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^2
rˆ
= ˆr∇^2 Vr+
2 φˆ
r^2
∂Vr
∂φ
−
Vr
r^2
ˆr (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^2
ˆφ. (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·∇Vgiving
V·∇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