ABSTRACT
When studying different stationary (time-independent) processes, we very often encounter elliptic partial differential equations. One of the most common is Laplace’s equation:
∇2u = 0. (7.1) The operator ∇2 is a scalar product of two gradient operators, ∇2 = ∇ · ∇ (a review of vector calculus is given in Appendix D). In orthogonal Cartesian coordinates,
∇ = ˆi ∂ ∂x
+ ˆj ∂
∂y + ˆk
∂
∂z ,
and Laplace’s equation is
∂2u
∂x2 + ∂2u
∂y2 + ∂2u
∂z2 = 0.
