Appendix B 521Application 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∂
∂φˆφ=−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^2ˆr (B.31)∇^2 ˆφ = −1
r^2ˆφ. (B.32)These results can now be used to calculate∇^2 (Vrˆr)and∇^2(
Vφφˆ)
which can then be