Harmonic Function: Three-Dimensional

M.M. Yovanovich

3DHARMONIC1.MWS

> restart:

Three-dimensional harmonic function [Maple Math] with [Maple Math] , and [Maple Math] . The point source coordinates [Maple Math] are fixed. The harmonic function satisfies the three-dimensional Laplace equation: [Maple Math] .

> u:= C/r;

[Maple Math]

> r:= sqrt((x-x1)^2 + (y-y1)^2 + (z-z1)^2);

[Maple Math]

> u_x:= diff(u,x): u_xx:= diff(u_x,x):

> u_y:= diff(u,y): u_yy:= diff(u_y,y):

> u_z:= diff(u,z): u_zz:= diff(u_z,z):

> simplify(u_xx + u_yy + u_zz);

[Maple Math]

The given harmonic function satisfies the three-dimensional Laplace equation in cartesian coordinates.