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 scalarThe 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