Example 8-20.
The Laplacian in rectangular coordinates (x, y, z)is given by
∇^2 U=∂^2 U
∂x^2 +∂^2 U
∂y^2 +∂^2 U
∂z^2 (8 .101)The Laplacian in cylindrical coordinates (r, θ, z)is given by
∇^2 U=^1
r∂
∂r(
r∂U
∂r)
+^1
r^2∂^2 U
∂θ^2+∂(^2) U
∂z^2
∇^2 U=∂
(^2) U
∂r^2
+^1
r
∂U
∂r
+^1
r^2
∂^2 U
∂θ^2
+∂
(^2) U
∂z^2
(8 .102)
The Laplacian in spherical coordinates (ρ, θ, φ )is given by
∇^2 U=1
ρ^2∂
∂ρ(
ρ^2∂U
∂ρ)
+1
ρ^2 sin θ∂
∂θ(
sin θ∂U
∂θ)
+1
ρ^2 sin^2 θ∂^2 U
∂φ^2∇^2 U=∂^2 U
∂ρ^2 +2
ρ∂U
∂ρ +1
ρ^2∂^2 U
∂θ^2 +cot θ
ρ^2∂U
∂θ +1
ρ^2 sin^2 θ∂^2 U
∂φ^2(8 .103)Special cases of the Laplace equation ∇^2 U= 0 are easy to solve.
1. If y =z= 0 in rectangular coordinates, the Laplace equation, with Laplacian
(8.101), reduces to d
(^2) U
dx^2
= 0. One integration produces dU
dx=C 1 , where C 1 is
a constant of integration. Another integration gives U=C 1 x+C 2 , where C 2 is
another constant of integration.
2. If θ =z = 0 in cylindrical coordinates, the Laplace equation, with Laplacian
(8.102), reduces to
1
rd
dr(
rdU
dr)= 0. An integration of this equation gives r
dU
dr =C 1 , where C 1 is a constant of integration. Separate the variables in this equations
and integrate again to show the solution of the special Laplace equation is given
by U=C 1 ln r+C 2 , where C 2 is another constant of integration.
3. If θ = φ = 0 in spherical coordinates, the Laplace equation, with Laplacian
(8.103), reduces to ρ^12
(
ρ^2 dUdρ)= 0. An integration of this equation gives
ρ^2 dU
dρ