Appendix B 521
Application to cylindrical coordinates
{x 1 ,x 2 ,x 3 }={r,φ,z};{h 1 ,h 2 ,h 3 }={ 1 ,r, 1 }
∇ψ = ˆr
∂ψ
∂r
+
ˆφ
r
∂ψ
∂φ
+ˆz
∂ψ
∂z
(B.22)
∇·V =
1
r
∂
∂r
(rVr)+
1
r
∂Vφ
∂φ
+
∂Vz
∂z
(B.23)
∇×V =
(
1
r
∂Vz
∂φ
−
∂Vφ
∂z
)
rˆ+
(
∂Vr
∂z
−
∂Vz
∂r
)
ˆφ
+
1
r
(
∂
∂r
(rVφ)−
∂Vr
∂φ
)
zˆ (B.24)
∇^2 ψ =
1
r
∂
∂r
(
r
∂ψ
∂r
)
+
1
r^2
∂^2 ψ
∂φ^2
+
∂^2 ψ
∂z^2
(B.25)
Laplacian of a vector in cylindrical coordinates
Before proceeding, note that the cylindrical coordinate unit vectors can be expressed in
terms Cartesian unit vectors as
ˆr = ˆxcosφ+ˆysinφ (B.26)
ˆφ = −xˆsinφ+ˆycosφ (B.27)
so
∂
∂φ
ˆr=φ,ˆ
∂
∂φ
ˆφ=−ˆr. (B.28)
Thus
∇rˆ =
(
rˆ
∂
∂r
+
ˆφ
r
∂
∂φ
+ˆz
∂
∂z
)
rˆ=
φˆφˆ
r
(B.29)
∇φˆ =
(
rˆ
∂
∂r
+
ˆφ
r
∂
∂φ
+ˆz
∂
∂z
)
φˆ=−
φˆrˆ
r
(B.30)
and
∇^2 ˆr = −
1
r^2
ˆr (B.31)
∇^2 ˆφ = −
1
r^2
ˆφ. (B.32)
These results can now be used to calculate∇^2 (Vrˆr)and∇^2
(
Vφφˆ
)
which can then be