ABSTRACT
When studying different stationary (time-independent) processes very often we meet Laplace’s equation
∇ =2 0u . (14.1)
A function is said to be harmonic in a certain region if it satisfies Laplace’s equation there and its first and second derivatives are continuous in that region. For example, the functions 2xy, x y2 2− and e yx− cos satisfy the two-dimensional Laplace’s equation
∂ ∂
+ ∂ ∂
=
2 0 u x
u y (14.2)
in the entire x, y-plane. The nonhomogeneous equation
∇ = − 2u f (14.3)
with a given function f of the coordinates is called Poisson’s equation. Laplace’s and Poisson’s partial differential equations are of elliptic type. Let us present the Laplace’s operator in different coordinate systems. Cylindrical coor-
dinates r, φ, z are related with Cartesian coordinates as
x = rcosφ, y = rsinφ, z = z
the Laplacian is
∇ = ∂∂
∂ ∂
+
∂ ∂
+ ∂ ∂
2 1 1 r r
r r r zϕ
.
