Begin2.DVI

In a cylindrical (r, θ, z )coordinate system , the Laplace equation takes the form

∇^2 U=∂

(^2) U
∂r^2
+^1
r
∂U
∂r
+^1
r^2
∂^2 U
∂θ^2
+∂
(^2) U
∂z^2
= 0, U =U(r, θ, z) (9 .76)

and in a spherical (ρ, θ, φ )coordinate system , the Laplace equation is represented has

the form

∇^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 = 0, U =U(ρ, θ, φ ) (9 .80)

Two-dimensional Representations

In a two-dimensional (x, y )coordinate system the Laplace equation is represented

∇^2 U=

∂^2 U

∂x^2 +

∂^2 U

∂y^2 = 0, U =U(x, y ) (9 .78)

In a polar (r, θ)coordinate system the Laplace equation becomes

∇^2 U=∂

(^2) U
∂r^2
+^1
r
∂U
∂r
+^1
r^2
∂^2 U
∂θ^2
= 0, U =U(r, θ) (9 .79)

In spherical coordinates, where there is symmetry with respect to the variable φ, the

Laplacian is represented

∇^2 U=∂

(^2) U
∂ρ^2
+^2
ρ
∂U
∂ρ
+^1
ρ^2
∂^2 U
∂θ^2
+cot θ
ρ^2
∂U
∂θ
= 0, U =U(ρ, θ) (9 .80)

One-dimensional Representations

In one-dimension, the Laplace equation becomes

d^2 U

dx^2

=0 , U =U(x) rectangular

d^2 U

dr^2

+^1

r

dU

dr

=^1

r

d

dr

(

rdU

dr

)

=0 , U =U(r) polar

d^2 U

dρ^2 +

2

ρ

dU

dρ =

1

ρ^2

d

dρ

(

ρ^2

dU

dρ

)

=0 , U =U(ρ) spherical

(9 .162)

Three-dimensional Conservative Vector Fields

Analogous to what has been done in studying two-dimensional vector fields, one

can state that if a three-dimensional vector field

F
 =F
(x, y, z) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z)ˆe 2 +F 3 (x, y, z)ˆe 3

is derivable from a potential function φ(x, y, z )such that F
= grad φ,, then the family

of surfaces φ(x, y, z ) = care called equipotential surfaces. The differential equation