Appendix B 523Application to spherical coordinates
{x 1 ,x 2 ,x 3 }={r,θ,φ};{h 1 ,h 2 ,h 3 }={ 1 ,r,rsinθ}
∇ψ = ˆr
∂ψ
∂r+
ˆθ
r∂ψ
∂θ+
φˆ
rsinθ∂ψ
∂φ(B.37)
∇·V =
1
r^2∂
∂r(
r^2 Vr)
+
1
rsinθ∂
∂θ(sinθVθ)+1
rsinθ∂
∂φ(Vφ) (B.38)∇×V =
1
rsinθ(
∂
∂θ(Vφsinθ)−
∂Vθ
∂φ)
rˆ+
1
rsinθ(
∂Vr
∂φ−
∂
∂r(Vφrsinθ))
ˆθ+
1
r(
∂
∂r(Vθr)−∂Vr
∂φ)
ˆφ (B.39)∇^2 ψ =1
r^2∂
∂r(
r^2
∂ψ
∂r)
+
1
r^2 sinθ∂
∂θ(
sinθ
∂ψ
∂θ)
+
1
r^2 sin^2 θ∂^2 ψ
∂φ^2