522 Appendix B
used to construct the full Laplacian. The first calculation gives
∇^2 (Vrˆr) = ∇·∇(Vrˆr)
= ∇·((∇Vr)ˆr+Vr∇ˆr)
= ˆr∇^2 Vr+(∇Vr)·∇ˆr+∇·(Vr∇ˆr)
= ˆr∇^2 Vr+2∇Vr·∇ˆr+Vr∇^2 ˆr= ˆr∇^2 Vr+
2 φˆφˆ
r·∇Vr+
Vr
r^2∂^2
∂φ^2rˆ= ˆr∇^2 Vr+
2 φˆ
r^2∂Vr
∂φ−
Vr
r^2ˆr (B.33)while the second calculation gives
∇^2
(
Vφφˆ)
= ∇·∇
(
Vφφˆ)
= ∇·
(
(∇Vφ)ˆφ+Vφ∇φˆ)
= ˆφ∇^2 Vφ+2∇Vφ·∇ˆφ+Vφ∇^2 ˆφ= ˆφ∇^2 Vφ−
2ˆr
r^2∂Vφ
∂φ−
Vφ
r^2ˆφ. (B.34)Since∇^2 (Vzzˆ)= ˆz∇^2 Vzit is seen that the Laplacian of a vector in cylindrical coordi-
nates is
∇^2 V = ˆr(
∇^2 Vr−2
r^2∂Vφ
∂φ−
Vr
r^2)
+ˆφ(
∇^2 Vφ+2
r^2∂Vr
∂φ−
Vφ
r^2)
+ˆz∇^2 Vz. (B.35)Equation (B.28) can also be used to calculateV·∇VgivingV·∇V =
(
Vr∂
∂r+
Vφ
r∂
∂φ+Vz∂
∂z)(
Vrrˆ+Vφφˆ+Vzˆz)
= ˆr(
Vr∂Vr
∂r+
Vφ
r∂Vr
∂φ+Vz∂Vr
∂z−
Vφ^2
r)
+ˆφ(
Vr∂Vφ
∂r+
Vφ
r∂Vφ
∂φ
+Vz∂Vφ
∂z+
VφVr
r)
+ˆz(
Vr∂Vz
∂r+
Vφ
r∂Vz
∂φ+Vz∂Vz
∂z