^2
+
∂^2 V
∂y^2
+
∂^2 V
∂z^2